Analytic geometry

David Pierce

December pm

Mathematics Department

Mimar Sinan Fine Arts University

dpiercemsgsuedutr

httpmatmsgsuedutr~dpierce

Contents

Introduction

The problem

Failures of rigor

A standard of rigor

Why rigor

A book from the s

Abscissas and ordinates

The geometry of the conic sections

A book from the s

Geometry to algebra

Algebra to geometry

Bibliography

List of Figures

An isosceles triangle in a vector space An isosceles triangle An isosceles triangle in a coordinate plane

Euclidrsquos Proposition I

Axial triangle and base of a cone A conic section Axial triangle and base of a cone

Proposition I of Apollonius Proposition I of Apollonius Menaechmusrsquos finding of two mean proportionals The quadratrix

Multiplication of lengths Euclidrsquos Proposition I Multiplication of lengths Associativity of multiplication of lengths

Introduction

The writing of this report was originally provoked both by frus-tration with the lack of rigor in analytic geometry texts and by abelief that this problem can be remedied by attention to mathe-maticians like Euclid and Descartes who are the original sources ofour collective understanding of geometry Analytic geometry arosewith the importing of algebraic notions and notations into geome-try Descartes at least justified the algebra geometrically Now it ispossible to go the other way using algebra to justify geometry Text-book writers of recent times do not make it clear which way they aregoing This makes it impossible for a student of analytic geometryto get a correct sense of what a proof is I find this unacceptable

If it be said that analytic geometry is not concerned with proofI would respond that in this case the subject pushes the studentback to a time before Euclid but armed with many more unexam-ined presuppositions Students today have the idea that every linesegment has a length which is a positive real number of units andconversely every positive real number is the length of some line seg-ment The latter presupposition is quite astounding since the realnumbers compose an uncountable set Euclidean geometry can infact be done in a countable space as David Hilbert points out

I made notes on some of these matters The notes grew intothis report as I found more and more things that seemed worthsaying There are still many more avenues to explore Some of thenotes here are just indications of what can be investigated furthereither in mathematics itself or in the existing literature about itMeanwhile the contents of the numbered chapters of this reportmight be summarized as follows

The logical foundations of analytic geometry as it is oftentaught are unclear The subject is not presented rigorously

What is rigor anyway I consider some modern examples

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where rigor is lacking Rigor is not an absolute notion butmust be defined in terms of the audience being addressed

Ancient mathematicians like Euclid and Archimedes still setthe standard for rigor

I have suggested how they are rigorous but why are they rigor-ous I donrsquot know But we still expect rigor from our studentsif only because we expect them to be able to justify their an-swers to the problems that we assign to themmdashor if we donrsquotexpect this we ought to

I look at an old analytic geometry textbook that I learnedsomething from as a child but that I now find mathematicallysloppy

Because that book uses the odd terms abscissa and ordinatewithout explaining their origin I provide an explanationwhich involves the conic sections as presented by Apollonius

The material on conic sections spills over to another chap-ter Here we start to look in more detail at the geometry ofDescartes How Apollonius himself works out his theoremsremains mysterious Descartesrsquos methods do not seem to illu-minate those theorems

I look at an analytic geometry textbook that I once taughtfrom It is more sophisticated than the textbook from mychildhood This makes its failures of rigor more frustrating tome and possibly more dangerous for the student

I spell out more details of the justification of the use of algebrain geometry Descartes acknowledges the need to provide thisjustification

Finally I review how the algebra of certain ordered fields canbe used to obtain a Euclidean plane

My scope here is the whole history of mathematics Obviously Icannot give this a thorough treatment I am not prepared to try todo this To come to some understanding of a mathematician onemust read him or her but I think one must read with a sense ofwhat it means to do mathematics and with an awareness that thissense may well differ from that of the mathematician whom one isreading This awareness requires experience in addition to the mere

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

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Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Contents

Introduction

The problem

Failures of rigor

A standard of rigor

Why rigor

A book from the s

Abscissas and ordinates

The geometry of the conic sections

A book from the s

Geometry to algebra

Algebra to geometry

Bibliography

List of Figures

An isosceles triangle in a vector space An isosceles triangle An isosceles triangle in a coordinate plane

Euclidrsquos Proposition I

Axial triangle and base of a cone A conic section Axial triangle and base of a cone

Proposition I of Apollonius Proposition I of Apollonius Menaechmusrsquos finding of two mean proportionals The quadratrix

Multiplication of lengths Euclidrsquos Proposition I Multiplication of lengths Associativity of multiplication of lengths

Introduction

The writing of this report was originally provoked both by frus-tration with the lack of rigor in analytic geometry texts and by abelief that this problem can be remedied by attention to mathe-maticians like Euclid and Descartes who are the original sources ofour collective understanding of geometry Analytic geometry arosewith the importing of algebraic notions and notations into geome-try Descartes at least justified the algebra geometrically Now it ispossible to go the other way using algebra to justify geometry Text-book writers of recent times do not make it clear which way they aregoing This makes it impossible for a student of analytic geometryto get a correct sense of what a proof is I find this unacceptable

If it be said that analytic geometry is not concerned with proofI would respond that in this case the subject pushes the studentback to a time before Euclid but armed with many more unexam-ined presuppositions Students today have the idea that every linesegment has a length which is a positive real number of units andconversely every positive real number is the length of some line seg-ment The latter presupposition is quite astounding since the realnumbers compose an uncountable set Euclidean geometry can infact be done in a countable space as David Hilbert points out

I made notes on some of these matters The notes grew intothis report as I found more and more things that seemed worthsaying There are still many more avenues to explore Some of thenotes here are just indications of what can be investigated furthereither in mathematics itself or in the existing literature about itMeanwhile the contents of the numbered chapters of this reportmight be summarized as follows

The logical foundations of analytic geometry as it is oftentaught are unclear The subject is not presented rigorously

What is rigor anyway I consider some modern examples

December pm

where rigor is lacking Rigor is not an absolute notion butmust be defined in terms of the audience being addressed

Ancient mathematicians like Euclid and Archimedes still setthe standard for rigor

I have suggested how they are rigorous but why are they rigor-ous I donrsquot know But we still expect rigor from our studentsif only because we expect them to be able to justify their an-swers to the problems that we assign to themmdashor if we donrsquotexpect this we ought to

I look at an old analytic geometry textbook that I learnedsomething from as a child but that I now find mathematicallysloppy

Because that book uses the odd terms abscissa and ordinatewithout explaining their origin I provide an explanationwhich involves the conic sections as presented by Apollonius

The material on conic sections spills over to another chap-ter Here we start to look in more detail at the geometry ofDescartes How Apollonius himself works out his theoremsremains mysterious Descartesrsquos methods do not seem to illu-minate those theorems

I look at an analytic geometry textbook that I once taughtfrom It is more sophisticated than the textbook from mychildhood This makes its failures of rigor more frustrating tome and possibly more dangerous for the student

I spell out more details of the justification of the use of algebrain geometry Descartes acknowledges the need to provide thisjustification

Finally I review how the algebra of certain ordered fields canbe used to obtain a Euclidean plane

My scope here is the whole history of mathematics Obviously Icannot give this a thorough treatment I am not prepared to try todo this To come to some understanding of a mathematician onemust read him or her but I think one must read with a sense ofwhat it means to do mathematics and with an awareness that thissense may well differ from that of the mathematician whom one isreading This awareness requires experience in addition to the mere

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

List of Figures

An isosceles triangle in a vector space An isosceles triangle An isosceles triangle in a coordinate plane

Euclidrsquos Proposition I

Axial triangle and base of a cone A conic section Axial triangle and base of a cone

Proposition I of Apollonius Proposition I of Apollonius Menaechmusrsquos finding of two mean proportionals The quadratrix

Multiplication of lengths Euclidrsquos Proposition I Multiplication of lengths Associativity of multiplication of lengths

Introduction

The writing of this report was originally provoked both by frus-tration with the lack of rigor in analytic geometry texts and by abelief that this problem can be remedied by attention to mathe-maticians like Euclid and Descartes who are the original sources ofour collective understanding of geometry Analytic geometry arosewith the importing of algebraic notions and notations into geome-try Descartes at least justified the algebra geometrically Now it ispossible to go the other way using algebra to justify geometry Text-book writers of recent times do not make it clear which way they aregoing This makes it impossible for a student of analytic geometryto get a correct sense of what a proof is I find this unacceptable

If it be said that analytic geometry is not concerned with proofI would respond that in this case the subject pushes the studentback to a time before Euclid but armed with many more unexam-ined presuppositions Students today have the idea that every linesegment has a length which is a positive real number of units andconversely every positive real number is the length of some line seg-ment The latter presupposition is quite astounding since the realnumbers compose an uncountable set Euclidean geometry can infact be done in a countable space as David Hilbert points out

I made notes on some of these matters The notes grew intothis report as I found more and more things that seemed worthsaying There are still many more avenues to explore Some of thenotes here are just indications of what can be investigated furthereither in mathematics itself or in the existing literature about itMeanwhile the contents of the numbered chapters of this reportmight be summarized as follows

The logical foundations of analytic geometry as it is oftentaught are unclear The subject is not presented rigorously

What is rigor anyway I consider some modern examples

December pm

where rigor is lacking Rigor is not an absolute notion butmust be defined in terms of the audience being addressed

Ancient mathematicians like Euclid and Archimedes still setthe standard for rigor

I have suggested how they are rigorous but why are they rigor-ous I donrsquot know But we still expect rigor from our studentsif only because we expect them to be able to justify their an-swers to the problems that we assign to themmdashor if we donrsquotexpect this we ought to

I look at an old analytic geometry textbook that I learnedsomething from as a child but that I now find mathematicallysloppy

Because that book uses the odd terms abscissa and ordinatewithout explaining their origin I provide an explanationwhich involves the conic sections as presented by Apollonius

The material on conic sections spills over to another chap-ter Here we start to look in more detail at the geometry ofDescartes How Apollonius himself works out his theoremsremains mysterious Descartesrsquos methods do not seem to illu-minate those theorems

I look at an analytic geometry textbook that I once taughtfrom It is more sophisticated than the textbook from mychildhood This makes its failures of rigor more frustrating tome and possibly more dangerous for the student

I spell out more details of the justification of the use of algebrain geometry Descartes acknowledges the need to provide thisjustification

Finally I review how the algebra of certain ordered fields canbe used to obtain a Euclidean plane

My scope here is the whole history of mathematics Obviously Icannot give this a thorough treatment I am not prepared to try todo this To come to some understanding of a mathematician onemust read him or her but I think one must read with a sense ofwhat it means to do mathematics and with an awareness that thissense may well differ from that of the mathematician whom one isreading This awareness requires experience in addition to the mere

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Introduction

The writing of this report was originally provoked both by frus-tration with the lack of rigor in analytic geometry texts and by abelief that this problem can be remedied by attention to mathe-maticians like Euclid and Descartes who are the original sources ofour collective understanding of geometry Analytic geometry arosewith the importing of algebraic notions and notations into geome-try Descartes at least justified the algebra geometrically Now it ispossible to go the other way using algebra to justify geometry Text-book writers of recent times do not make it clear which way they aregoing This makes it impossible for a student of analytic geometryto get a correct sense of what a proof is I find this unacceptable

If it be said that analytic geometry is not concerned with proofI would respond that in this case the subject pushes the studentback to a time before Euclid but armed with many more unexam-ined presuppositions Students today have the idea that every linesegment has a length which is a positive real number of units andconversely every positive real number is the length of some line seg-ment The latter presupposition is quite astounding since the realnumbers compose an uncountable set Euclidean geometry can infact be done in a countable space as David Hilbert points out

I made notes on some of these matters The notes grew intothis report as I found more and more things that seemed worthsaying There are still many more avenues to explore Some of thenotes here are just indications of what can be investigated furthereither in mathematics itself or in the existing literature about itMeanwhile the contents of the numbered chapters of this reportmight be summarized as follows

The logical foundations of analytic geometry as it is oftentaught are unclear The subject is not presented rigorously

What is rigor anyway I consider some modern examples

December pm

where rigor is lacking Rigor is not an absolute notion butmust be defined in terms of the audience being addressed

Ancient mathematicians like Euclid and Archimedes still setthe standard for rigor

I have suggested how they are rigorous but why are they rigor-ous I donrsquot know But we still expect rigor from our studentsif only because we expect them to be able to justify their an-swers to the problems that we assign to themmdashor if we donrsquotexpect this we ought to

I look at an old analytic geometry textbook that I learnedsomething from as a child but that I now find mathematicallysloppy

Because that book uses the odd terms abscissa and ordinatewithout explaining their origin I provide an explanationwhich involves the conic sections as presented by Apollonius

The material on conic sections spills over to another chap-ter Here we start to look in more detail at the geometry ofDescartes How Apollonius himself works out his theoremsremains mysterious Descartesrsquos methods do not seem to illu-minate those theorems

I look at an analytic geometry textbook that I once taughtfrom It is more sophisticated than the textbook from mychildhood This makes its failures of rigor more frustrating tome and possibly more dangerous for the student

I spell out more details of the justification of the use of algebrain geometry Descartes acknowledges the need to provide thisjustification

Finally I review how the algebra of certain ordered fields canbe used to obtain a Euclidean plane

My scope here is the whole history of mathematics Obviously Icannot give this a thorough treatment I am not prepared to try todo this To come to some understanding of a mathematician onemust read him or her but I think one must read with a sense ofwhat it means to do mathematics and with an awareness that thissense may well differ from that of the mathematician whom one isreading This awareness requires experience in addition to the mere

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

where rigor is lacking Rigor is not an absolute notion butmust be defined in terms of the audience being addressed

Ancient mathematicians like Euclid and Archimedes still setthe standard for rigor

I have suggested how they are rigorous but why are they rigor-ous I donrsquot know But we still expect rigor from our studentsif only because we expect them to be able to justify their an-swers to the problems that we assign to themmdashor if we donrsquotexpect this we ought to

I look at an old analytic geometry textbook that I learnedsomething from as a child but that I now find mathematicallysloppy

Because that book uses the odd terms abscissa and ordinatewithout explaining their origin I provide an explanationwhich involves the conic sections as presented by Apollonius

The material on conic sections spills over to another chap-ter Here we start to look in more detail at the geometry ofDescartes How Apollonius himself works out his theoremsremains mysterious Descartesrsquos methods do not seem to illu-minate those theorems

I look at an analytic geometry textbook that I once taughtfrom It is more sophisticated than the textbook from mychildhood This makes its failures of rigor more frustrating tome and possibly more dangerous for the student

I spell out more details of the justification of the use of algebrain geometry Descartes acknowledges the need to provide thisjustification

Finally I review how the algebra of certain ordered fields canbe used to obtain a Euclidean plane

My scope here is the whole history of mathematics Obviously Icannot give this a thorough treatment I am not prepared to try todo this To come to some understanding of a mathematician onemust read him or her but I think one must read with a sense ofwhat it means to do mathematics and with an awareness that thissense may well differ from that of the mathematician whom one isreading This awareness requires experience in addition to the mere

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Introduction

will to have itI have been fortunate to read old mathematics both as a student

and as a teacher in classrooms where everybody is working throughthis mathematics and presenting it to the class I am currently in thethird year of seeing how new undergraduate mathematics studentsrespond to Book I of Euclidrsquos Elements I continue to be surprisedby what the students have to say Mostly what I learn from thestudents themselves is how strange the notion of proof can be tosome of them This impresses on me how amazing it is that theElements was produced in the first place It may remind me thatwhat Euclid even means by a proof may be quite different from whatwe mean today

But the students alone may not be able to impress on me somethings Some students are given to writing down assertions whosecorrectness has not been established These studentsrsquo proofs mayend up as a sequence of statements whose logical interconnectionsare unclear This is not the case with Euclid And yet Eucliddoes begin each of his propositions with a bare assertion He doesnot preface this enunciation or protasis (πρότασις) with the wordldquotheoremrdquo or ldquoproblemrdquo as we might today He does not have thetypographical means that Heiberg uses in his own edition of Euclidto distinguish the protasis from the rest of the proposition Nothe protasis just sits there not even preceded by the ldquoI say thatrdquo(λέγω ὅτι) that may be seen further down in the proof For me tonotice this naiumlve students were apparently not enough but I hadalso to read Fowlerrsquos Mathematics of Platorsquos Academy [ (e)pp ndash]

Unfortunately some established mathematicians use the same style in theirown lectures

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The problem

Textbooks of analytic geometry do not make their logical founda-tions clear Of course I can speak only of the books that I have beenable to consult these are from the last century or so Descartesrsquosoriginal presentation [] in the th century is clear enough In anabstract sense Descartes may be no more rigorous than his succes-sors He does get credit for actually inventing his subject or atleast for introducing the notation we use today minuscule lettersfor lengths with letters from the beginning of the alphabet used forknown lengths and letters from the end for unknown lengths Asfor his mathematics itself Descartes explicitly bases it on an an-cient tradition that culminates in the th century with Pappus ofAlexandria

More recent analytic geometry books start in the middle of thingsbut they do not make it clear what those things are I think this isa problem The chief aim of these notes is to identify this problemand its solution

How can analytic geometry be presented rigorously Rigor is nota fixed standard but depends on the audience Still it puts somerequirements on any work of mathematics as I shall discuss in Chap-ter In my own mathematics department students of analytic ge-ometry have had a semester of calculus and a semester of syntheticgeometry from its own original source Book I of Euclidrsquos Elements[ ] They are the audience that I especially have in mind in myconsiderations of rigor But I would suggest that any students ofanalytic geometry ought to come to the subject similarly preparedat least on the geometric side

Plane analytic geometry can be seen as the study of the Euclideanplane with the aid of a sort of rectangular grid that can be laid overthe plane as desired Alternatively the subject can be seen as adiscovery of geometric properties in the set of ordered pairs of real

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The problem

numbers I propose to call these two approaches the geometric andthe algebraic respectively Either approach can be made rigorousBut a course ought to be clear which approach is being taken

Probably most courses of analytic geometry take the geometricapproach relying on students to know something of synthetic geom-etry already Then the so-called Distance Formula can be justifiedby appeal to the Pythagorean Theorem However even in such acourse students might be asked to use algebraic methods to provefor example that the base angles of an isosceles triangle are equal

Perhaps what would be expected is something like the following

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively as in Figure These

b

βc

γ

a

Figure An isosceles triangle in a vector space

angles are given by

(aminus b) middot (cminus b) = |aminus b| middot |cminus b| middot cosβ(aminus c) middot (bminus c) = |aminus c| middot |bminus c| middot cos γ

We assume the triangle is isosceles and in particular

|aminus b| = |aminus c|I had a memory that this problem was assigned in an analytic geometry course

that I was once involved with However I cannot find the problem in my filesI do find similar problems such as to prove that the line segment bisectingtwo sides of triangle is parallel to the third side and is half its length or toprove that in an isosceles triangle the median drawn to the third side is justits perpendicular bisector In each case the student is explicitly required touse analytic methods

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

Then

(aminus c) middot (bminus c) = (aminus c) middot (bminus a+ aminus c)

= (aminus c) middot (bminus a) + (aminus c) middot (aminus c)

= (aminus c) middot (bminus a) + (aminus b) middot (aminus b)

= (cminus a) middot (aminus b) + (aminus b) middot (aminus b)

= (cminus b) middot (aminus b)

and so cosβ = cos γ

If one has the Law of Cosines the argument is simpler

Proof Suppose the vertices of a triangle are a b and c and theangles at b and c are β and γ respectively again as in Figure By the Law of Cosines

|aminus c|2 = |aminus b|2 + |cminus b|2 minus 2 middot |aminus b| middot |cminus b| middot cosβ

cosβ =|cminus b|2 middot |aminus b|

and similarly

cos γ =|bminus c|2 middot |aminus c|

If |aminus b| = |aminus c| then cosβ = cos γ so β = γ

In this last argument though the vector notation is a needlesscomplication We can streamline things as follows

Proof In a triangle ABC let the sides opposite A B and C havelengths a b and c respectively and let the angles as B and C be βand γ respectively as in Figure If b = c then

b2 = c2 + a2 minus 2ca cosβcosβ =

a

2c=

a

2b= cos γ

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The problem

B

β

C

γ

A

c b

a

Figure An isosceles triangle

Possibly this is not considered analytic geometry though sincecoordinates are not used even implicitly We can use coordinatesexplicitly laying down our grid conveniently

Proof Suppose a triangle has vertices (0 a) (b0) and (c0) asin Figure We assume a2 + b2 = a2 + c2 and so b = minusc In this

b c

a

Figure An isosceles triangle in a coordinate plane

case the cosines of the angles at (b0) and (c0) must be the same

namely |b|radic

a2 + b2

In any case as a proof of what is actually Euclidrsquos Proposition Ithis whole exercise is logically worthless assuming we have taken thegeometric approach to analytic geometry By this approach we shallhave had to show how to erect perpendiculars to given straight linesas in Euclidrsquos Proposition I whose proof relies ultimately on I

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

One could perhaps develop analytic geometry on Euclidean princi-ples without proving Euclidrsquos I as an independent proposition Forthe equality of angles that it establishes can be established immedi-ately by means of I the Side-Angle-Side theorem by the methodattributed to Pappus by Proclus [ pp ndash] In this case onecan still see clearly that I is true without needing to resort to anyof the analytic methods suggested above

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Failures of rigor

The root meaning of the word rigor is stiffness Rigor in a piece ofmathematics is what makes it able to stand up to questioning Rigorin mathematics education requires helping students to see what ques-tions might be asked about a piece of mathematics

An education in mathematics will take the student through sev-eral passes over the same subjects With each pass the studentrsquosunderstanding should deepen At an early stage the student neednot and cannot be told all of the questions that might be raisedat a later stage But if the mathematics of an early course resem-bles that of a different later course it ought to be equally rigorousOtherwise the older student might assume wrongly that the earliermathematics could in fact stand up to the same scrutiny that thelater mathematics stands up to Concepts in an earlier course mustnot be presented in such a way that they will be misunderstood ina later course

By this standard students of calculus need not master the epsilon-delta definition of limit If the students later take an analysis coursethen they will fill in the logical gaps from the calculus course Thestudents are not going to think that everything was already proved incalculus class so that epsilons and deltas are a needless complicationThey may think there is no reason to prove everything but that isanother matter If students of calculus never study analysis butbecome engineers perhaps or teachers of school mathematics theyare not likely to have false beliefs about what theorems can be provedin mathematics they just will not have a highly developed notion of

It might be counted as a defect in my own education that I did not have un-dergraduate courses in algebra and topology before taking graduate versionsof these courses Graduate analysis was for me a continuation of my highschool course which had been based on Spivakrsquos Calculus [] and (in smallpart) Apostolrsquos Mathematical Analysis []

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

proof

By introducing and using the epsilon-delta definition of limit atthe very beginning of calculus the teacher might actually violate therequirements of rigor if he or she instills the false notion that thereis no rigorous alternative definition of limits How many calculusteachers ignorant of Robinsonrsquos ldquononstandardrdquo analysis [] willtry to give their students some notion of epsilons and deltas out ofa misguided notion of rigor when the intuitive approach by meansof infinitesimals can be given full logical justification

On the other hand in mathematical circles I have encountereddisbelief that the real numbers constitute the unique complete or-dered field Since every valued field has a completion it is possibleto suppose wrongly that every ordered field has a completion Thisconfusion might be due to a lack of rigor in education somewherealong the way There are (at least) two ways to obtain R fromQ One way is to take the quotient of the ring of Cauchy sequencesof rational numbers by the ideal of such sequences that convergeto 0 Another way is to complete the ordering of Q as by takingDedekind cuts and then to show that this completion can be madeinto a field The first construction can be generalized to apply tonon-Archimedean ordered fields the second cannot More preciselythe second construction can be applied to a non-Archimedean fieldbut the result is not a field It might be better to say that while thefirst construction achieves the completeness of R as a metric spacethe second achieves the continuity of R as an ordered field At anyrate continuity is the word Dedekind uses for what his constructionis supposed to accomplish []

In an elementary course the student may learn a theorem ac-cording to which certain conditions on certain structures are logi-cally equivalent But the theorem may use assumptions that are notspelled out This is a failure of rigor In later courses the studentlearns logical equivalences whose assumptions are spelled out Thestudent may then assume that the earlier theorem is like the later

I may have been saved from this confusion by Spivakrsquos final chapter ldquoUnique-ness of the real numbersrdquo [ ch ]

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Failures of rigor

ones It may not be and failure to appreciate this may cause thestudent to overlook some lovely pieces of mathematics The wordstudent here may encompass all of us

The supposed theorem that I have in mind is that in numbertheory the principles of induction and well ordering are equivalent

Proofs of two implications may be offered to back up this claimthough one of the proofs may be left as an exercise The proofs willbe of the standard form They will look like other proofs And yetstrictly speaking they will make no logical sense becausebull Induction is a property of algebraic structures in a signature

with a constant such as 1 and a singulary function symbolsuch as prime (ldquoprimerdquo) for the operation of adding 1

bull Well ordering is a property of ordered structuresWhen well ordering is used to prove induction a set A is takenthat contains 1 and is closed under adding 1 and it is shown thatthe complement of A cannot have a least element For the leastelement cannot be 1 and if the least element is n+ 1 then n isin Aso n + 1 isin A contradicting n + 1 isin A It is assumed here thatn lt n+ 1 The correct conclusion is not that the complement of Ais empty but that if it is not then its least element is not 1 and isnot obtained by adding 1 to anything Thus what is proved is thefollowing

Theorem Suppose (S1 lt) is a well-ordered set with least ele-ment 1 and with no greatest element so that S is also equipped withthe operation prime given by

nprime = minx n lt x

If

(lowast) forallx existy (x = 1 or yprime = x)

then (S1 prime) admits induction

ldquoEither principle may be considered as a basic assumption about the naturalnumbersrdquo [ ch p ] I use this book as an example because it isotherwise so admirable

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

The condition (lowast) is not redundant Every ordinal number in vonNeumannrsquos definition is a well-ordered set and every limit ordinal isclosed under the operation prime but only the least limit ordinal which is012 or ω admits induction in the sense we are discussing

The next limit ordinal which is ωcupωω+1ω+2 or ω+ωdoes not admit induction neither do any of the rest for the samereason they are not ω but they include it Being well-ordered isequivalent to admitting transfinite induction but that is somethingelse

Under the assumption n lt n + 1 ordinary induction does implywell ordering That is we have the following

Theorem Suppose (S1 prime lt) admits induction is linearly or-dered and satisfies

(dagger) forallx x lt xprime

Then (Slt) is well ordered

However the following argument is inadequate

Standard proof If a subset A of S has no least element we can letB be the set of all n in S such that no element of x x lt n belongsto A We have 1 isin B since no element of the empty set belongsto A If n isin B then n isin A since otherwise it would be the leastelement of A so nprime isin B By induction B contains everything in Sand so A contains nothing

We have tacitly used

Lemma Under the conditions of the theorem

forallx 1 6 x

An ordinal in von Neumannrsquos definition is a set rather than the isomorphism-class of well-ordered sets that it was understood to be earlier Von Neumannrsquosoriginal paper from is [] One can read it but one must allow for somedifferences in notation from what is customary now This is one difficulty ofrelying on special notation to express mathematics

The least element of ω is usually denoted not by 1 but by 0 because it is theempty set

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Failures of rigor

This is easily proved by induction Trivially 1 6 1 Moreover if1 6 x then 1 lt xprime since x lt xprime (and orderings are by definitiontransitive) But the standard proof of Theorem also uses

x x lt nprime = x x lt n cup n

that is

x lt nprime hArr x 6 n

The reverse implication is immediate the forward is the following

Lemma Under the conditions of the theorem

forallx forally (x lt yprime rArr x 6 y)

This is not so easy to establish although there are a couple ofways to do it The first method assumes we have established thestandard properties of the set N of natural numbers perhaps byusing the full complement of Peano Axioms as in Landau []

Proof By recursion we define a homomorphism h from (N1 prime)to (S1 prime) By induction in N h is order-preserving and thereforeinjective By induction in S h is surjective Thus h is an isomor-phism Since (N lt) has the desired property so does (Slt)

The foregoing proof can serve by itself as a proof of Theorem An alternative direct proof of Lemma is as follows I do not knowa simpler argument

Proof We name two formulas

ϕ(x y) x lt y rArr xprime 6 y

ψ(x y) x lt yprime rArr x 6 y

So we want to prove forallx existy ψ(x y) Since lt is a linear ordering wehave

(Dagger) ϕ(x y) hArr ψ(y x)

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

We can prove also as a lemma

(sect) ϕ(x y) and ψ(x y) rArr ϕ(x yprime)

Indeed assume ϕ(x y) and ψ(x y) and x lt yprime Then we have

x 6 y

x = y or x lt y

xprime = yprime or xprime 6 y

xprime 6 yprime

[by ψ(x y)]

[by ϕ(x y)]

This gives us (sect) We shall use this to establish by induction

(para) forally forallx(

ϕ(x y) and ψ(x y))

As the base of the induction first we prove

() forallx(

ϕ(x1) and ψ(x1))

Vacuously forallx ϕ(x1) so by (Dagger) we have ψ(1 x) and in particularψ(11) Using ψ(1 x) and putting (1 x) for (x y) in (sect) gives us

ϕ(1 x) rArr ϕ(1 xprime)

ψ(x1) rArr ψ(xprime1)

By induction we have forallx ψ(x1) and hence () Now suppose wehave for some y

(lowastlowast) forallx(

ϕ(x y) and ψ(x y))

By (sect) we get forallx ϕ(x yprime) Finally we establish forallx ψ(x yprime) by induc-tion From () we have ϕ(yprime1) hence ψ(1 yprime) Suppose ψ(x yprime)Then ϕ(yprime x) But from ϕ(x yprime) we have ψ(yprime x) Hence by (sect) wehave ϕ(yprime xprime) that is ψ(xprime yprime) Thus assuming (lowastlowast) we have

forallx(

ϕ(x yprime) and ψ(x yprime))

By induction then we have (para) From this we extract forally forallx ψ(x y)as desired

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Failures of rigor

In Theorem the condition (dagger) is not redundant It is false that astructure that admits induction must have a linear ordering so that(dagger) is satisfied It is true that all counterexamples to this claim arefinite It may seem that there is no practical need to use inductionin a finite structure since the members can be checked individuallyfor their satisfaction of some property However this checking mayneed to be done for every member of every set in an infinite familySuch is the case for Fermatrsquos Theorem as I have discussed elsewhere

Indeed in the Disquisitiones Arithmeticae of [ para] whichis apparently the origin of our notion of modular arithmetic Gaussreports that Eulerrsquos first proof of Fermatrsquos Theorem was as followsLet p be a prime modulus Trivially 1p equiv 1 (with respect to p orindeed any modulus) If ap equiv a (modulo p) for some a then since(a + 1)p equiv ap + 1 we conclude (a + 1)p equiv a + 1 This can beunderstood as a perfectly valid proof by induction in the ring withp elements that we denote by ZpZ we have then proved ap = afor all a in this ring []

In analysis one learns some form of the following possibly associ-ated with the names of Heine and Borel

Theorem The following are equivalent conditions on an intervalI of R

I is closed and bounded All continuous functions from I to R are uniformly continuous

Such a theorem is pedagogically useful both for clarifying theorder of quantification in the definitions of continuity and uniformcontinuity and for highlighting (or at least setting the stage for)the notion of compactness A theorem about the equivalence ofinduction and well ordering serves no such useful purpose If it isloaded up with enough conditions so that it is actually correct asin Theorems and above then there is only one structure (upto isomorphism) that meets the equivalent conditions and this isjust the usual structure of the natural numbers If however theextra conditions are left out as being a distraction to the immature

Henkin investigates them in []

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

student then that student may later be insensitive to the propertiesof structures like ω+ω or ZpZ Thus the assertion that inductionand well ordering are equivalent is nonrigorous in the worst senseNot only does its proof require hidden assumptions but the hidingof those assumptions can lead to real mathematical ignorance

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A standard of rigor

The highest standard of rigor might be the formal proof verifiableby computer But this is not a standard that most mathematiciansaspire to Normally one tries to write proofs that can be checkedand appreciated by other human beings In this case ancient Greekmathematicians such as Euclid and Archimedes set an unsurpassedstandard

How can I say this Two sections of Morris Klinersquos MathematicalThought from Ancient to Modern Times [] are called ldquoThe Meritsand Defects of the Elementsrdquo (ch sect p ) and ldquoThe Defects inEuclidrdquo (ch sect p ) One of the supposed defects is ldquohe usesdozens of assumptions that he never states and undoubtedly did notrecognizerdquo I have made this criticism of modern textbook writersin the previous chapters and I shall do so again in later chaptersHowever it is not really a criticism unless the critic can show thatbad effects follow from ignorance of the unrecognized assumptions Ishall address these assumptions in Euclid a bit later in this chapter

Meanwhile I think the most serious defect mentioned by Klineis the vagueness and pointlessness of certain definitions in EuclidHowever some if not all of the worst offenders were probably addedto the Elements after Euclid was through with it Euclid apparentlydid not need these definitions In any case I am not aware that poordefinitions make any proofs in Euclid confusing

Used to prove the Side-Angle-Side Theorem (Proposition I) fortriangle congruence Euclidrsquos method of superposition is considereda defect However if you do not want to use this method thenyou can just make the theorem an axiom as Hilbert does in TheFoundations of Geometry [] (Hilbertrsquos axioms are spelled outin Chapter below) I myself do not object to Euclidrsquos proof bysuperposition If two line segments are given as equal what else can

See in particular Russorsquos lsquoFirst Few Definitions in the Elementsrsquo [ ]

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

this mean but that one of them can be superimposed on the otherOtherwise equality would seem to be a meaningless notion Likewisefor angles Euclid assumes that two line segments or two angles ina diagram can be given as equal Hilbert assumes not only this butsomething stronger a given line segment can be copied to any otherlocation and likewise for an angle Euclid proves these possibilitiesas his Propositions I and

In his first proposition of all where he constructs an equilateraltriangle on a given line segment Euclid uses two circles each cen-tered at one endpoint of the segment and passing through the otherendpoint as in Figure The two circles intersect at a point that

Α Β

Γ

∆ Ε

Figure Euclidrsquos Proposition I

is the apex of the desired triangle But why should the circles in-tersect It is considered a defect that Euclid does not answer thisquestion Hilbert avoids this question by not mentioning circles inhis axioms

Hilbertrsquos axioms can be used to show that desired points on circlesdo exist It is not a defect of rigor that Euclid does things differentlyThe original meaning of geometry in Greek is surveying Herodotus[ II] traces the subject to ancient Egypt where the amountof land lost to the annual flooding of the Nile had to be measuredThe last two propositions of Book I of Euclid are the Pythagorean

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A standard of rigor

Theorem and its converse but perhaps the climax of Book I comestwo propositions earlier with number Here it is shown that for aplot of land having any number of straight sides an equal rectangularplot of land with one given side can be found The whole point ofBook I is to work out rigorously what can be done with tools suchas a surveyor or perhaps a carpenter might have

() a tool for drawing and extending straight lines() a tool for marking out the points that are equidistant from a

fixed point and() a set square not for drawing right angles but for justifying

the postulate that all right angles are equal to one anotherIn the th century it is shown that the same work can be accom-plished with even less This possibly reveals a defect of Euclid butI do not think it is a defect of rigor

There is however a danger in reading Euclid today The dangerlies in a hidden assumption but it is an assumption that we makenot Euclid We assume that with his postulates he is doing thesame sort of thing that Hilbert is doing with his axioms He is notHilbert has to deal with the possibility of non-Euclidean geometryHilbert can contemplate models that satisfy some of his axioms butviolate others For Euclid there is just one model this world

If mathematicians never encountered structures other than thenatural numbers that were well ordered or admitted induction thenthere might be nothing wrong with saying that induction and wellordering were equivalent But then again even to speak of equiva-lence is to suggest the possibility of different structures that satisfythe conditions in question This is not a possibility that Euclid hasto consider

If Euclid is not doing what modern mathematicians are doingwhat is the point of reading him I respond that he is obviouslydoing something that we can recognize as mathematics If he is juststudying the world so are mathematicians today it is just a worldthat we have made more complicated I suggested in the beginningthat students are supposed to come to an analytic geometry classwith some notion of synthetic geometry As I observe below in Chap-ter for example students are also supposed to have the notion that

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

every line segment has a length which is a so-called real numberThis is a notion that has been added to the world

Nonetheless the roots of this notion can be found in the Elementsin the theory of ratio and proportion beginning in Book V Accord-ing to this theory magnitudes A B C and D are in proportionso that the ratio of A to B is the same as the ratio of C to D if forall natural numbers k and m

kA gt mB hArr kC gt mD

In this case we may write

A B C D

though Euclid uses no such notation What is expressed by thisnotation is not the equality but the identity of two ratios Equalityis a possible property of two nonidentical magnitudes Magnitudesare geometric things ratios are not Euclid never draws a ratio orassigns a letter to it

According to a scholium to Book V ldquoSome say that the book is the discovery ofEudoxus the pupil of Platordquo [ p ] The so-called Euclidean Algorithmas used in Proposition X of the Elements may be a remnant of anothertheory of proportion Apparently this possibility was first recognized in [ pp ndash] The idea is developed in []

I am aware of one possible counterexample to this claim The lastpropositionmdashnumber mdashin Book VII is to find the number that is the least

of those that will have given parts The meaning of this is revealed in theproof which begins ldquoLet the given parts be Α Β and Γ Then it is requiredto find the number that is the least of those that will have the parts Α Βand Γ So let ∆ Ε and Ζ be numbers homonymous with the parts Α Β andΓ and let the least number Η measured by ∆ Ε and Ζ be takenrdquo Thus Η isthe least common multiple of ∆ Ε and Ζ which can be found by PropositionVII Also if for example ∆ is the number n then Α is an nth consideredabstractly it is not given as an nth part of anything in particular ThenΑ might be considered as the ratio of 1 to n Possibly VII was addedlater to Euclidrsquos original text although Heathrsquos note [ p ] suggestsno such possibility If indeed VII is a later addition then so probablyare the two previous propositions on which it relies they are that if n | rthen r has an nth part and conversely But Fowler mentions Propositions and seemingly being as typical or as especially illustrative examplesof propositions from Book VII [ p ]

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A standard of rigor

In any case in the definition it is assumed that A and B have aratio in the first place in the sense that some multiple of either ofthem exceeds the other and likewise for C and D In this case thepair

(m

k kA gt mB

m

k kA 6 mB

)

is a cut (of positive rational numbers) in the sense of Dedekind [p ] Dedekind traces his definition of irrational numbers to theconviction that

an irrational number is defined by the specification of all ratio-nal numbers that are less and all those that are greater than thenumber to be defined if one regards the irrational number asthe ratio of two measurable quantities then is this manner of de-termining it already set forth in the clearest possible way in thecelebrated definition which Euclid gives of the equality of two ra-tios [ pp ndash]

In saying this Dedekind intends to distinguish his account of thecompleteness or continuity of the real number line from some otheraccounts Dedekind does not literally define an irrational numberas a ratio of two ldquomeasurable quantitiesrdquo the definition of cuts suchas the one above does not require the use of magnitudes such asA and B Dedekind observes moreover that Euclidrsquos geometricalconstructions do not require continuity of lines ldquoIf any one shouldsayrdquo writes Dedekind

that we cannot conceive of space as anything else than continuousI should venture to doubt it and to call attention to the fact thata far advanced refined scientific training is demanded in orderto perceive clearly the essence of continuity and to comprehendthat besides rational quantitative relations also irrational andbesides algebraic also transcendental quantitative relations areconceivable [ pp ]

Modern geometry textbooks (as in Chapter below) assume con-tinuity in this sense but without providing the ldquorefined scientifictrainingrdquo required to understand what it means Euclid does pro-vide something of this training starting in Book V of the Elementsbefore this he makes no use of continuity in Dedekindrsquos sense

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

In the Muslim and Christian worlds Euclid has educated mathe-maticians for centuries He shows the world what it means to provethings One need not read all of the Elements today But Book Ilays out the basics of geometry in a beautiful way If you want stu-dents to learn what a proof is I think you can do no better thentell them ldquoA proof is something like what you see in Book I of theElementsrdquo

I have heard of textbook writers who informed of errors decide toleave them in their books anyway to keep the readers attentive Theperceived flaws in Euclid can be considered this way The Elementsmust not be treated as a holy book If it causes the student to thinkhow things might be done better this is good

The Elements is not a holy book it is one of the supreme achieve-ments of the human intellect It is worth reading for this reasonjust as say Homerrsquos Iliad is worth reading

The rigor of Euclidrsquos Elements is astonishing Students in schooltoday learn formulas like A = πr2 for the area of a circle Thisformula encodes the following

Theorem (Proposition XII of Euclid) Circles are to one an-other as the squares on the diameters

One might take this to be an obvious corollary of

Theorem (Proposition XII of Euclid) Similar polygons in-scribed in circles are to one another as the squares on the diameters

And yet Euclid gives an elaborate proof of XII by what is todaycalled the Method of Exhaustion

Euclidrsquos proof of Theorem in modern notation Suppose a circleC1 with diameter d1 is to a circle C2 with diameter d2 in a lesserratio than d1

2 is to d22 Then d1

2 is to d22 as C1 is to some fourth

proportional R that is smaller than C2 More symbolically

C1 C2 lt d12 d2

2

d12 d2

2 C1 R

R lt C2

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A standard of rigor

By inscribing in C2 a square then an octagon then a 16-gon and soforth eventually (by Euclidrsquos Proposition X) we obtain a 2n-gonthat is greater than R The 2n-gon inscribed in C1 has (by Theo-rem that is Euclidrsquos XII) the same ratio to the one inscribed inC2 as d1

2 has to d22 Then

d12 d2

2 lt C1 R

which contradicts the proportion above

Such is Euclidrsquos proof in modern symbolism Euclid himself doesnot refer to a 2n-gon as such His diagram must fix a value for 2nand the value fixed is 8 I do not know if anybody would considerthis a lack of rigor as if rigor is achieved by symbolism I wonderhow often modern symbolism is used to give only the appearance ofrigor (See Chapter below)

A more serious problem with Euclidrsquos proof is the assumption ofthe existence of the fourth proportional R Kline does not men-tion this as a defect in the sections of his book cited above he doesmention the assumption elsewhere (on page his ) but not criti-cally Heath mentions the assumption in his own notes [ v p ] though he does not supply the following way to avoid theassumption

Second proof of Theorem We assume instead that if two unequalmagnitudes have a ratio in the sense of Book V of the Elementsthen their difference has a ratio with either one of them In thenotation above since C1 C2 lt d1

2 d22 there are some natural

numbers m and k such that

mC1 lt kC2 md12gt kd2

2

Let r be a natural number such that r(kC2 minusmC1) gt C2 Then

rmC1 lt (rk minus 1)C2 rmd12gt rkd2

2

This is the postulate of Archimedes ldquoamong unequal [magnitudes] the greaterexceeds the smaller by such a [difference] that is capable added itself to itselfof exceeding everything set forth (of those which are in a ratio to one anotherrdquo[ p ]

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

Assuming 2nminus1 gt rk let P1 be the 2n-gon inscribed in C1 and P2in C2 Then

C2 minus P2 lt1

2nminus1C2 6

1rkC2

rmP1 lt rmC1 lt (rk minus 1)C2 lt rkP2

But also P1 P2 d12 d2

2 so that rmP1 gt rkP2 which isabsurd

Does this second proof of Theorem supply a defect in Euclidrsquosproof In his note Heath quotes Simson to the effect that assumingthe mere existence of a fourth proportional does no harm even ifthe fourth proportional cannot be constructed I see no reason whyEuclid could not have been aware of the possibility of avoiding thisassumption although he decided not to bother his readers with thedetails

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Why rigor

As used by Euclid and Archimedes the Method of Exhaustion servesno practical purpose Archimedes has an intuitive method [] forfinding equations of areas and volumes He uses this method todiscover that

() a section of a parabola is a third again as large as the trianglewith the same base and height

() if a cylinder is inscribed in a prism with square base then thepart of the cylinder cut off by a plane through a side of thetop of the prism and the center of the base of the cylinder is asixth of the prism and

() the intersection of two cylinders is two thirds of the cube inwhich this intersection is inscribed

However Archimedes does not believe that his method provides arigorous proof of his equations He supplies proofs after the equa-tions themselves are discovered Why does he do this After allhe believes Democritus should be credited for discovering that thepyramid is the third part of the prism with the same base and heighteven though it was Eudoxus who later actually gave a proof (Thetheorem is a corollary to Proposition XII of Euclidrsquos Elements)

Although Heath translated Euclid faithfully into English appar-ently he thought the rigor of Archimedes was too much for modernmathematicians to handle so he paraphrased Archimedes with mod-ern symbolism [] This symbolism is a way to avoid keeping toomany ideas in onersquos head at once When one wants to use a theoremfor some practical purpose then this labor-saving feature of symbol-ism is perhaps desirable But if the whole point of a theorem is tosee and appreciate something then perhaps symbolism gets in theway of this

I do not think Archimedes really explains his compulsion for math-ematical rigor Being the originator of the ldquomerciless telegram stylerdquo

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

that Landau [ p xi] for example writes in Euclid does not ex-plain anything at all in the Elements he just does the mathematicsI suppose the rigor of this mathematics at least regarding propor-tions is to be explained as a remnant of the discovery of incommen-surable magnitudes This discovery necessitates such a theory ofproportion as is attributed to Eudoxus of Cnidus and is presented inBook V of the Elements In any case given a theory of proportionif Euclid is going to assert Proposition XII he is duty-bound toprove it in accordance with the theory at hand

Modern mathematicians are likewise duty-bound to respect cur-rent standards of rigor As suggested above in Chapter this doesnot mean that a textbook has to prove everything from first prin-ciples but at least some idea ought to be given of what those firstprinciples are This standard is set by Euclid and respected byArchimedes It is not so much respected by modern textbooks ofanalytic geometry In Chapters and I look at a couple of exam-ples of these

It may be said that the purpose of an analytic geometry text isto teach the student how to do certain things how to solve certainproblems as detailed in the first quotation in Chapter below Thepurpose is not to teach proof However as suggested in Chapter previous Book I of the Elements is also concerned with doingthings with the help of such tools as a surveyor or carpenter mightuse What makes the Elements mathematics is that it justifies themethods it gives for doing things It provides proofs

I suppose that when a student of mathematics is given a problemeven a numerical problem she or he is expected to be able to comeup with a solution and not just an answer A solution is a proofthat the answer is correct It tells why the answer is correct Thusit gives the reader the means to solve other problems An analyticgeometry text ought to prove that its methods are correct At leastit ought to give some indication of how the proofs might be supplied

Fowler [ p n ] refers to Landau as the ldquopremier exponentrdquo of ldquoamore recent German style of setting out mathematics generally called lsquoSatz-Beweisrsquo style that has some affinities with protasis-stylerdquo that is Euclidrsquosstyle

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

According to the Preface of Nelson Folley and Borgmanrsquos volume Analytic Geometry [ pp iiindashiv]

This text has been prepared for use in an undergraduate coursein analytic geometry which is planned as preparation for thecalculus rather than as a study of geometry In order that it maybe of maximum value to the future student of the calculus thebasic sciences and engineering considerable attention is given totwo important problems of analytic geometry They are (a) giventhe equation of a locus to draw the curve or describe it geomet-rically (b) given the geometric description of a locus to find itsequation that is to translate a verbal description of a locus intoa mathematical equation

Inasmuch as the studentrsquos ability to use analytic geometry asa tool depends largely on his understanding of the coordinate sys-tem particular attention has been given to producing as thorougha grasp as possible He must appreciate for example that thepoint (a b) is not necessarily located in the first quadrant andthat the equation of a curve may be made to take a simple form ifthe coordinate axes are placed with forethought

By referring to judicious placement of axes the authors reveal theirworking hypothesis that there is already a geometric plane beforeany coordinatization It is not clear what students are expected toknow about this plane The book proper begins on page

Directed Line Segments If A B and C (Fig ) are threepoints which are taken in that order on an infinite straight line

I have this book only because my mother used it in college When I was twelveI used this book in order to plot graphs of conic sections I also used the tablesat the back to plot graphs of the trigonometric and logarithmic functionsI donrsquot remember whether I tried to read the book from the beginning if Idid I must not have got very far

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

then in conformity with the principles of plane geometry we maywrite

() AB +BC = AC

For the purposes of analytic geometry it is convenient to haveequation () valid regardless of the order of the points A B and Con the infinite line The conventional way of accomplishing this

A B C

Fig

is to select a positive direction on the line and then define thesymbol AB to mean the number of linear units between A and Bor the negative of that number according as we associate withthe segment AB the positive or the negative direction With thisunderstanding the segment AB is called a directed line segment

In any given problem such a segment possesses an intrinsic signdecided in advance through the arbitrary selection of a positivedirection for the infinite line of which the segment is a part

Thus it is assumed that the student knows what a ldquonumber of linearunitsrdquo means I suppose the student has been trained in high schoolto believe that () every line segment has a length and () thislength is a number of some unit But probably the student has noidea of how numbers in the original sensemdashnatural numbersmdashcan beused to create all of the numbers that might be needed to designategeometrical lengths

Instead of making the unexamined assumption that every linesegment has a numerical length we could take congruence as thefundamental notion and without defining length itself we couldsay that congruent line segments have the same length One whoknew about equivalence relations could then define a length as acongruence class of segments but this need not be made explicit

Alternatively and more in line with the approach of Nelson amp alwe can fix a unit line segment in the manner of Descartes in the

This quotation has almost exactly the same visual appearance as in the originaltext In particular the lines and the placement of the figure are the same

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

Geometry [] Then by using the definition of proportion found inBook V of Euclidrsquos Elements and discussed in Chapter above wecan define the length of an arbitrary line segment as the ratio of thissegment to the unit segment This gives us lengths rigorously as realnumbers if we use Dedekindrsquos definition of the latter As Dedekindobserved though and as we repeated there is no need to assumethat every positive real number is the length of some segment

Perhaps these details need not be rehearsed with the student Butneither is it necessary to introduce lengths at all in order to justifythe equation AB +BC = AC in the text It need only be said thatan expression like AB no longer represents merely a line segmentbut a directed line segment Then BA is the negative of AB andwe can write

BA = minusAB AB = minusBA AB +BA = 0()

as indeed Nelson amp al do later in their sect (on their page ) Thusdirected segments are understood to compose an abelian group In-deed although not a word is said about the commutativity or asso-ciativity of addition of segments these properties might be under-stood to follow from the ldquoprinciples of plane geometryrdquo mentioned inthe text

If a positive direction is fixed for the straight line containing A andB then AB itself is understood as positive if B is further than A inthe positive direction otherwise AB is negative Thus the abeliangroup of directed segments becomes an ordered group I quote thenext section of the text in its entirety

Length Distance The length of a directed line segment isthe number of linear units which it contains The symbol |AB|will be used to designate the length of the segment AB or thedistance between the points A and B

Occasionally the symbol AB will be used to represent the linesegment as a geometric entity but if a numerical measure is impliedthen it stands for the directed segment AB or the directed distance

from A to B

Two directed segments of the same line or of parallel lines areequal if they have equal lengths and the same intrinsic signs

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

Again we see the unexamined assumption that the reader knowswhat a ldquonumber of linear unitsrdquo means when a length as such canbe defined simply as a congruence class Congruence of directed seg-ments of a given straight line can be understood as a congruence ofsegments that is established by translation only without reflectionThe text defines this kind of congruence as equality

An expression like AB in the text can now have any of threemeanings It can mean

() the segment bounded by A and B() the directed segment from A to B or() the directed distance from A to B which we might identify

with the class of directed segments that are equal to the di-rected segment from A to B

In fact there will be yet another possible meaning of the expressionAB a meaning that will be used without comment in a quotationgiven below AB can mean

() the infinite straight line containing A and BThe second and third meanings of expressions like AB are used inexercises on page of the book

In Problems ndash the consecutive points A B C D E F Gare spaced one inch apart on an infinite line which is positivelydirected from A to B

Verify that BD +GA = BA+GD Verify that DB +GA = DA+GB Verify that BG+ FC = BC + FG = 2DE Verify that 12 (EA+ EG) = ED

Of the five equations here the first three can be understood as equa-tions of directed segments the remaining two are equations of di-rected distances The preamble to the problems here refers not justto abstract units but to inches which have no mathematical defi-nition Every mathematical statement about inches is just as true ifinches are replaced with miles This goes unmentioned

In another problem the authors display their assumption thatnumbers of units can be irrational and even transcendental

On a coordinate axis where the unit of measure is one

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

inch plot the points whose coordinates are 2 minus 53 π 3minusradic5 and

3radicminus16 respectively

There is no properly geometric reason to introduce transcendentalor even nonquadratic lengths However as noted in the quotationfrom the preface the book is for students of the calculus The bookitself has a chapter called ldquoGraphs of Single-Valued TranscendentalFunctionsrdquo Meanwhile after a coordinatization of a straight line isconsidered in sect of Chapter the plane is considered

Rectangular Coordinates The coordinate system of thepreceding article may be generalized so as to enable us to de-

X

Y

O

bP

x

y

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Fig

scribe the location of a point in a the plane Through any point O(Fig ) select two mutually perpendicular directed infinite straightlines OX and OY thus dividing the plane into four parts calledquadrants which are numbered as shown in the figure The pointO is the origin and the directed lines are called the x-axis and they-axis respectively A unit of measure is selected for each axisUnless the contrary is stated the units selected will be the samefor both axes

The directed distance from OY to any point P in the planeis the x-coordinate or abscissa of P the directed distance from

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

OX to the point is its y-coordinate or ordinate Together theabscissa and ordinate of a point are called its rectangular coordi-

nates When a letter is necessary to represent the abscissa x ismost frequently used y is used to represent the ordinate

Consequently we may represent a point by its coordinatesplaced in parentheses (the abscissa always first) and refer to thissymbol as the point itself For example we may refer to the pointP1 of [the omitted figure] as the point (35) Sometimes it is con-venient to use both designations we then write P1(35) Whena coordinate of a point is an irrational number a decimal approx-imation is used in plotting the point

Here OX and OY are not segments but infinite straight lines Nel-son amp al evidently do not want to give a name such as R to theset of all numbers under consideration Hence they cannot say thatthey identify the geometrical plane with R times R they can say onlythat they identify individual points with pairs of numbers This isfine except that it leaves unexamined the assumption that lengthsare numbers of units

How many generations of students have had to learn the wordsabscissa and ordinate without being given their etymological mean-ings Nelson amp al do not discuss them even in their chapter onconic sections although the terms are the Latin translations of Greekwords used by Apollonius of Perga in the Conics [] I consider theiroriginal meaning in Chapter below

Meanwhile I just have to wonder whether an analytic geometrytextbook cannot be more enticing than that of Nelson amp al If thepurpose of the subject is to solve problems why not present someof the actual problems that the subject was invented to solve Apossible example is the duplication of the cube discussed at the endof Chapter

One complaint I have about my own education is having had to learn technicalterms without their etymology In a literature class in high school how mucheasier it would have been to learn what zeugma was if only it had beenpointed out to us that the Greek word was cognate with the Latin-derivedjoin and the Anglo-Saxon yoke

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Abscissas and ordinates

In the first of the eight books of the Conics [] Apollonius derivesproperties of the conic sections that can be used to write their equa-tions in rectangular or oblique coordinates I review these propertieshere because () they have intrinsic interest () they are the reasonwhy Apollonius gave to the three conic sections the names that theynow have and () the vocabulary of Apollonius is a source for manyof our technical terms including abscissa and ordinate

Apollonius did not invent any of his terms these were just ordi-nary words used in a certain way When we carry those wordsmdashortheir Latin versionsmdashover into our own language we create somedistance between ourselves and mathematics When I first learnedthat a conic section had a latus rectum I understood that there was awhole theory of conic sections that was not being revealed althoughits existence was hinted at by this peculiar Latin term Had thelatus rectum been called an upright side as in Apollonius it wouldhave been easier to ask ldquoWhat is an upright siderdquo In turn textbookwriters might have felt more obliged to explain what it was In anycase I am going to give an explanation below

English does borrow foreign words freely this is a characteristicof the language A large lexicon is not a bad thing A choice fromamong two or more synonyms can help establish the register of apiece of speech If distinctions between near-synonyms are carefullymaintained then subtlety of expression is possible Circle and cycleare Latin and Greek words for the same thing but the Greek word

The first four books survive in Greek the next three in Arabic translationthe last book is lost

In the s the Washington Post described a best-selling book called Color

Me Beautiful as offering ldquothe color-wheel approach to female pulchrituderdquoThe New York Times just said the book provided ldquobeauty tips for womenrdquoThe register of the Post was mocking the Times neutral

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

is used more abstractly in English and it would be bizarre to referto a finite group of prime order as being ldquocircularrdquo

However mathematics can be done in any language Greek doesmathematics without a specialized vocabulary It is worthwhile toconsider what this is like

For Apollonius a cone (ὁ κῶνος ldquopine-conerdquo) is a solid figure de-termined by () a base (ἡ βάσις) which is a circle and () a vertex(ἡ κορυφή ldquosummitrdquo) which is a point that is not in the plane of thebase The surface of the cone contains all of the straight lines drawnfrom the vertex to the circumference of the base A conic surface(ἡ κωνικὴ ἐπιφάνεια) consists of such straight lines not bounded bythe base or the vertex but extended indefinitely in both directions

The straight line drawn from the vertex of a cone to the centerof the base is the axis (ὁ ἄξων ldquoaxlerdquo) of the cone If the axis isperpendicular to the base then the cone is right (ὀρθός) otherwiseit is scalene (σκαληνός ldquounevenrdquo) Apollonius considers both kindsof cones indifferently

A plane containing the axis intersects the cone in a triangle Sup-pose a cone with vertex A has axial triangle ABC Then the baseBC of this triangle is a diameter of the base of the cone Let anarbitrary chord DE of the base of the cone cut the base BC of theaxial triangle at right angles at a point F as in Figure In theaxial triangle let the straight line FG be drawn from the base tothe side AC This straight line FG may but need not be parallelto the side BA It is not at right angles to DE unless the plane ofthe axial triangle is at right angles to the plane of the base of thecone In any case the two straight lines FG and DE meeting at F are not in a straight line with one another and so they determinea plane This plane cuts the surface of the cone in such a curve

The word ἐπιφάνεια means originally ldquoappearancerdquo and is the source of theEnglish ldquoepiphanyrdquo

Although it is the source of the English cord and chord [] Apollonius doesnot use the word ἡ χορδή although he proves in Proposition I that thestraight line joining any two points of a conic section is a chord in the sensethat it falls within the section The Greek χορδή means gut hence anything

made with gut be it a lyre-string or a sausage []

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Abscissas and ordinates

A

B CF

G

B C

D

E

F

Figure Axial triangle and base of a cone

DGE as is shown in Figure Apollonius refers to such a curve

D EF

G

Figure A conic section

first (in Proposition I) as a section (ἡ τομή) in the surface of thecone and later (I) as a section of a cone All of the chords of thissection that are parallel to DE are bisected by the straight line GF Therefore Apollonius calls this straight line a diameter (ἡ διάμετρος[γραμμή]) of the section

The parallel chords bisected by the diameter are said to be drawnto the diameter in an orderly way The Greek adverb here isτεταγμένως [] from the verb τάσσω which has meanings like ldquoto

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

draw up in order of battlerdquo [] A Greek noun derived from this verbis τάξις which is found in English technical terms like taxonomy andsyntax [] The Latin adverb corresponding the Greek τεταγμένωςis ordinate from the verb ordino From the Greek expression forthe straight line drawn in an orderly way Apollonius will elide themiddle part leaving the in-an-orderly-way This term will refer tohalf of a chord bisected by a diameter Similar elision in the Latinleaves us with the word ordinate for this half-chord [] Descartesrefers to ordinates as [lignes] qui srsquoappliquent par ordre [au] diametre[ p ]

The point G at which the diameter GF cuts the conic sectionDGE is called a vertex (κορυφής as before) The segment of thediameter between the vertex and an ordinate has come to be calledin English an abscissa but this just the Latin translation of Apol-loniusrsquos Greek for being cut off (ἀπολαμβανομένη ldquotakenrdquo)

Apollonius will show that every point of a conic section is thevertex for some unique diameter If the ordinates corresponding toa particular diameter are at right angles to it then the diameterwill be an axis of the section Meanwhile in describing the relationbetween the ordinates and the abscissas of conic section there arethree cases to consider

The parabola

Suppose the diameter of a conic section is parallel to a side of thecorresponding axial triangle For example suppose in Figure that FG is parallel to BA The square on the ordinate DF is equalto the rectangle whose sides are BF and FC (by Euclidrsquos PropositionIII) More briefly DF 2 = BF middot FC But BF is independent of

Heath [ p clxi] translates τεταγμένως as ordinate-wise Taliaferro [ p ]as ordinatewise But this usage strikes me as anachronistic The term or-

dinatewise seems to mean in the manner of an ordinate but ordinates arejust what we are trying to define when we translate τεταγμένως

I note the usage of the Greek participle in [ I p ] Its general usagefor what we translate as abscissa is confirmed in [] although the generalsense of the verb is not of cutting but of taking

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Abscissas and ordinates

the choice of the point D on the conic section That is for any suchchoice (aside from the vertex of the section) a plane containing thechosen point and parallel to the base of the cone cuts the cone inanother circle and the axial triangle cuts this circle along a diameterand the plane of the section cuts this diameter at right angles intotwo pieces one of which is equal to BF The square on DF thusvaries as FC which varies as FG That is the square on an ordinatevaries as the abscissa (I) Hence there is a straight line GH suchthat

DF 2 = FG middotGHand GH is independent of the choice of D

This straight line GH can be conceived as being drawn at rightangles to the plane of the conic section DGE Apollonius calls GHthe upright side (ὀρθία [πλευρά]) and Descartes accordingly callsit le costeacute droit [ p ] Apollonius calls the conic section itself aparabola (ἡ παραβολή) that is an application presumably becausethe rectangle bounded by the abscissa and the upright side is theresult of applying (παραβάλλω) the square on the abscissa to theupright side Such an application is made for example in PropositionI of Euclidrsquos Elements where a parallelogram equal to a giventriangle is applied to a given straight line (This proposition is alemma for Proposition mentioned in Chapter above as theclimax of Book I of the Elements)

The Latin term for upright side is latus rectum This term is alsoused in English In the Oxford English Dictionary the earliest quo-tation illustrating the use of the term is from a mathematical dictio-nary published in Evidently the quotation refers to Apolloniusand gives his meaning

App Conic Sections In a Parabola the Rectangle of the Diam-eter and Latus Rectum is equal to the rectangle of the Segmentsof the double Ordinate []

I assume the ldquosegments of the double ordinaterdquo are the two halvesof a chord so that each of them is what we are calling an ordinateand the rectangle contained by them is equal to the square on oneof them

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

The textbook by Nelson amp al considered in Chapter above de-fines the parabola in terms of a focus and directrix The possibilityof defining all of the conic sections in this way is demonstrated byPappus [ p ] and was probably known to Apollonius Ac-cording to Nelson amp al

The chord of the parabola which contains the focus and is perpen-dicular to the axis is called the latus rectum Its length is of valuein estimating the amount of ldquospreadrdquo of the parabola

The first sentence here defines the latus rectum so that it is four timesthe length of Apolloniusrsquos The second sentence correctly describesthe significance of the latus rectum However the juxtaposition ofthe two sentences may mislead somebody who knows just a littleLatin The Latin adjective latus -a -um does mean ldquobroad widespacious extensiverdquo it is the root of the English noun latitudeHowever the Latin adjective latus is unrelated to the noun latus-eris ldquoside flankrdquo [] which is found in English in the adjectivelateral and this noun is what is used in the phrase latus rectum

Each conic section can be understood as the locus of a point whose distancefrom a given point has a given ratio to its distance from a given straight lineAs Heath [ pp xxxvindashxl] explains Pappus proves this theorem becauseEuclid did not supply a proof in his now-lost treatise on surface loci Euclidmust have omitted the proof because it was already well known and Euclidpredates Apollonius Kline [ p ] summarizes all of this by saying thatthe focus-directrix property ldquowas known to Euclid and is stated and proved byPappusrdquo Later (on his page ) Kline gives the precise reference to Pappusit is Proposition [in Hultschrsquos numbering] of Book VII ldquoAs noted in thepreceding chapterrdquo he says ldquoEuclid probably knew itrdquo

In latus rectum the adjective rectus -a -um ldquostraight uprightrdquo is given theneuter form because the noun latus is neuter The plural of latus rectum

is latera recta The neuter plural of the adjective latus would be lata Thedictionary writes the adjective as latus with a long ldquoardquo but the ldquoardquo in thenoun is unmarked and therefore short As far as I can tell the adjectiveis to be distinguished from another Latin adjective with the same spelling(and the same long ldquoardquo) but with the meaning of ldquocarried bornerdquo used forthe past participle of the verb fero ferre tulı latum This past participleappears in English in words like translate while fer- appears in transfer

The American Heritage Dictionary [] traces latus ldquobroadrdquo to an Indo-European root stel- and gives latitude and dilate as English derivatives latus

ldquocarriedrdquo comes from an Indo-European root tel- and is found in English

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Abscissas and ordinates

Denoting abscissa by x and ordinate by y and latus rectum by ℓwe have for the parabola the modern equation

(lowast) y2 = ℓx

The hyperbola

The second possibility for a conic section is that the diameter meetsthe other side of the axial triangle when this side is extended beyondthe vertex of the cone In Figure the diameter FG crossing one

A

B CF

G

K

H

L

M

B C

D

E

F

Figure Axial triangle and base of a cone

side of the axial triangle ABC at G crosses the other side extended

words like translate and relate but also dilatory Thus dilatory is not tobe considered as a derivative of dilate A French etymological dictionary[] implicitly confirms this under the adjacent entries dilater and dilatoire

The older Skeat [] does give dilatory as a derivative of dilate Howeverunder latitude he traces latus ldquobroadrdquo to the Old Latin stlatus while undertolerate he traces latum ldquobornerdquo to tlatum In his introduction Skeat says hehas collated his dictionary ldquowith the New English Dictionary [as the Oxford

English Dictionary was originally called] from A to H (excepting a smallportion of G)rdquo In fact the OED distinguishes two English verbs dilate onefor each of the Latin adjectives latus But the dictionary notes ldquoThe senselsquoprolongrsquo comes so near lsquoenlargersquo lsquoexpandrsquo or lsquoset forth at lengthrsquo that thetwo verbs were probably not thought of as distinct wordsrdquo

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

at K Again DF 2 = BF middot FC but the latter product now varies asKF middot FG The upright side GH can now be defined so that

BF middot FC KF middot FG GH GK

We draw KH and extend to L so that FL is parallel to GH andwe extend GH to M so that LM is parallel to FG Then

FL middot FG KF middot FG FL KF

GH GK

BF middot FC KF middot FG

and so FL middot FG = BF middot FC Thus

DF 2 = FG middot FL

Apollonius calls the conic section here an hyperbola (ἡ ὑπερβολή)that is an exceeding because the square on the ordinate is equal to arectangle whose one side is the abscissa and whose other side is ap-plied to the upright side but this rectangle exceeds (ὑπερβάλλω) therectangle contained by the abscissa and the upright side by anotherrectangle This last rectangle is similar to the rectangle containedby the upright side and GK Apollonius calls GK the transverseside (ἡ πλαγία πλευρά) of the hyperbola Denoting its length by aand the other segments as before we have the modern equation

(dagger) y2 = ℓx+ℓ

ax2

The ellipse

The last possibility is that the diameter meets the other side of theaxial triangle when this side is extended below the base All of thecomputations will be as for the hyperbola except that now if it isconsidered as a directed segment as in Chapter the transverse sideis negative and so the modern equation is

(Dagger) y2 = ℓxminus ℓ

ax2

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Abscissas and ordinates

In this case Apollonius calls the conic section an ellipse (ἡ ἔλλειψις)that is a falling short because again the square on the ordinate isequal to a rectangle whose one side is the abscissa and whose otherside is applied to the upright side but this rectangle now falls short(ἐλλείπω) of the rectangle contained by the abscissa and the uprightside by another rectangle Again this last rectangle is similar to therectangle contained by the upright and transverse sides

Thus the terms abscissa and ordinate are ultimately translationsof Greek words that merely describe certain line segments that canbe used to describe points on conic sections For Apollonius theyare not required to be at right angles to one another

Descartes generalizes the use of the terms slightly In one example[ p ] he considers a curve derived from a given conic sectionin such a way that if a point of the conic section is given by anequation of the form

y2 = x

then a point on the new curve is given by

y2 = xprime

where xxprime is constant But Descartes just describes the new curvein words

toutes les lignes droites appliqueacutees par ordre a son diametre estantesgales a celles drsquoune section conique les segmens de ce diametrequi sont entre le sommet amp ces lignes ont mesme proportion aune certaine ligne donneacutee que cete ligne donneacutee a aux segmensdu diametre de la section conique auquels les pareilles lignes sontappliqueacutees par ordre

There is still no notion that an arbitrary point might have two co-ordinates called abscissa and ordinate respectively

ldquoAll of the straight lines drawn in an orderly way to its diameter being equalto those of a conic section the segments of this diameter that are betweenthe vertex and these lines have the same ratio to a given line that this givenline has to the segments of the diameter of the conic section to which theparallel lines are drawn in an orderly wayrdquo

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

For an hyperbola or ellipse the center (κέντρον) is the midpointof the transverse side In Book I of the Conics Apollonius showsthat the diameters of () an ellipse are the straight lines throughits center () an hyperbola are the straight lines through its centerthat actually cut the hyperbola () a parabola are the straightlines that are parallel to the axis Moreover with respect to a newdiameter the relation between ordinates and abscissas is as beforeexcept that the upright and transverse sides may be different

I do not know of an efficient way to prove these theorems by Carte-sian analytic methods Descartes opens his Geometry by saying

All problems in geometry can easily be reduced to such terms thatone need only know the lengths of certain straight lines in orderto solve them

However Apollonius proves his theorems about diameters by meansof areas Areas can be reduced to products of straight lines but thereduction in the present context seems not to be particularly easyFor example to shift the diameter of a parabola Apollonius will usethe following

Lemma (Proposition I of Apollonius) In Figure it isassumed that () the parabola GDB has diameter AB () AG istangent to the parabola at G () GJ is an ordinate and () GJBHis a parallelogram Moreover () the point D is chosen at randomon the parabola and () triangle EDZ is drawn similar to AGJ Itfollows that

the triangle EDZ is equal to the parallelogram HZ

If the hyperbola is considered together with its conjugate hyperbola then allstraight lines through the center are diameters except the asymptotes

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

A

B

G

D

E

Z

H

J

Figure Proposition I of Apollonius

Proof The proof relies on knowing (from I) that AB = BJ Therefore AGJ = HJ Thus the claim follows when D is just thepoint G In general we have

EDZ HJ EDZ AGJ [Euclid V]

DZ2 GJ2 [Euclid VI]

BZ BJ [Apollonius I]

HZ HJ [Euclid VI]

and so EDZ = HZ by Euclid V The relative positions of D andG on the parabola are irrelevant to the argument

Then the diameter of a parabola can be shifted by the following

Theorem (Proposition I of Apollonius) In Figure it isassumed that () KDB is a parabola () its diameter is MBG() GD is tangent to the parabola and () through D parallel to BGstraight line ZDN is drawn Moreover () the point K is chosen atrandom on the parabola () through K parallel to GD the straightline KL is drawn and () BR is drawn parallel to GD It follows

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

M

B

G

D

Z

NK

L

RM

B

G

D

Z

NK

L

P

R

X

E

Figure Proposition I of Apollonius

that

KL2 BR2 DL DR

Proof Let ordinate DX be drawn and let BZ be drawn parallel toit Then

GB = BX [Apollonius I]

= ZD [Euclid I]

and so (by Euclid I amp )

EGB = EZD

Let ordinate KNM be drawn Adding to either side of the lastequation the pentagon DEBMN we have the trapezoid DGMNequal to the parallelogram ZM (that is ZBMN)

Let KL be extended to P By the lemma above the parallelogramZM is equal to the triangle KPM Thus

DGMN = KPM

Apollonius also finds the upright side corresponding to the new diameter DN it is H such that ED DZ H 2GD

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

Subtracting the trapezoid LPMN gives

KLN = LG

We have alsoBRZ = RG

(as by adding the trapezoid DEBR to the equal triangles EZD andEGB) Therefore

KL2 BR2 KLN BRZ

LG RG

LD RD

The proof given above works when K is to the left of D Theargument can be adapted to the other case Then as a corollary wehave that DN bisects all chords parallel to DG In fact Apolloniusproves this independently in Proposition I

Again I do not see how the foregoing arguments can be improvedby expressing all of the areas involved in terms of lengths Rule Fourin Descartesrsquos Rules for the Direction of the Mind [] is ldquoWe needa method if we are to investigate the truth of thingsrdquo Descarteselaborates

So useful is this method that without it the pursuit of learn-ing would I think be more harmful than profitable Hence I canreadily believe that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone This is ourexperience in the simplest of sciences arithmetic and geometrywe are well aware that the geometers of antiquity employed a sortof analysis which they went on to apply to the solution of everyproblem though they begrudged revealing it to posterity At thepresent time a sort of arithmetic called ldquoalgebrardquo is flourishingand this is achieving for numbers what the ancients did for fig-ures But if one attends closely to my meaning one will readilysee that ordinary mathematics is far from my mind here that it isquite another discipline I am expounding and that these illustra-tions are more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics I think

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

in Pappus and Diophantus who though not of that earliest an-tiquity lived many centuries before our time But I have cometo think that these writers themselves with a kind of perniciouscunning later suppressed this mathematics as notoriously manyinventors are known to have done where their own discoveries areconcerned In the present age some very gifted men have tried torevive this method for the method seems to me to be none otherthan the art which goes by the outlandish name of ldquoalgebrardquomdashorat least it would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it and insteadpossessed that abundance of clarity and simplicity which I believetrue mathematics ought to have

Descartes does not mention Apollonius among the ancient mathe-maticians and I do not believe that in his Geometry he has managedto recover the method whereby Apollonius proves all of his theorems

On the other hand Descartes may have recovered one methodused by ancient mathematicians because perhaps some of thesemathematicians did solve problems by considering equations of poly-nomial functions of lengths only An example is Menaechmus ldquoapupil of Eudoxus and a contemporary of Platordquo [ p xix]

Apollonius did not discover the conic sections Menaechmus isthought to have done this if only because his is the oldest nameassociated with the conic sections According to the commentary byEutocius on Archimedes Menaechmus had two methods for findingtwo mean proportionals to two given straight lines each of thesemethods uses conic sections One of the methods is illustrated byFigure apparently Menaechmusrsquos own diagram was just like this[ p ] Given the lengths Α and Ε we want to find Β and Γ sothat

Α Β Β Γ Γ Ε

or equivalently

Β2 = Α middot Γ Β middot Γ = Α middot Ε(lowast)Eutocius flourished around ce and his commentary was revised by

Isadore of Miletus [ p ] who along with Anthemius of Tralles wasa master-builder of Justinianrsquos Ayasofya []

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

ΘΚ

∆ Ζ Η

Figure Menaechmusrsquos finding of two mean proportionals

In the special case where Α is twice Ε we shall have that the cubewith side Γ is double the cube with side Ε In any case it is sufficientif () Β is an ordinate and Γ the corresponding abscissa of theparabola with upright side Α whose axis is ∆Η in the diagram and() Β and Γ are the coordinates of a point on the hyperbola whoseasymptotes are ∆Κ and ∆Η in the diagram and which also passesthrough the points with coordinates Α and Ε For then we shallhave (lowast) as desired Thus if Θ is the intersection of the parabolaand hyperbola we can let Β be ΖΘ and let Γ be ∆Ζ

We have used the property proved by Apollonius in his Proposi-tion II that the rectangle bounded by the straight lines drawnfrom a point on an hyperbola to the asymptotes has constant areaHeath has an idea of how Menaechmus proved this [ xxvndashxxviii]In any case by the report of Eutocius Menaechmusrsquos other methodof finding two mean proportionals was to use two parabolas withorthogonal axes

I referred to Β and Γ as coordinates but this is an anachronismAccording to one historian

Since this material has a strong resemblance to the use of coor-dinates as illustrated above it has sometimes been maintainedthat Menaechmus had analytic geometry Such a judgment is war-ranted only in part for certainly Menaechmus was unaware thatany equation in two unknown quantities determines a curve Infact the general concept of an equation in unknown quantities

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

was alien to Greek thought It was shortcomings in algebraic no-tations that more than anything else operated against the Greekachievement of a full-fledged coordinate geometry [ pp ndash]

Boyer evidently considers analytic geometry as the study of thegraphs of arbitrary equations but this would seem to be withinthe purview of calculus rather than geometry The book of Nelsonamp al discussed in Chapter does have chapters on graphs of single-valued algebraic functions single-valued transcendental functionsand multiple-valued functions as well as on parametric equationsbut this fits the explicit purpose of the text as a preparation forcalculus

Did Descartes have a ldquofull-fledged analytic geometryrdquo in the senseof Boyer In the Geometry [ pp ndash] Descartes rejects thestudy of curves like the quadratrix which today can be defined bythe equation

tan(

π

2middot y

)

=y

x

or more elaborately by the pair of equations

θ

y=

π

2 tan θ =

y

x

the variables being as in Figure Descartes does not write downan equation for the quadratrix but an equation is not needed forproving theorems about this curve Pappus [ pp ndash] definesthe quadratrix as being traced in a square by the intersection of twostraight lines one horizontal and moving from the top edge BG tothe bottom edge AD the other swinging about the lower left cornerA from the left edge AB to the bottom edge AD Assuming there is

In fact what Boyer refers to as ldquothis materialrdquo is the properties of the conicsections given by equations (lowast) (dagger) and (Dagger) in the previous chapter Boyerwill presently give the method of cube-duplication using two parabolas andthen say ldquoIt is probable that Menaechmus knew that the duplication couldbe achieved also by the use of a rectangular hyperbola and a parabolardquo It isnot clear why he says ldquoIt is probable thatrdquo unless he questions the authorityof Eutocius

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

x

y

θA

B G

DH

Figure The quadratrix

a point H where the quadratrix meets the lower edge of the squarewe have

BD AB AB BH

where BD is the circular arc centered at A Then a straight lineequal to this arc can be found and so the circle can be squared Thisis why the curve is called the quadratrix (τετραγωνίζουσα) Pappusdemonstrates this property while pointing out that we have no wayto construct the quadratrix without knowing where the point H isin the first place Today we have a notation for its position if Dis one unit away from A then the length of AH is what we call2π However this notation does not give us the location of H anybetter than Pappusrsquos description of the quadratrix does Today wecan prove that π (and hence 2π) is transcendental but this is nota topic of analytic geometry

Is ldquothe general concept of an equation in unknown quantitiesrdquosomething that is ldquoalien to Greek thoughtrdquo Perhaps it is alien to ourown thought According to Boyer ldquoany equation in two unknownquantities determines a curverdquo But this would seem to be an ex-aggeration unless an arbitrary subset S of the plane R times R is tobe considered a curve For if χS is the characteristic function of S

Pappus attributes this criticism to one Sporus about whom we apparentlyhave no source but Pappus himself [ p n ]

It is however a topic included in Spivakrsquos Calculus [ ch ]

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

then S is the solution-set of the equation

χS(x y) = 1

Probably Boyer does not have in mind equations with parameterslike S but equations whose only parameters are real numbers andin particular equations that are expressed by means of polynomialtrigonometric logarithmic and exponential functions

If Menaechmus neglects to study all such functions it is not forlack of adequate algebraic notation but lack of interest He solvesthe problem of finding two mean proportionals to two given line seg-ments If a numerical approximation is wanted this can be foundas close as desired therefore by the continuity of the real line es-tablished by Dedekind an exact solution exists But Menaechmuswants a geometric solution and he finds one evidently by usingthe kind of mathematics that we refer to today as analytic geom-etry Indeed Heath suspects that Menaechmus first came up withthe equations (lowast) and then discovered that curves defined by theseequations could be obtained as conic sections [ p xxi] Figure could appear at the beginning of any analytic geometry text as anillustration of what the subject is about

Pappus [ pp ndash] reports three kinds of geometry problemplane as being solved by means straight lines and circles only whichlie in a plane solid as requiring also the use of conic sections whichin particular are sections of a solid figure and linear as involvingmore complicated lines that is curves such as the quadratrix Per-haps justly Descartes criticizes this analysis as simplistic He showsthat curves given by polynomial equations have a heirarchy deter-mined by the degrees of the polynomials This hierarchy could havebeen meaningful for Pappus since lower-degree curves can be usedto construct higher-degree curves by methods more precise than theconstruction of the quadratrix

One solid problem described by Pappus [ pp ndash] is the four-line locus problem find the locus of points such that the rectanglewhose dimensions are the distances to two given straight lines bearsa given ratio to the rectangle whose dimensions are the distances

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

The geometry of the conic sections

to two more given straight lines According to Pappus theoremsof Apollonius were needed to solve this problem but it is not clearwhether Pappus thinks Apollonius actually did work out a full solu-tion By the last three propositions namely ndash of Book III of theConics of Apollonius it is implied that the conic sections are three-line loci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works out thedetails and derives the theorem that the conic sections are four-lineloci

Descartes works out a full solution to the four-line locus problemHe also solves a particular five-line locus problem the solution is acurve obtained as the intersection of a sliding parabola and a straightline through two points one fixed the other sliding along with theparabola

Thus Descartes would seem to have made progress along an an-cient line of research rather than just heading off in a differentdirection As Descartes observes Pappus [ pp -] could for-mulate the 2n-line locus problem for arbitrary n If n gt 3 the ratioof the product of n segments with the product of n segments canbe understood as the ratio compounded of the respective ratios ofsegment to segment That is given 2n segments A1 An B1 Bn we can understand the ratio of the product of the Ak to theproduct of the Bk as the ratio of A1 to Cn where

A1 C1 A1 B1

C1 C2 A2 B2

C2 C3 A3 B3

Cnminus1 Cn An Bn

Descartes expresses the solution of the 2n-line locus problem as annth-degree polynomial equation in x and y where y is the distancefrom the point to one of the given straight lines and x is the distancefrom a given point on that line to the foot of the perpendicular fromthe point of the locus

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

In fact Descartes does not use the perpendicular as such but astraight line drawn at an arbitrarily given angle to the given lineFor the original 2n-line problem literally involves not distances tothe given lines but lengths of straight lines drawn at given anglesto the given lines For the methods of Descartes the distinction istrivial For Apollonius the distinction would seem not to be trivial

The question remains If Descartes can express the solution ofa locus problem in terms that would make sense to Apollonius orPappus would the ancient mathematician accept Descartesrsquos proofa proof that involves algebraic manipulations of symbols

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

In in Ankara with two colleagues I taught a first-year first-semester undergraduate analytic geometry course from a locally pub-lished text that was undated but had apparently been produced in [] The preface of that text begins

This book is meant as a basic text book for a course in AnalyticGeometry

Throughout the book the connections and interrelations be-tween algebra and geometry are emphasized the notions of Lin-ear Algebra are introduced and applied simultaneously with moretraditional topics of Analytic Geometry Some of the notions ofLinear Algebra are used without mentioning them explicitly

The preface continues with brief descriptions of the eight chaptersand two appendices and it concludes with acknowledgements Thetextrsquos Chapter ldquoFundamental Principle of Analytic Geometryrdquohas five sections

Set Theory Relations Functions Families of Sets Fundamental Principle of Analytic Geometry

Thus the book appears more sophisticated than the book dis-cussed in Chapter Possibly this shows the influence of the inter-vening New Math in the US if the text draws on American sourcesbut here I am only speculating The authorrsquos acknowledgements in-clude no written sources and the book has no bibliography Theintroduction to Chapter reads

Analytic Geometry is a branch of mathematics which studies ge-ometry through the use of algebra It was Rene Descartes (ndash) who introduced the subject for the first time Analyticgeometry is based on the observation that there is a one-to-one

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

correspondence between the points of a straight line and the realnumbers (see sect) This fact is used to introduce coordinate sys-tems in the plane or in three space so that a geometric object canbe viewed as a set of pairs of real numbers or as a set of triples ofreal numbers

In this chapter we list notations review set theoretic notionsand give the fundamental principle of analytic geometry

The reference to Descartes is too vague to be meaningful Descartesdoes not observe but he tacitly assumes that there is a one-to-onecorrespondence between lengths and positive numbers He assumestoo that numbers can be multiplied by one another but in casethere is any question about this assumption he proves that thismultiplication is induced by a geometrically meaningful notion Hisproof is discussed further below in Chapter

As spelled out on pages and in the book under review theFundamental Principle of Analytic Geometry is that for everystraight line ℓ there is a function P from R to ℓ such that

a) P (0) 6= P (1)b) for every positive integer n the points P (plusmnn) are n times as far

away from P (0) as P (1) is and are on the same and oppositesides of P (0) respectively

c) similarly for the points P (plusmnk) and P (kn) when k is also apositive integer

d) if x lt y then the direction from P (x) to P (y) is the same asfrom P (0) to P (1)

It follows that any choice of distinct points P (0) and P (1) uniquelydetermines such a function P

In a more rudimentary form this Fundamental Principle is calledthe CantorndashDedekind Axiom on Wikipedia I would analyzethis Principle or Axiom into two parts

For a point O on a straight line for one of the two sides ofO the line has the structure of an ordered (abelian) group in

Article of that name accessed October It says ldquothe phrase Cantorndash

Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometryrdquo No preservationof algebraic structure is discussed

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

which the chosen point is the neutral element and the positiveelements are on the chosen side of O If A and B are arbitrarypoints on the line then A + B is that point C such that thesegments OB and AC are congruent and C is on the same sideof A that B is of O

If a particular point U of the line is chosen on the chosenside of O then there is an isomorphism P from the orderedgroup of real numbers to the ordered group of the line in whichP (1) = U

The first part here defines addition of points compatibly with theaddition of segments defined in the text of Nelson amp al discussed inChapter above The second part establishes continuity of straightlines in the sense of Dedekind discussed in Chapter above

In fact this two-part formulation of the Fundamental Principle isstrictly stronger than what the text gives By the text version themap P is not a group homomorphism only its restriction to Q is agroup homomorphism As noted in Chapter above by Dedekindrsquosconstruction R is initially obtained from Q as a linear order aloneit must be proved to have field-operations extending those of QBy the second part of the two-part formulation of the FundamentalPrinciple addition on R is geometrically meaningful This is left outof the Principle as formulated in the text under review

As Dedekind observes and as was repeated in Chapter abovecontinuity is not necessary for doing geometry Thus the so-calledFundamental Principle is not necessary It is not even sufficient fordoing geometry for it provides no clue about what happens awayfrom a given straight line The Principle holds for every Riemannianmanifold with no closed geodesics In such a manifold a chosen pointon a geodesic and a chosen direction along the geodesic determinethe structure of an ordered group on the geodesic and this orderedgroup is isomorphic with R as an ordered group The bijectionfrom R to the geodesic induces a multiplication on the geodesicbut this multiplication is not generally of significance within themanifold It is of significance in a Euclidean manifold where wehave an equation between the product of two lengths and the areaof a rectangle whose dimensions are those lengths This equation

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

is fundamental to analytic geometry but the so-called FundamentalPrinciple of Analytic Geometry does not give it to us

The first theorem in the book under review is that the usual for-mula for the distance between two points is correct The proofappeals to the Pythagorean Theorem without further explanationThis is a failure of rigor Proposition I of Euclidrsquos Elements givesus the Pythagorean Theorem as an equation of certain linear com-binations of areas To do analytic geometry we need to be able tounderstand this as an equation of certain polynomial functions oflengths If all lengths are commensurable this is easy Since notall lengths are commensurable more work is needed which will bediscussed in the next chapter

The second theorem in the book under review is that every straightline is defined by a linear equation as follows

A vertical straight line is defined by an equation x = a A horizontal straight line is defined by an equation y = b If the line is inclined then any two points of the line give the

same slope for the line and so the line is defined by an equationy = m(xminus x1) + y1

This last conclusion is justified by similarity of triangles The pos-sibility of distinguishing straight lines as vertical horizontal oroblique is asserted without explanation The meaning of straightnessis not discussed

Each of the equations that have been found for a line can be putinto the form Ax + By + C = 0 The third theorem of the textis the converse If one of A and B is not 0 then the equationAx+By + C = 0 defines

() the vertical line defined by x = minusCA if B = 0() the straight line through (0minusCB) having slope minusAB if

B 6= 0According to the text the equations x = a y = b and y = m(x minus x1) + y1

are called defining equations of the corresponding lines and an equation ofthe form Ax+ By + C = 0 is called a linear equation The second theoremis given as ldquoThe defining equation of any straight line is a linear equationrdquoTaken literally this is trivial

This condition is omitted from the text

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

A book from the s

This assumes that a point and a slope determine a lineIt seems to me that these first three theorems are founded on

notions from high-school mathematics but add no rigor to thesenotions It would be more honest to say something like

As we know from high school straight lines are just graphs ofequations of the form Ax + By + C = 0 where at least one of Aand B is not 0

To write this out as formal numbered theorems with proofs labelledas ldquoProofrdquo and ended with boxes mdashthis is a failure of rigor unlessthe axioms that the proofs rely on are made explicit We havealready seen that the Fundamental Principle of Analytic Geometryis an insufficient axiomatic foundation

I argued in Chapter above that Euclidrsquos Proposition I ldquoSide-Angle-Siderdquo is reasonably treated as a theorem rather than a pos-tulate even though it relies on no postulates But Euclid is notworking within a formal system He has no such notion Today wehave the notion and a textbook of mathematics ought to give atleast a nod to the reader who is familiar with the notion

Descartes is more rigorous than the book considered here eventhough he does not write out any theorems as such It is clear thathis logical basis is Euclidean geometry as used by the ancient Greekmathematicians

In the book under review the first three theorems get the numbers and The first and third have proofs ended with boxes The secondis preceded by two pages of discussion and diagrams followed by ldquoWe havethus provedrdquo

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Geometry to algebra

To consider the matter of rigor in more detail I propose to comparethe so-called Fundamental Principle of Analytic Geometry in theprevious chapter with the axioms of David Hilbert [] The latterare given in five groups For plane geometry the axioms can besummarized as follows

I Connectionndash Two distinct points lie on a unique straight line

A line contains at least two pointsII Order

ndash The points of a straight line are densely linearly orderedwithout extrema

(Paschrsquos Axiom) A straight line intersecting one side of atriangle intersects one of the other two sides or meetstheir common vertex

III Parallels (Euclidrsquos Axiom)Through a given point exactly one parallel to a given straightline can be drawn

IV Congruence Every segment can be uniquely laid off upon a given side

of a given point of a given straight linendash Congruence of segments is transitive and additive

Hilbert writes this axiom as two axioms although the distinction between thetwo is obscure If the two axioms are to be logically independent from oneanother then I think the first axiom should be understood as being that forany two distinct points A and B there is at least one straight line calledAB or BA that contains them Then the second axiom is that if A B andC are distinct points and AB = AC then AB = BC It would then bea theorem that two distinct points lie on a unique straight line HoweverHilbert does not mention such a theorem Perhaps he considers it to be animmediate consequence of his first axiom But in that case his second axiomwould appear to be redundant

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Geometry to algebra

Every angle can be uniquely laid off upon a given side ofa given ray

Congruence of angles is transitive ldquoSide-Angle-Siderdquo (Euclidrsquos Proposition I)

V Continuity (Archimedean Axiom)Some multiple of one segment exceeds another

Hilbert gives an additional Axiom of Completeness that nolarger system satisfies the axioms

In the two-part formulation of the Fundamental Principle of An-alytic Geometry given in Chapter previous the first part is equiv-alent to Hilbertrsquos Order and Congruence Axioms as restricted to asingle straight line

Granted that Hilbertrsquos axioms allow the construction of an or-dered field K as discussed below Paschrsquos axiom ensures that ldquospacerdquohas at most two dimensions The Completeness Axiom then ensuresthat space has exactly two dimensions Then the Completeness andContinuity Axioms together ensure that the ordered field K is RIndeed these two axioms in the presence of the others are equiv-alent to the second part of the Fundamental Principle of AnalyticGeometry

Hilbert shows that the Axiom of Parallels and the ldquoSide-Angle-Siderdquo axiom respectively are independent from all of the other ax-ioms In particular then the Fundamental Principle of AlgebraicGeometry is not sufficient for doing geometry We have already citedDedekind to the effect that continuity is not necessary for doing ge-ometry

What is needed for doing analytic geometry is what Descartesobserves Given an ordered-group isomorphism P from a field Kto a straight line with a distinguished point and direction we mustbe able to obtain P (x middot y) from P (x) and P (y) in a geometricallymeaningful way For Descartes this grounds algebra in geometryand so makes algebra rigorous

Hilbert (or his translator) says ldquohalf-rayrdquoStrictly Hilbert leaves the Completeness Axiom out of his arguments but

leaving it in would not affect his independence proofs

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

Having picked a unit length Descartes defines a multiplicationof lengths by means of the theory of proportionmdashpresumably thetheory of Book V of the Elements In particular Descartes implicitlyuses Euclidrsquos Proposition VI that a straight line parallel to thebase of a triangle divides the sides proportionally If one side isdivided into parts of lengths 1 and a then the other side is dividedinto parts of lengths b and ab for some b as in Figure and a and

1

a

b

ab

Figure Multiplication of lengths

b can be chosen in advanceAs developed in Euclid the theory of proportion uses the Archi-

medean Axiom Hilbert shows how to avoid using this Axiom butthe arguments are somewhat complicated Hartshorne [] has amore streamlined approach using properties of circles in Book IIIof the Elements

In fact Euclidrsquos Book I alone provides a sufficient basis for definingmultiplication It was suggested in Chapter above that Proposition is the climax of that book It was observed in Chapter thatProposition is a lemma for this proposition This lemma in turnrelies heavily on Proposition

In any parallelogram the complements of parallelograms about thediameter are equal to one another

In particular in rectangle ΑΒΓ∆ in Figure the diagonal ΑΓ istaken and rectangles ΕΘ and ΗΖ are drawn sharing this diagonalThen the complementary rectangles ΒΚ and Κ∆ are equal to oneanother because they are the same linear combinations of trianglesthat are respectively equal to one another

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Geometry to algebra

Ε

Β

Θ ∆Α

ΓΗ

ΖΚ

Figure Euclidrsquos Proposition I

Now we can define the product of two lengths a and b as in Fig-ure so that ab is the width of a rectangle of unit height that is

1

a

b ab

Figure Multiplication of lengths

equal to a rectangle of dimensions a and b Given the theorem (easilyproved) that all rectangles of the same dimensions are equal mul-tiplication is automatically commutative Easily too it distributesover addition and there are multiplicative inverses Associativitytakes a little more work In Figure by definition of ab cb anda(cb) we have

(lowast)

A+B = E + F +H +K

C = G

A = D + E +G+H

Also a(cb) = c(ab) if and only if

C +D + E = K

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

1

a

b ab

c

cb a(cb)

A B

C D E F

G H K

Figure Associativity of multiplication of lengths

From (lowast) we compute

D + C +B = F +K

We finish by noting that by Euclidrsquos Proposition I

B = E + F

Thus we can establish by geometric means alone that lengths arethe positive elements of an ordered field This is what makes analyticgeometry possible

Descartes apparently does not recognize any need to establishcommutativity distributivity and associativity of multiplicationStill he should be credited with the observation that a geometricdefinition of multiplication of lengths is what is needed for the ap-plication of algebra to geometry It is a shame that textbooks shouldcite Descartes as the creator of analytic geometry when his funda-mental insight is forgotten

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Algebra to geometry

One can work the other way One can start with an ordered fieldK and one can interpret the product K timesK as a Euclidean planeHilbert sketches the argument in order to prove that his axioms forsuch a plane are consistent [ sect pp -]

In fact an arbitrary ordered field is not sufficient To provide amodel for Hilbertrsquos axioms K must also be Pythagorean that is

closed under the operation x 7rarrradic

1+ x2 This allows hypotenusesof right triangles to have lengths in the field

For Euclidrsquos postulates K must be Euclidean that is closed un-der taking square roots of all positive elements Indeed Descartesdefines square roots geometrically so that

radica is in effect the ordi-

nate corresponding to the abscissa 1 in a circle of diameter 1 + aHilbert shows [ sect pp -] that the Euclidean condition isstrictly stronger than the Pythagorean condition The Pythagoreanclosure of Q contains the conjugates of all of its elements But theconjugate of a positive element need not be positive For exampleradic2minus1 is positive but its conjugate minusradic

2minus1 is not Thusradic2minus1

has no square root in the Pythagorean closure of Q

It is perhaps odd that Hilbert should feel the need to prove hisaxioms consistent One might consider them as self-evidently con-sistent Hilbert bases his consistency argument on the existence ofa Pythagorean ordered field One might argue that we believe suchfields exist only because of geometric demonstrations like Descartesrsquosas discussed in Chapter previous

On the other hand we can obtain the ordered field Q withoutgeometry and Dedekind shows how to do the same for R Thestudent may know something about R from calculus class Thenthe student also knows about the product RtimesR from calculus but

Hilbert does not use this terminology

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

mainly as the setting for graphs of functions With this backgroundhow can the student recover Euclidean geometry

By means of the structure of R as an ordered field we can giveRtimes R the structure of an inner product space

Rtimes R has the abelian group structure induced from R itself By the standard multiplication R acts on RtimesR as a field that

is R embeds in the (unital associative) ring of endomorphismsof R times R as an abelian group Thus R times R is a vector spaceover R

The function (xy) 7rarr x middot y from Rtimes R to R given by

x middot y = x0y0 + x1y1

is a real inner product function that is a positive-definitesymmetric bilinear function

Hence there is a norm function x 7rarr |x| given by

|x| =radicx middot x

We declare that the distance between two points a and b in RtimesR

is |b minus a| We must check that distance has the geometrical prop-erties that we expect The most basic of these properties are foundin the definition of a metric Easily the distance function is sym-metric Also the distance between distinct points is positive whilethe distance between identical points is zero We can establish theTriangle Inequality by means of the CauchyndashSchwartz Inequality

To this end we first define two elements a and b of R times R to beparallel writing

a b

if the equationxa+ yb = 000

has a nonzero solution Given arbitrary a and b in R times R wherea 6= 000 we have the identity

sum

ilt2

(aix+ bi)2 = |a|2x2 + 2(a middot b)x+ |b|2

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Algebra to geometry

If a b there is exactly one zero to this quadratic polynomialand so the discriminant is zero Otherwise there are no zeros so thediscriminant is negative Therefore we have the CauchyndashSchwartzInequality

|a middot b| 6 |a| middot |b|

with equality if and only if a b Hence

(lowast)

|a+ b|2 = (a+ b) middot (a+ b)

= |a|2 + 2a middot b+ |b|2

6 |a|2 + 2|a| middot |b|+ |b|2

= (|a|+ |b|)2

with equality if and only if a b and a middot b gt 0 This condition isthat a and b are in the same direction We obtain the TriangleInequality

|a+ b| 6 |a|+ |b|

with equality if and only if a and b are in the same directionWe can now define the line segment with distinct endpoints a

and b to be the set of points x such that

|bminus a| = |bminus x|+ |xminus a|

This is just the set of x such that bminus x and xminus a are in the samedirection We may say in this case that x is between a and bWe should note that this definition is symmetric in a and b Thesegment can be denoted indifferently by ab or ba The distancebetween a and b is the length of this segment Two segments areequal if they have the same length

The (straight) line containing distinct points a and b consistsof those x such that

xminus a bminus a

The points a and b determine this line Any two distinct points ofthe line determine it

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

The circle with center a passing through b consists of all x suchthat

|xminus a| = |bminus a|The radius of this circle is |bminus a|

It appears we now have Euclidrsquos first three postulates But now iswhen the accusation that Euclid uses hidden assumptions becomesmeaningful In the present context we do not automatically haveequilateral triangles by Euclidrsquos Proposition I In the proof asdescribed through Figure (in Chapter ) above one must checkthat the two circles do indeed intersect Here one would use that R

containsradic3

For Proposition I we need the notion of angle We can defineit as the union of two line segments that share a common endpointprovided that the the other endpoints are not collinear with thecommon endpoint The union of the segments ab and ac can bedenoted indifferently by bac or cab If we have the cosine functioncos from the interval (0π) to (minus11) along with its inverse arccosdefined analytically by power series we can define the measure ofthe angle bac as

arccos(cminus a) middot (bminus a)

|cminus a| middot |bminus a|

Then two angles are equal if they have the same measure If d ande are distinct from a and bminusa and dminusa are in the same directionand likewise for cminusa and eminusa then angles bac and dae are equal

Our earlier computations now give us the Law of Cosines If ab and c are three noncollinear points so that they are vertices of atriangle and if the measure of angle bac is θ then by (lowast) we have

|bminus c|2 = |bminus a|2 + |aminus c|2 minus 2|bminus a| middot |aminus c| middot cos θ

Hence the lengths of the sides ab and ac and the measure of theangle they include determine the length of the opposite side bcConversely the lengths of all three sides determine the measure ofeach angle So we have Euclidrsquos I and also I (ldquoSide-Side-Siderdquo)

We have relied on the cosine function and its inverse althoughbeing defined by power series they are not algebraic their use here

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Algebra to geometry

assumes that our ordered field is not arbitrary but is indeed RHowever we need not actually find measures of angles Two anglesare equal if and only if their cosines are equal and these cosines aredefined algebraically from the sides of the angles

Hilbert seems to allude to a different approach apparently theapproach pioneered by Felix Klein [ p ] We can just declareany two subsets of Rtimes R to be congruent if one can be carried tothe other by a translation

x 7rarr x+ a

followed by a rotation

(x y) 7rarr (x cos θ minus y sin θ x sin θ + y cos θ)

or a reflection

(x y) 7rarr (x cos θ + y sin θ x sin θ minus y cos θ)

Then two line segments will be equal if and only if they are congru-ent The same will go for angles provided we define these as unionsof rays rather than segments

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on ConicSections University Press Cambridge UK Edited by TL Heath in modern notation with introductions including anessay on the earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei qvae Graece exstant cvmcommentariis antiqvis volume I Teubner Edidit etLatine interpretatvs est I L Heiberg

[] Apollonius of Perga Conics Books IndashIII Green Lion PressSanta Fe NM revised edition Translated and with anote and an appendix by R Catesby Taliaferro With a prefaceby Dana Densmore and William H Donahue an introductionby Harvey Flaumenhaft and diagrams by Donahue Edited byDensmore

[] Tom M Apostol Mathematical analysis Addison-Wesley Pub-lishing Co Reading Mass-London-Don Mills Ont secondedition

[] Archimedes The two books On the sphere and the cylindervolume I of The works of Archimedes Cambridge UniversityPress Cambridge Translated into English together withEutociusrsquo commentaries with commentary and critical editionof the diagrams by Reviel Netz

[] Carl B Boyer A History of Mathematics John Wiley amp SonsNew York

[] Richard Dedekind Essays on the theory of numbers I Con-tinuity and irrational numbers II The nature and meaning ofnumbers authorized translation by Wooster Woodruff BemanDover Publications Inc New York

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Bibliography

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Pub-lications Inc New York Translated from the French andLatin by David Eugene Smith and Marcia L Latham with afacsimile of the first edition of

[] Reneacute Descartes The Philosophical Writings of Descartes vol-ume I Cambridge University Press translated by JohnCottingham Robert Stoothoff and Dugald Murdoch

[] Jean Dubois Henri Mitterand and Albert Dauzat Diction-naire drsquoEacutetymologie Larousse Paris First edition

[] Euclid The thirteen books of Euclidrsquos Elements translated fromthe text of Heiberg Vol I Introduction and Books I II VolII Books IIIndashIX Vol III Books XndashXIII and Appendix DoverPublications Inc New York Translated with introduc-tion and commentary by Thomas L Heath nd ed

[] Euclid The Bones Green Lion Press Santa Fe NM Ahandy where-to-find-it pocket reference companion to EuclidrsquosElements

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The ThomasL Heath translation edited by Dana Densmore

[] David Fowler The mathematics of Platorsquos academy ClarendonPress Oxford second edition A new reconstruction

[] Carl Friedrich Gauss Disquisitiones Arithmeticae Springer-Verlag New York Translated into English by Arthur AClarke revised by William C Waterhouse

[] Robin Hartshorne Geometry Euclid and beyond Undergrad-uate Texts in Mathematics Springer-Verlag New York

[] T L Heath The Works of Archimedes Edited in Modern Nota-tion With Introductory Chapters University Press CambridgeUK

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

[] Thomas Heath A history of Greek mathematics Vol II DoverPublications Inc New York From Aristarchus to Dio-phantus Corrected reprint of the original

[] Thomas L Heath editor The Method of Archimedes Re-cently Discovered by Heiberg A supplement to The Works ofArchimedes University Press Cambridge UK

[] Leon Henkin On mathematical induction Amer MathMonthly ndash

[] Herodotus The Persian Wars Books IndashII volume of LoebClassical Library Harvard University Press Cambridge Mas-sachusetts and London England Translation by A DGodley first published revised

[] David Hilbert The foundations of geometry Authorized trans-lation by E J Townsend Reprint edition The Open CourtPublishing Co La Salle Ill Project Gutenberg editionreleased December (wwwgutenbergnet)

[] T F Hoad editor The Concise Oxford Dictionary of EnglishEtymology Oxford University Press Oxford and New York Reissued in new covers

[] H I Karakaş Analytic Geometry M oplus V [Matematik Vakfı][Ankara] nd []

[] Morris Kline Mathematical thought from ancient to moderntimes Oxford University Press New York

[] Edmund Landau Foundations of Analysis The Arithmetic ofWhole Rational Irrational and Complex Numbers ChelseaPublishing Company New York NY third edition translated by F Steinhardt first edition first Germanpublication

[] Henry George Liddell and Robert Scott A Greek-English Lex-icon Clarendon Press Oxford revised and augmented

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

Bibliography

throughout by Sir Henry Stuart Jones with the assistance ofRoderick McKenzie and with the cooperation of many scholarsWith a revised supplement

[] William Morris editor The Grolier International DictionaryGrolier Inc Danbury Connecticut two volumes appearsto be the American Heritage Dictionary in a different cover

[] James Morwood editor The Pocket Oxford Latin DictionaryOxford University Press First edition published byRoutledge amp Kegan Paul

[] Murray et al editors The Compact Edition of the OxfordEnglish Dictionary Oxford University Press

[] Alfred L Nelson Karl W Folley and William M BorgmanAnalytic Geometry The Ronald Press Company New York

[] Pappus Pappus Alexandrini Collectionis Quae Supersunt vol-ume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit Frid-ericus Hultsch

[] David Pierce Induction and recursion The De Morgan Journal()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] Proclus Procli Diadochi in primum Euclidis Elementorum li-brum commentarii Bibliotheca scriptorum Graecorum et Ro-manorum Teubneriana In aedibus B G Teubneri Exrecognitione Godofredi Friedlein

[] Procopius On Buildings volume of Loeb Classical LibraryHarvard University Press Cambridge Massachusetts and Lon-don England revised edition with an English translationby H B Dewing

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA

December pm

[] Abraham Robinson Non-standard analysis Princeton Land-marks in Mathematics Princeton University Press PrincetonNJ Reprint of the second () edition With a forewordby Wilhelmus A J Luxemburg

[] Lucio Russo The forgotten revolution Springer-Verlag Berlin How science was born in BC and why it had to bereborn translated from the Italian original by Silvio Levy

[] R W Sharpe Differential geometry volume of Gradu-ate Texts in Mathematics Springer-Verlag New York Cartanrsquos generalization of Kleinrsquos Erlangen program With aforeword by S S Chern

[] Walter W Skeat A Concise Etymological Dictionary of theEnglish Language Perigee Books New York First edition original date of this edition not given

[] Michael Spivak Calculus nd ed Berkeley California Publishor Perish Inc XIII pp

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol I From Thales to Euclid volume of LoebClassical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections illustrating the history of Greekmathematics Vol II From Aristarchus to Pappus volume of Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] John von Neumann On the introduction of transfinite numbers() In Jean van Heijenoort editor From Frege to Goumldel Asource book in mathematical logic ndash pages ndashHarvard University Press Cambridge MA