allen, donald - the origin of greek mathematics

The Origins of Greek Mathematics1

Though the Greeks certainly borrowed from other civilizations, theybuilt a culture and civilization on their own which is

• The most impressive of all civilizations,• The most influential in Western culture,• The most decisive in founding mathematics as we know it.

The impact of Greece is typified by the hyperbole of Sir Henry Main:“Except the blind forces of nature, nothing moves in this world whichis not Greek in its origin.”2 Including the adoption of Egyptian andother earlier cultures by the Greeks, we find their patrimony in allphases of modern life. Handicrafts, mining techniques, engineering,trade, governmental regulation of commerce and more have all comedown to use from Rome and from Rome through Greece. Especially,our democrasies and dictatorships go back to Greek exemplars, as welldo our schools and universities, our sports, our games. And there ismore. Our literature and literary genres, our alphabet, our music, oursculpture, and most particularly our mathematics all exist as facets of

1 c°2000, G. Donald Allen2Rede Lecture for 1875, in J.A. Symonds, Studies of Greek Poets, London, 1920.

Origins of Greek Mathematics 2

the Greek heritage. The detailed study of Greek mathematics revealsmuch about modern mathematics, if not the modern directions, then thelogic and methods.

The best estimate is that the Greek civilization dates back to 2800BCE — just about the time of the construction of the great pyramidsin Egypt. The Greeks settled in Asia Minor, possibly their originalhome, in the area of modern Greece, and in southern Italy, Sicily, Crete,Rhodes, Delos, and North Africa. About 775 BCE they changed froma hieroglyphic writing to the Phoenician alphabet. This allowed themto become more literate, or at least more facile in their ability to ex-press conceptual thought. The ancient Greek civilization lasted untilabout 600 BCE Originally, the Egyptian and Babylonian influence wasgreatest in Miletus, a city of Ionia in Asia Minor and the birthplace ofGreek philosophy, mathematics and science.

From the viewpoint of its mathematics, it is best to distinguishbetween the two periods: the classical period from about 600 BCE to300 BCE and the Alexandrian or Hellenistic period from 300 BCEto 300 A.D. Indeed, from about 350 BCE the center of mathematicsmoved from Athens to Alexandria (in Egypt), the city built by PtolemyI, a Macedonian general in the army of Alexander the Great (358-323 BCE). It remained a center of mathematics for most of the nextmillennium, until the library was sacked by the Muslims in about 700A.D.

1 The Sources of Greek Mathematics

In actual fact, our direct knowledge of Greek mathematics is less reliablethan that of the older Egyptian and Babylonian mathematics, becausenone of the original manuscripts are extant.

There are two sources:

• Byzantine Greek codices (manuscript books) written 500-1500years after the Greek works were composed.

• Arabic translations of Greek works and Latin translations of theArabic versions. (Were there changes to the originals?)


Moreover, we do not know even if these works were made from theoriginals. For example, Heron made a number of changes in Euclid’sElements, adding new cases, providing different proofs and converses.Likewise for Theon of Alexandria (400 A. D.).

The Greeks wrote histories of Mathematics:

• Eudemus (4th century BCE), a member of Aristotle’s school wrotehistories3 of arithmetic, geometry and astronomy (lost),

• Theophrastus (c. 372 - c. 287 BCE) wrote a history of physics(lost).

• Pappus (late 3rdcentury CE) wrote the Mathematical Collection,an account of classical mathematics from Euclid to Ptolemy (ex-tant).

• Pappus wrote Treasury of Analysis, a collection of the Greekworks themselves (lost).

• Proclus (410-485 CE) wrote the Commentary, treating Book I ofEuclid and contains quotations due to Eudemus (extant).

• various fragments of others.

2 The Major Schools of Greek Mathematics

The Classical Greek mathematics can be neatly divided in to severalschools, which represent a philosophy and a style of mathematics. Cul-minating with The Elements of Euclid, each contributed in a real wayimportant facets to that monumental work. In some cases the influencewas much broader. We begin with the Ionian School.

2.1 The Ionian School

The Ionian School was founded by Thales (c. 643 - c. 546 BCE).Students included Anaximander4 (c. 610 - c. 547 BCE) and

3Here the most remarkable fact must be that knowledge at that time must have beensufficiently broad and extensive to warrant histories.

4Anaximander further developed the air, water, fire theory as the original and primaryform of the body, arguing that it was unnecessary to fix upon any one of them. He preferred


Anaximenes (c. 550 - c. 480 BCE),actually a student of Anaximander. Heregarded air as the origin and used theterm ‘air’ as god. Thales is the first ofthose to write on physics physiologia,which was on the principles of beingand developing in things. His workwas enthusiastically advanced by hisstudent Anaximander. Exploring theorigins of the universe, Axaximanderwrote that the first principle was avast Indefinite-Infinite (apeiron), a boundless mass possessing no spe-cific qualities. By inherent forces, it gradually developed into the uni-verse. In his system, the animate and eternal but impersonal Infinite isthe only God, and is unvarying and everlasting.

Thales is sometimes credited with having given the first deductiveproofs. He is credited with five basic theorems in plane geometry, onebeing that the every triangle inscribed in a semicircle is a right triangle.Another result, that the diameter bisects a circle appears in The Elementsas a definition. Therefore, it is doubtful that proofs provided by Thalesmatch the rigor of logic based on the principles set out by Aristotle andclimaxed in The Elements. Thales is also credited with a number ofremarkable achievements, from astronomy to mensuration to businessacumen, that will be taken up another chapter.

The importance of the Ionian School for philosophy and the philosophyof science is without dispute.

2.2 The Pythagorean School

The Pythagorean School was founded by Pythagoras in about 455 BCEMore on this later. A brief list of Pythagorean contributions includes:

1. Philosophy.

2. The study of proportion.

3. The study of plane and solid geometry.the boundless as the source and destiny of all things.


4. Number theory.

5. The theory of proof.

6. The discovery of incommensurables.5

For another example, Hippocrates of Chios (late 5th century BCE), com-puted the quadrature of certain lunes. This, by the way, is the first cor-rect proof of the area of a curvilinear figure, next to the circle, thoughthe issue is technical. He also was able to duplicate the cube by findingtwo mean proportionals. That is, take a = 1 and b = 2 in equationof two mean proportionals a : x = x : y = y : b. Solve for x to getx = 3

√2.)

2.3 The Eleatic School

The Eleatic School from the southern Italian city of Elea was founded byXenophanes of Colophon, but its chief tenets appear first in Parmenides,the second leader of the school. Melissus was the third and last leader ofthe school. Zeno of Elea (c. 495 - c. 430 BCE), son of Teleutagoras andpupil and friend of Parmenides, no doubt strongly influenced the school.Called by Aristotle the inventor of dialectic, he is universally knownfor his four paradoxes. These, while perplexing generations of thinkers,contributed substantially to the development of logical and mathematicalrigor. They were regarded as insoluble until the development of preciseconcepts of continuity and infinity.

It remains controversial that Zeno was arguing against the Pythagore-ans who believed in a plurality composed of numbers that were thoughtof as extended units. The fact is that the logical problems which hisparadoxes raise about a mathematical continuum are serious, fundamen-tal, and were inadequately solved by Aristotle.

Zeno made use of three premises:

1. Any unit has magnitude

2. That it is infinitely divisible

3. That it is indivisible.5The discovery of incommensurables brought to a fore one of the principle difficulties in

all of mathematics – the nature of infinity.


Yet he incorporated arguments for each. In his hands, he had a verypowerful complex argument in the form of a dilemma, one horn ofwhich supposed indivisibility, the other infinite divisibility, both lead-ing to a contradiction of the original hypothesis who brought to thefore the contradictions between the discrete and the continuous, thedecomposable and indecomposable.

Zeno’s ParadoxesZeno constructed his paradoxes to illustrate that current notions of mo-tion are unclear, that whether one viewed time or space as continuousor discrete, there are contradictions. Paradoxes such as these arose be-cause mankind was attempting to rationally understand the notions ofinfinity for the first time. The confusion centers around what happenswhen the logic of the finite (discrete) is used to treat the infinite (in-finitesimal) and conversely, when the infinite is perceived within thediscrete logical framework. They are

Dichotomy. To get to a fixed point one must cover the halfwaymark, and then the halfway mark of what remains, etc.

Achilles. Essentially the same for a moving point.

Arrow. An object in flight occupies a space equal to itself but thatwhich occupies a space equal to itself is not in motion.

Stade. Suppose there is a smallest instant of time. Then time mustbe further divisible!

A

B

C

A

B

C

Stade

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 41 2 3 4

Now, the idea is this: if there is a smallest instant of time and ifthe farthest that a block can move in that instant is the length of oneblock, then if we move the set B to the right that length in the smallestinstant and the set C to the left in that instant, then the net shift ofthe sets B and C is two blocks. Thus there must be a smaller instant


of time when the relative shift is just one block. Note here the use ofrelative motion.

Aristotle gave these paradoxes considerable time, in his effort toresolve them. His resolutions are unsatisfactory from a modern view-point. However, even today the paradoxes puzzle and confound anyoneuninitiated in the ways of “limits, continuity, and infinity.”

Democritus of Abdera (ca. 460-370 BCE) should also be includedwith the Eleatics. One of a half a dozen great figures of this era,he was renown for many different abilities. For example, he was aproponent of the materialistic atomic doctrine. He wrote books onnumbers, geometry, tangencies, and irrationals. (His work in geometrywas said to be significant.) To demonstrate the clarity of thinking atthe time, both he and Protagoras puzzled over whether the tangent to acircle meets it at a point or a line. By the time of Euclid, this subtlepoint was eventually settled in favor of “a point”, but it is effectivelya definition — not a proposition. In addition, he discovered6 that thevolumes of a cone and a pyramid are 1/3 the volumes of the respectivecylinder and prism.

2.4 The Sophist School

The Sophist School (e480 BCE) was centered in Athens, just after thefinal defeat of the Persians.7 There were many sophists and for manyyears, say until 380 BCE, they were the only source of higher educa-tion in the more advanced Greek cities. Of course such services wereprovided for money. Their influence waned as the philosophic schools,such as Plato’s academy, grew in prestige. Chief among the sophistswere most important were Protagoras, Gorgias, Antiphon, Prodicus, andThrasymachus. In some regards Socrates must be considered amongthem, or at least a special category of one among them. Plato em-phasized, however, that Socrates never accepted money for knowledge.Greece

6as attested by Archimedes. However, he did not rigorously prove these results. Recallthat the formula for the volume pyramid was known to the Egyptians and the Babylonians.

7This was the time of Pericles. Athens became a rich trading center with a true democratictradition. All citizens met annually to discuss the current affairs of state and to vote forleaders. Ionians and Pythagoreans were attracted to Athens. This was also the time of theconquest of Athens by Sparta.


Because Athens was a democrasy, young men needed instruction inpolitics. Sophists provided that instruction, teaching men how to speakand what arguments to use in public debate. A Sophistic educationbecame popular among older families and the upwardly mobile withoutfamilies. Among the instruction given were ways to argue against tradi-tional values, which Plato thought unfair and unjustified. However, helearned that even to defend traditional values, one must use a reasonedargument, not appeals to tradition and unreflecting faith.

In the Sophist school, emphasis was given to abstract reasoningand to the goal of using reason to understand the universe. This schoolhad amongst its chief pursuits the use of mathematics to understand thefunction of the universe. At this time many efforts were made to solvethe three great problems of antiquity: doubling the cube, squaring thecircle, and trisecting an angle — with just a straight edge and compass.One member of this school who ventured a solution to the angle trisec-tion problem,was the mathematician Hippias of Elis (460 - 400 BCE).8For example, the Sophists Antiphon of Rhamnos (c. 480 - 411 BCE)and Bryson of Heraclea (b. 450?) considered the circle squaring prob-lem by comparing the circle to polygons inscribed within it. Anothersophist, Hippias of Elis lectured widely on mathematics and as well onpoetry, grammar, history, politics, archaeology, and astronomy. He wasa prolific writer, producing elegies, tragedies, and technical treatises inprose. His work on Homer was considered excellent. Nothing of hisremains except a few fragments.

According to Plato, Hippias whom he depicted in his dialogue Pro-tagoras, was a ’vain and boastful man’ (ca. 460 BCE) who discoveredthe trisectrix. The trisectrix, also known as the quadratrix, was a me-chanically generated curve which he showed could be used to trisectany angle.

The Trisectrix. Here is how to construct the trisectrix. A rotating armbegins at the vertical position and rotates clockwise as a constant rate tothe 3 o’clock position. A horizontal bar falls from the top (12 o’clockposition) to the x-axis at a constant rate, in the same time. The locusof points where the horizontal bar intersects the rotating arm traces thetrisectrix. (In the figure below, you may assume that radian measure isused with a (quarter) circle of radius π

2. Thus the time axis ranges in

[0, π2].)

8a city in the Peloponnesus


θ

Bar

RotatingArm

θ

Trisectrix

2.5 The Platonic School

The Platonic School, the most famous of all was founded by Plato (427- 347 BCE) in 387 BCE in Athens as an institute for the pursuit ofphilosophical and scientific teaching and research. Plato, though not amathematician, encouraged research in mathematics. Pythagorean fore-runners of the school, Theodorus9 of Cyrene and Archytas10 of Tar-entum, through their teachings,produced a strongPythagorean influence in the entirePlatonic school. Little is known ofPlato’s personality and little can beinferred from his writing. SaidAristotle, certainly his most ableand famous student, Plato is a man“whom it is blasphemy in the baseeven to praise.” This meant thateven those of base standing insociety should not mention hisname, so noble was he. Much ofthe most significant mathematicalwork of the 4th century was accomplished by colleagues or pupils of

9Theodorus proved the incommensurability of√3,√5,√7, ...,

√17.

10Archytas solved the duplication of the cube problem at the intersection of a cone, a torus,and a cylinder.


Plato. Members of the school included Menaechmus11 and his brotherDinostratus12 and Theaetetus13(c. 415-369 BCE) According to Pro-clus, Menaechmus was one of those who “made the whole of geometrymore perfect”. We know little of the details. He is also attributed thequote to Alexander

“O King, for traveling over the country there are royal roadsand roads for common citizens; but in geometry these is oneroad for all.”

This famous quotation, in slightly different forms, has been attributed toothers, as well, notably Euclid. As the inventor of the conics Menaech-mus no doubt was aware of many of the now familiar properties ofconics, including asymptotes. He was also probably aware of thesolution of the duplication of the cube problem by intersecting theparabola y2 = 2ax and the hyperbola xy = a2, for which the solu-tion is x = a 3

√2. For, solving both for x yields

x = y2/(2a) and x = a2/y

y2/(2a) = a2/y

y3 = 2a3

y =3√2 · a.

Another famous pupil/friend, Eudoxus of Cnidos removed his schoolfrom Cyzicus to Athens for the purpose of cooperating with Plato. Dur-ing one of Plato’s absences Eudoxus apparently acted as the head ofthe Academy.

The academy of Plato was much like a modern university. Therewere grounds, buildings, students, and formal course taught by Platoand his aides. During the classical period, mathematics and philosophywere favored. Plato was not a mathematician — but was a strongadvocate of all of mathematics. Plato believed that the perfect idealsof physical objects are the reality. The world of ideals and relationshipsamong them is permanent, ageless, incorruptible, and universal. ThePlatonists are credited with discovery of two methods of proof, the11Menaechmus invented the conic sections. Only one branch of the hyperbola was recognized

at this time.12Dinostratus showed how to square the circle using the trisectrix13Theaetetus proved that there are only five regular solids: the tetrahedron (4 sides, tri-

angles), cube (6 sides, squares, octahedron (8 sides, triangles), dodecahedron (12 sides, pen-tagons), and icosahedron (20 sides, hexagons). Theaetetus was a student of Theodorus


method of analysis14 and the reductio ad absurdum.15

Plato affirmed the deductive organization of knowledge, and wasfirst to systematize the rules of rigorous demonstration.

The academy was closed by the Christian emperor Justinian in A.D.529 because it taught “pagan and perverse learning”.

2.6 The School of Eudoxus

The School of Eudoxus founded by Eudoxus (c. 408 BCE), the mostfamous of all the classical Greek mathematicians and second only toArchimedes.

• Eudoxus developed the theory of proportion, partly to account forand study the incommensurables (irrationals).

• He produced many theorems in plane geometry and furthered thelogical organization of proof.

• He also introduced the notion of magnitude.• He gave the first rigorous proof on the quadrature of the circle.(Proposition. The areas of two circles are as the squares of theirdiameters. 16)

14where what is to be proved is regarded as known and the consequences deduced until aknown truth or a contradiction is reached. A contradiction renders the proposition to be false.15where what is to be proved is taken and false and consequences are deduced until a

contradiction is produced, thus proving the proposition. This, the indirect method, is alsoattributed to Hippocrates.16At this time there is still no apparent concept of a formula such as A = πr2.


2.7 The School of Aristotle

The School of Aristotle, called the Lyceum, founded by Aristotle (384-322 BCE) followed the Platonic school. It had a garden, a lecture room,and an altar to the Muses.Of his character more is known thanfor others we have considered. Heseems to have been wealthy with hold-ings from Stagira. Therefore, he hadthe leisure to study. He apparentlyused sums of money to purchase books.So many books did he read that Platoreferred to him as the “reader”, indi-cating a bit of contempt or perhaps ri-valry. While still a member of Plato’sacademy,his early writings works were dialogues were concerned with thoughtsof the next world and the worthlessness of this one, themes familiar tohim from Plato’s writing (e.g. Phaedo). Anecdotes about him showhim as a kindly and affectionate. They show hardly a trace of theself-importance that some scholars claim to detect in his works. Hiswill has survived and exhibits the same kindly traits; he referencesa happy family life and takes solicitous care of his children, as wellas his servants. His apparent joy of life is reflected in the literary OnPhilosophy, which was completed in about 348. Afterwards, he devotedhis energies to research, teaching, and writing of technical treatises.

After Plato’s death in about 348, his (Plato’s) nephew Speusippuswas appointed head of the Academy. Shortly thereafter Aristotle leftAthens, possibly as some claim because of not being appointed Plato’ssuccessor. He travelled, with friend Xenocrates to Assus where heenjoyed the patronship of Hermeias of Atarneus, a Greek soldier of for-tune. There he was a prinipal at the new Assus Academy. Here he wrotemuch including On Politics and On Kinship (now lost). After just threeyears at the Assus Academy, Aristotle moved to the island of Lesbosand settled in Mytilene, the capital city. With his friend Theophrastushe established a philosophical circle similar to the Athenian Academy.In late 343 at the age of 42, Aristotle was invited by Philip II of Mace-donia to his capital at Pella to tutor his 13-year-old son, Alexander.As the leading intellectual figure of the day, Aristotle was instructed toprepare Alexander for his future role as a military leader and king.


After three years in Pella, Aristotle returned to Stagira and re-mained there until 335, when at almost 50 years of age, he returned toAthens. There he opened his own academy, the Lyceum, a gymnasiumattached to the temple of Apollo Lyceus. It was situated in a grovejust outside Athens. It was a place frequented by teachers, includingpossibly Socrates. It was only after Aristotle’s death that the school,under Theophrastus, acquired extensive property. His instruction, givenin the peripatos, or covered walkway, of the gymnasium, was the sourceof the namesake of Peripatetic.

In 323, with the death of Alexander, there was some anti-Macedoniansentiment throughout Athens. Well connected to the Macedonians throughAlexander, Aristotle fled the city to his mother’s estates in Chalcis onthe island of Euboea where within a year at the age of 62 or 63 he diedfrom a stomach illness. Referring obviously to Socrates, it was reportedthat he left Athens in order to save the Athenians from sinning twiceagainst philosophy.

Aristotle’s writings fall into two groups. The first group is com-prised of those works published by Aristotle and now lost. The secondgroup consists of those not published nor intended for publication byAristotle but collected and preserved by others. Finally, the writingsthat have survived, termed “acroamatic,” or treatises, were intended foruse in Aristotle’s school and were written in a concise and individ-ualistic style. In later antiquity Aristotle’s collected writings totalledhundreds of rolls. Today the surviving 30 works fill more than 2,000printed pages. Ancient catalogs list more than 170 separate works byAristotle.

Aristotle set the philosophy of physics, mathematics, and realityon a foundations that would carry it to modern times. He viewed thesciences as being of three types — theoretical (math physics, logic andmetaphysics), productive (the arts), and the practical (ethics, politics).


He contributed little to mathematics however,

...his views on the nature of mathematics and its rela-tions to the physical world were highly influential. WhereasPlato believed that there was an independent, eternally ex-isting world of ideas which constituted the reality of theuniverse and that mathematical concepts were part of thisworld,17 Aristotle favored concrete matter or substance18.

Aristotle regards the notion of definition19 as a significant aspectof argument. He required that a definition may not reference priorobjects. The following definition,

’A point is that which has no part’,

which is the first definition from the first book of Euclid’s Elements20,would be unacceptable. Aristotle also treats the basic principles ofmathematics, distinguishing between axioms and postulates.

• Axioms include the laws of logic, the law of contradiction, etc.• The postulates need not be self-evident, but their truth must besustained by the results derived from them.

Euclid uses this distinction. Aristotle explored the relation of the pointto the line — again the problem of the indecomposable and decompos-able.

Aristotle makes the distinction between potential infinity and ac-tual infinity21. He states only the former actually exists, in all regards.

Aristotle is credited with the invention of logic, through the syllo-gism. He cites two laws studied by every student.

1. The law of contradiction. (A statement may not be T and F)17This is still an issue of debate and contention18Morris Kline, Mathematical Thought From Ancient to Modern Times19and hence also undefined terms20This self-referential use of language in definintion and statement is the logician’s bane of

language, having caused paradoxes and other problems well into the 20th century21This is still a problem today.


2. The law of the excluded middle. (A statement must be T or F,there is no other alternative.)

His logic remained unchallenged until the 19th century. Even Aristo-tle regarded logic as an independent subject that should precede scienceand mathematics.

Aristotle’s influence has been immeasurably vast.

3 Exercises

1. Show how to trisect a line segment.

2. Show how to trisect an angle using the trisectrix. (Hint. Firsttrisect the line segement projected to the right margin by the angleto the angle that is to be trisected.)

Thales of Miletus1

1 Thales – Life and Accomplishments

Little is known of Thales. Born about 624 BC in Miletus, Asia Minor(now Turkey), he was the son of Examyes and Cleobuline. He diedabout 546 BC in Miletos, Turkey.

The bust shown above is in the Capitoline Museum in Rome, but is notcontemporary with Thales. Indeed, though there are statues and otherimages of significant people of the time, there is little assurance of theirauthenticity.

Some impression and highlights of his life and work follow:1 c°2000, G. Donald Allen

Thales 2

• Thales of Miletus was the first known Greek philosopher, scientistand mathematician. Some consider him the teacher of Pythagoras,though it may be only be that he advised Pythagoras to travel toEgypt and Chaldea.

• From Eudemus of Rhodes (fl ca. 320 B.C) we know that hestudied in Egypt and brought these teachings to Greece. He isunanimously ascribed to have introduced the mathematical andastronomical sciences into Greece.

• He is unanimously regarded as having been unusally clever—bygeneral agreement the first of the Seven Wise Men2 , a pupil ofthe Egyptians and the Chaldeans.

• None of his writing survives and no contemporary sources exist;thus, his achievements are difficult to assess, particularly his phi-losophy and mathematical discoveries. Indeed, many mathematicaldiscoveries of this early period have been attributed to others, oftencenturies later. In addition one must consider the ancient practiceof crediting particular discoveries to men with a reputation forwisdom. This is no doubt certainly true in Pythagoras’ case.

• There is, of course, the story, related by Aristotle, of his successfulspeculation in olive oil presses after he had concluded there wouldbe a bountiful harvest — as testament to his practical businessacumen.

• The Greek writer Xenophanes claimed that he predicted an eclipseof the Sun on May 28, 585 BC, startling all of Ionia and therebystopping the battle between the Lydian Alyattes and the MedianCyaxares. Modern scholars, in further analysis, believe this storyis apocryphal, that he could not have had the knowledge to predictan eclipse so accurately. Herodotus spoke of his foretelling theyear only. However, the fact that the eclipse was nearly total andoccurred during this significant battle possibly contributed to hisreputation as an astronomer. It is very likely he used Babyolonianastronomy in his prediction.

• He is also said to have used his knowledge of geometry to measurethe Egyptian pyramids, though it is equally likely he brought thisknowledge back from his studies in Egypt. The reputed method

2Besides Thales, the other of the Seven Wise Men were Bias of Priene, Chilon of Sparta,Cleobulus of Lindus, Pittacus of Mytilene, Solon of Athens and Periander of Corinth

Thales 3

was to use their shadows and therefore proportionality of the sidesof similar triangles.

• He is credited with five theorems of elementary geometry. Thiswill form the center of our study.

• Being asked what was very difficult, he answered, in a famousaphorism, “To Know Thyself.” Asked what was very easy, he an-swered, “To give advice.” To the question, what/who is God?, heanswered, “That which has no beginning or no end.” (The infi-nite!?!) Asked how men might live most virtuously, he answered,“If we never do ourselves what we blame in others.”According to Diogenes Laertius, Thales died “while present as aspectator at a gymnastic contest, being worn out with heat andthirst and weakness, for he was very old.”

From W. K. C. Guthrie3 we have

The achievement of Thales, has been represented by histo-rians in two entirely different lights: on the one hand, asa marvelous anticipation of modern scientific thinking, andon the other as nothing but a transparent rationalization of amyth.

According to Guthrie himself, one may say that “ideas of Thalesand other Milesians created a bridge between the two worlds–the worldof myth and the world of the mind.”

Thales believed that the Earth is a flat disk that floats on an endlessexpanse of water and all things come to be from water. This comes tous from Aristotle who suggested that Thales was the first to suggest asingle material substratum for the universe. But, more preciesly, Thalesand the Milesians proceeded from the assumption of a fundamentalunity of all material things that is to be found behind their apparentdiversity. This is the first recorded monism, or monistic cosmology, inhistory. He also regards the world as alive and thus life and matter to beinseparable. Even plants he feels have a immortal “soul”. (This form ofpantheism, present in the early stages of Greek philosophy, held that the

3W. K. C. Guthrie, A History of Greek Philosophy, vol. 1 (Cambridge: Cambridge Uni-versity Press, 1962), p. 70.

Thales 4

divine is one of the elements in the world whose function is to animatethe other elements that constitute the world is called hylozoism.)

Perhaps the essense of this contribution is that it marks an awaken-ing in mankind the desire to investigate the universe on a non-religiousbasis, to “model” the universe and to derive conclusions from the model.There would be many models considered, studies, and rejected beforeour contemporary one. So the new task of philosophers was to estab-lish what exactly provided this unity: one said it was water; another,the Boundless; yet another, air. (The goal is, of course, the quest forrational understanding of the world. Answering “The Big Questions”is the most difficult.))

Thales is believed to have been the teacher of Anaximander; he isthe first natural philosopher in the Ionian (Milesian) School. AccordingAnaximander of Miletus (mid-6th century), the word apeiron meant un-bounded, infinite, indefinite, or undefined. Originally used to referencethe unlimited and that which preceded the separation into contrastingqualities, such as hot and cold, wet and dry, it thus represented theprimitive unity of all phenomena. Thus, for the Greeks, the originalchaos out of which the world was formed was apeiron. Evidence existsthat he wrote treatises on geography, astronomy, and cosmology thatsurvived for several centuries. He is also said to have made a mapof the known world. A rationalist, he prized symmetry and introducedgeometry and mathematical proportions into his efforts to map the heav-ens, a tradition that has continued throughout history. We may concludethat his theories departed from earlier, mystical models of the universeand anticipated the achievements of later astronomers.

Continuing on a linear path from Thales, Anaximander’s successor,Anaximenes of Miletus (second half of the 6th century), taught that airwas the origin of all things. Anaximenes, supplementing the theoriesof Thales and Anaximander specified the way in which the other thingsarose out of the water or apeiron. He claimed that the other types ofmatter arose out of air by condensation and rarefaction. In this way, wesee another building block in the development complete model, that ofthe explanation of phenomena, and this remained essentially the samethrough all of its transmutations.

Thales 5

2 Thales’ Mathematics

Thales is also said to have discovered a method of measuring the dis-tance to a ship at sea. There are several theories that conjecture hisexplanation. The first is to use sightings on short and to construct acongruent triangle on shore and measure the unknown distance. Thisexplanation would have little utility, because the time of measurementof a distant ship would be so long as not to be of use.

Ship

Shore

Using similarity, a bit of geometry known at the time and in partic-ular to Thales if the pyramid story is anywhre near accurate, an alternateexplanation uses two shore observation points, where the angles to theship are measured. This information is given to the observatory wherethe actual ship-to-shore distance is determined by creating a triangle(∆CDE) similar to triangle ∆ABS.

Shore observation pointsA B

α β

Distance

Observatory modelC D

E

α β

ProportionalDistance

S

The actual distance is computed by proportionality. That is

Actual distanceProportional distance

=|AB||CD|

Thales 6

Five basic propositions with proofs4 of plane geometry are at-tributed to Thales.

Proposition. A circle is bisected by any diameter.5

Proposition. The base angles of an isosceles triangle are equal.

Proposition. The angles between two intersecting straight lines areequal.

Proposition. Two triangles are congruent if they have two angles andthe included side equal.

Proposition. An angle in a semicircle is a right angle.

On the next page we prove the last of these, referred to as Thales’Theorem.

4Exactly why Thales needed to prove theorems and propostions that the Egyptians andBabylonians accepted without justification is even now a subject of conjecture. Many standardphilosophical explanations have been given. Clearly, with the new attempts to formulate ascientific model for the world, philosophers might be inclined to establish that existing truthsfit within its framework. Truths from geometry may have been the first to fall under suchanalysis.

5These propositions, particularly this one comes from Proclus. But we should be carefulwith what “proof” means here. Even Euclid states this as a fact, i.e. a postulate.

Thales 7


Proof. From the picture below, note that /DEC is two right angles6minus /ACE minus /ADE and because the triangles 4ACE and4AED are isoceles this equals /AEC plus /DEA. Hence /ACE plus/ADE and therefore /DEC is one right angle. In the proof shown inthe picture below, the same proof is given using angle notation.

AC

D

E

α β

γ

γ

δ

δ

α + β = 180

2γ + α = 180

2δ + β = 180

2(δ + γ) + α + β = 360δ + γ = 90

Since there was no clear theory of angles at that time, the second ver-sion is probably not the proof furnished by Thales. As to the firstversion, the result indicates that Thales knew that the sum of the an-gles of a triangle equals two right angles. This was probably known.A simple indicator of why is shown in the illustration to the right.Consider the triangle 4ABC.Construct a line DE parallel to ABthrough C. Given knowledge of theresult about alternating interiorangles formed by a transversalcutting two parallels, we have/BCE = /ABC and/BAC = /ACD. Therefore A B

CD E

the results follows. Another indication proof shown below was fur-nished by Heath. One first constructs the inscribed triangle, then drawsthe other diagonal. It is then easy to show that the angles of the resultingquadrilateral are all equal.

6Note, the modern way to state this is that the sum of the angles of a right triangle is180o.

Thales 8

A

B

C

D

O

Alternate approach withthe "Thales rectangle",as proposed by Heath.

Preknowledge that thesum of the angles ofa triangle is two rightangles is not needed.


After Thales, the torch of mathematical investication was passedto Pythagoras.

Additional References:

1. Apostle, H. G., Aristotle’s Philosphy of Mathematics, Universityof Chicago Press, Chicago, 1952.

2. Dictionary of Scientific Biography

3. Encyclopaedia Britannica

4. Ueberweg, F., A History of Philosophy, from Thales to the PresentTime (1972) (2 Volumes).

5. Guthrie,W. K. C., A History of Greek Philosophy, vol. 1, Cam-bridge University Press,Cambridge, 1962.

6. Dicks, D. R., lassical Quarterly 9 (1959), 294-309.

7. Heath, Thomas L., A History of Greek Mathematics I, OxfordUniversity Press, Oxford,1921.

Pythagoras and the Pythagoreans1

Historically, the name Pythagoras means much more than thefamiliar namesake of the famous theorem about right triangles. Thephilosophy of Pythagoras and his school has become a part of the veryfiber of mathematics, physics, and even the western tradition of liberaleducation, no matter what the discipline.

The stamp above depicts a coin issued by Greece on August 20,1955, to commemorate the 2500th anniversary of the founding of thefirst school of philosophy by Pythagoras. Pythagorean philosophy wasthe prime source of inspiration for Plato and Aristotle whose influenceon western thought is without question and is immeasurable.

1 c°G. Donald Allen, 1999

Pythagoras and the Pythagoreans 2

1 Pythagoras and the Pythagoreans

Of his life, little is known. Pythagoras (fl 580-500, BC) was born inSamos on the western coast of what is now Turkey. He was reportedlythe son of a substantial citizen, Mnesarchos. He met Thales, likely as ayoung man, who recommended he travel to Egypt. It seems certain thathe gained much of his knowledge from the Egyptians, as had Thalesbefore him. He had a reputation of having a wide range of knowledgeover many subjects, though to one author as having little wisdom (Her-aclitus) and to another as profoundly wise (Empedocles). Like Thales,there are no extant written works by Pythagoras or the Pythagoreans.Our knowledge about the Pythagoreans comes from others, includingAristotle, Theon of Smyrna, Plato, Herodotus, Philolaus of Tarentum,and others.

SamosMiletus

Cnidus

Pythagoras lived on Samos for many years under the rule ofthe tyrant Polycrates, who had a tendency to switch alliances in timesof conflict � which were frequent. Probably because of continualconflicts and strife in Samos, he settled in Croton, on the eastern coastof Italy, a place of relative peace and safety. Even so, just as he arrived


in about 532 BCE, Croton lost a war to neighboring city Locri, butsoon thereafter defeated utterly the luxurious city of Sybaris. This iswhere Pythagoras began his society.

2 The Pythagorean School

The school of Pythagoras was every bit as much a religion as a schoolof mathematics. A rule of secrecy bound the members to the school,and oral communication was the rule. The Pythagoreans had numerousrules for everyday living. For example, here are a few of them:

� To abstain from beans.

� Not to pick up what has fallen.

� Not to touch a white cock.

� Not to stir the fire with iron.

...

� Do not look in a mirror beside a light.

Vegetarianism was strictly practiced probably because Pythago-ras preached the transmigration of souls2.

What is remarkable is that despite the lasting contributions of thePythagoreans to philosophy and mathematics, the school of Pythagorasrepresents the mystic tradition in contrast with the scientific. Indeed,Pythagoras regarded himself as a mystic and even semi-divine. SaidPythagoras,

�There are men, gods, and men like Pythagoras.�

It is likely that Pythagoras was a charismatic, as well.

Life in the Pythagorean society was more-or-less egalitarian.

� The Pythagorean school regarded men and women equally.2reincarnation


� They enjoyed a common way of life.

� Property was communal.

� Even mathematical discoveries were communal and by associationattributed to Pythagoras himself � even from the grave. Hence,exactly what Pythagoras personally discovered is difficult to as-certain. Even Aristotle and those of his time were unable to at-tribute direct contributions from Pythagoras, always referring to�the Pythagoreans�, or even the �so-called Pythagoreans�. Aristo-tle, in fact, wrote the book On the Pythagoreans which is nowlost.

The Pythagorean Philosophy

The basis of the Pythagorean philosophy is simply stated:

�There are three kinds of men and three sorts of peoplethat attend the Olympic Games. The lowest class is madeup of those who come to buy and sell, the next above themare those who compete. Best of all, however, are those whocome simply to look on. The greatest purification of all is,therefore, disinterested science, and it is the man who devoteshimself to that, the true philosopher, who has most effectuallyreleased himself from the �wheel of birth�.�3

The message of this passage is radically in conflict with modern values.We need only consider sports and politics.

? Is not reverence these days is bestowed only on the �super-stars�?

? Are not there ubiquitous demands for accountability.

The gentleman4, of this passage, has had a long run with thisphilosophy, because he was associated with the Greek genius, because

3Burnet, Early Greek Philosophy4How many such philosophers are icons of the western tradition? We can include Hume,

Locke, Descartes, Fermat, Milton, Gothe, Thoreau. Compare these names to Napoleon, Nel-son, Bismark, Edison, Whitney, James Watt. You get a different feel.


the �virtue of contemplation� acquired theological endorsement, andbecause the ideal of disinterested truth dignified the academic life.

The Pythagorean Philosophy ala Bertrand Russell

From Bertrand Russell,5, we have

�It is to this gentleman that we owe pure mathematics.The contemplative ideal � since it led to pure mathematics� was the source of a useful activity. This increased it�sprestige and gave it a success in theology, in ethics, and inphilosophy.�

Mathematics, so honored, became the model for other sciences.Thought became superior to the senses; intuition became superior toobservation. The combination of mathematics and theology began withPythagoras. It characterized the religious philosophy in Greece, in theMiddle ages, and down through Kant. In Plato, Aquinas, Descartes,Spinoza and Kant there is a blending of religion and reason, of moralaspiration with logical admiration of what is timeless.

Platonism was essentially Pythagoreanism. The whole conceptof an eternal world revealed to intellect but not to the senses can beattributed from the teachings of Pythagoras.

The Pythagorean School gained considerable influence in Crotonand became politically active � on the side of the aristocracy. Probablybecause of this, after a time the citizens turned against him and hisfollowers, burning his house. Forced out, he moved to Metapontum,also in Southern Italy. Here he died at the age of eighty. His school livedon, alternating between decline and re-emergence, for several hundredyears. Tradition holds that Pythagoras left no written works, but thathis ideas were carried on by eager disciples.

5A History of Western Philosophy. Russell was a logician, mathematician and philosopher fromthe Þrst half of the twentieth century. He is known for attempting to bring pure mathematicsinto the scope of symbolic logic and for discovering some profound paradoxes in set theory.


3 Pythagorean Mathematics

What is known of the Pythagorean school is substantially from a bookwritten by the Pythagorean, Philolaus (fl. c. 475 BCE) of Tarentum.However, according to the 3rd-century-AD Greek historian DiogenesLaertius, he was born at Croton. After the death of Pythagoras, dis-sension was prevalent in Italian cities, Philolaus may have fled first toLucania and then to Thebes, in Greece. Later, upon returning to Italy,he may have been a teacher of the Greek thinker Archytas. From hisbook Plato learned the philosophy of Pythagoras.

The dictum of the Pythagorean school was

All is number

The origin of this model may have been in the study of the constella-tions, where each constellation possessed a certain number of stars andthe geometrical figure which it forms. What this dictum meant wasthat all things of the universe had a numerical attribute that uniquelydescribed them. Even stronger, it means that all things which can beknown or even conceived have number. Stronger still, not only doall things possess numbers, but all things are numbers. As Aristotleobserves, the Pythagoreans regarded that number is both the princi-ple matter for things and for constituting their attributes and permanentstates. There are of course logical problems, here. (Using a basis to de-scribe the same basis is usually a risky venture.) That Pythagoras couldaccomplish this came in part from further discoveries such musical har-monics and knowledge about what are now called Pythagorean triples.This is somewhat different from the Ionian school, where the elementalforce of nature was some physical quantity such as water or air. Here,we see a model of the universe with number as its base, a rather abstractphilosophy.

Even qualities, states, and other aspects of nature had descriptivenumbers. For example,

� The number one : the number of reason.

� The number two: the first even or female number, the number ofopinion.

� The number three: the first true male number, the number ofharmony.


� The number four: the number of justice or retribution.

� The number five: marriage.

� The number six: creation

...

� The number ten: the tetractys, the number of the universe.

The Pythagoreans expended great effort to form the numbers froma single number, the Unit, (i.e. one). They treated the unit, which is apoint without position, as a point, and a point as a unit having position.The unit was not originally considered a number, because a measure isnot the things measured, but the measure of the One is the beginningof number.6 This view is reflected in Euclid7 where he refers to themultitude as being comprised of units, and a unit is that by virtue ofwhich each of existing things is called one. The first definition ofnumber is attributed to Thales, who defined it as a collection of units,clearly a derivate based on Egyptians arithmetic which was essentiallygrouping. Numerous attempts were made throughout Greek history todetermine the root of numbers possessing some consistent and satisfyingphilosophical basis. This argument could certainly qualify as one of theearliest forms of the philosophy of mathematics.

The greatest of the numbers, ten, was so named for several rea-sons. Certainly, it is the base of Egyptian and Greek counting. It alsocontains the ratios of musical harmonies: 2:1 for the octave, 3:2 for thefifth, and 4:3 for the fourth. We may also note the only regular figuresknown at that time were the equilateral triangle, square, and pentagon8were also contained by within tetractys. Speusippus (d. 339 BCE)notes the geometrical connection.

Dimension:

One point: generator of dimensions (point).

Two points: generator of a line of dimension one6Aristotle, Metaphysics7The Elements8Others such as the hexagon, octagon, etc. are easily constructed regular polygons with

number of sides as multiples of these. The 15-gon, which is a multiple of three and Þve sidesis also constructible. These polygons and their side multiples by powers of two were all thoseknown.


Three points: generator of a triangle of dimension two

Four points: generator of a tetrahedron, of dimension three.

The sum of these is ten and represents all dimensions. Note the ab-straction of concept. This is quite an intellectual distance from �fingersand toes�.

Classification of numbers. The distinction between even and oddnumbers certainly dates to Pythagoras. From Philolaus, we learn that

�...number is of two special kinds, odd and even, with athird, even-odd, arising from a mixture of the two; and ofeach kind there are many forms.�

And these, even and odd, correspond to the usual definitions, thoughexpressed in unusual way9. But even-odd means a product of two andodd number, though later it is an even time an odd number. Othersubdivisions of even numbers10 are reported by Nicomachus (a neo-Pythagorean∼100 A.D.).

� even-even � 2n

� even-odd � 2(2m+ 1)

� odd-even � 2n+1(2m+ 1)

Originally (our) number 2, the dyad, was not considered even,though Aristotle refers to it as the only even prime. This particulardirection of mathematics, though it is based upon the earliest ideasof factoring, was eventually abandoned as not useful, though even andodd numbers and especially prime numbers play a major role in modernnumber theory.

Prime or incomposite numbers and secondary or composite numbersare defined in Philolaus:

9Nicomachus of Gerase (ß 100 CE) gives as ancient the deÞnition that an even number isthat which can be divided in to two equal parts and into two unequal part (except two), buthowever divided the parts must be of the same type (i.e. both even or both odd).10Bear in mind that there is no zero extant at this time. Note, the �experimentation� with

deÞnition. The same goes on today. DeÞnitions and directions of approach are in a continualßux, then and now.


� A prime number is rectilinear, meaning that it can only be set outin one dimension. The number 2 was not originally regarded as aprime number, or even as a number at all.

� A composite number is that which is measured by (has a factor)some number. (Euclid)

� Two numbers are prime to one another or composite to oneanother if their greatest common divisor11 is one or greater thanone, respectively. Again, as with even and odd numbers there werenumerous alternative classifications, which also failed to surviveas viable concepts.12

For prime numbers, we have from Euclid the following theorem, whoseproof is considered by many mathematicians as the quintessentially mostelegant of all mathematical proofs.

Proposition. There are an infinite number of primes.

Proof. (Euclid) Suppose that there exist only finitely many primesp1 < p2 < ... < pr. Let N = (p1)(p2)...(pr) > 2. The integer N − 1,being a product of primes, has a prime divisor pi in common with N ;so, pi divides N − (N − 1) = 1, which is absurd!

The search for primes goes on. Eratsothenes (276 B.C. - 197 B.C.)13,who worked in Alexandria, devised a sieve for determining primes.This sieve is based on a simple concept:

Lay off all the numbers, then mark of all the multiples of 2, then3, then 5, and so on. A prime is determined when a number is notmarked out. So, 3 is uncovered after the multiples of two are markedout; 5 is uncovered after the multiples of two and three are marked out.Although it is not possible to determine large primes in this fashion,the sieve was used to determine early tables of primes. (This makes awonderful exercise in the discovery of primes for young students.)11in modern terms12We have

� prime and incomposite � ordinary primes excluding 2,

� secondary and composite � ordinary composite with prime factors only,

� relatively prime � two composite numbers but prime and incomposite to another num-ber, e.g. 9 and 25. Actually the third category is wholly subsumed by the second.

13Eratsothenes will be studied in somewhat more detail later, was gifted in almost everyintellectual endeavor. His admirers call him the second Plato and some called him beta,indicating that he was the second of the wise men of antiquity.


It is known that there is an infinite number of primes, but thereis no way to find them. For example, it was only at the end of the 19th

century that results were obtained that describe the asymptotic densityof the primes among the integers. They are relatively sparce as thefollowing formula

The number of primes ≤ n ∼ n

lnn

shows.14 Called the Prime Number Theorem, this celebrated resultswas not even conjectured in its correct form until the late 18th centuryand its proof uses mathematical machinery well beyond the scope ofthe entirety of ancient Greek mathematical knowledge. The history ofthis theorem is interesting in its own right and we will consider it in alater chapter. For now we continue with the Pythagorean story.

The pair of numbers a and b are called amicable or friendly ifthe divisors of a sum to b and if the divisors of b sum to a. The pair220 and 284, were known to the Greeks. Iamblichus (C.300 -C.350CE) attributes this discovery to Pythagoras by way of the anecdote ofPythagoras upon being asked �what is a friend� answered �Alter ego�,and on this thought applied the term directly to numbers pairs such as220 and 284. Among other things it is not known if there is infinite setof amicable pairs. Example: All primes are deficient. More interestingthat amicable numbers are perfect numbers, those numbers amicable tothemselves. Mathematically, a number n is perfect if the sum of itsdivisors is itself.

Examples: ( 6, 28, 496, 8128, ...)

6 = 1 + 2 + 3

28 = 1 + 2 + 4 + 7 + 14

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

There are no direct references to the Pythagorean study of thesenumbers, but in the comments on the Pythagorean study of amicablenumbers, they were almost certainly studied as well. In Euclid, we findthe following proposition.

Theorem. (Euclid) If 2p − 1 is prime, then (2p − 1)2p−1 is perfect.Proof. The proof is straight forward. Suppose 2p − 1 is prime. Weidentify all the factors of (2p − 1)2p−1. They are14This asymptotic result if also expressed as follows. Let P (n) = The number of primes ≤

n. Then limn→∞ P (n)/[ nlnn

] = 1.


1, 2, 4, . . . , 2p−1, and1 · (2p−1 − 1), 2 · (2p−1 − 1), 4 · (2p−1 − 1), , . . . , 2p−2 · (2p−1 − 1)

Adding we have15

p−1Xn=0

2n + (2p−1 − 1)p−2Xn=0

2n = 2p − 1 + (2p − 1)(2p−1 − 1)

= (2p − 1)2p−1

and the proof is complete.

(Try, p = 2, 3, 5, and 7 to get the numbers above.) There is justsomething about the word �perfect�. The search for perfect numberscontinues to this day. By Euclid�s theorem, this means the search is forprimes of the form (2p−1), where p is a prime. The story of and searchfor perfect numbers is far from over. First of all, it is not known if thereare an infinite number of perfect numbers. However, as we shall soonsee, this hasn�t been for a lack of trying. Completing this concept ofdescribing of numbers according to the sum of their divisors, the numbera is classified as abundant or deficient16 according as their divisorssums greater or less than a, respectively. Example: The divisors of 12are: 6,4,3,2,1 � Their sum is 16. So, 12 is abundant. Clearly all primenumbers, with only one divisor (namely, 1) are deficient.

In about 1736, one of history�s greatest mathematicians, LeonhardEuler (1707 - 1783) showed that all even perfect numbers must have theform given in Euclid�s theorem. This theorem stated below is singularlyremarkable in that the individual contributions span more than twomillenia. Even more remarkable is that Euler�s proof could have beendiscovered with known methods from the time of Euclid. The proofbelow is particularly elementary.

Theorem (Euclid - Euler) An even number is perfect if and only if ithas the form (2p − 1)2p−1 where 2p − 1 is prime.Proof. The sufficiency has been already proved. We turn to the neces-sity. The slight change that Euler brings to the description of perfectnumbers is that he includes the number itself as a divisor. Thus a per-fect is one whose divisors add to twice the number. We use this newdefinition below. Suppose that m is an even perfect number. Factor m15Recall, the geometric series

PN

n=0rn = rN+1−1

r−1 . This was also well known in antiquity

and is in Euclid, The Elements.16Other terms used were over-perfect and defective respectively for these concepts.


as 2p−1a, where a is odd and of course p > 1. First, recall that the sumof the factors of 2p−1, when 2p−1 itself is included, is (2p − 1) Then

2m = 2pa = (2p − 1)(a+ · · ·+ 1)where the term · · · refers to the sum of all the other factors of a. Since(2p−1) is odd and 2p is even, it follows that (2p−1)|a, or a = b(2p−1).First assume b > 1. Substituting above we have 2pa = 2p(2p− 1)b andthus

2p(2p − 1)b = (2p − 1)((2p − 1)b+ (2p − 1) + b+ · · ·+ 1)= (2p − 1)(2p + 2p b+ · · ·)

where the term · · · refers to the sum of all other the factors of a. Cancelthe terms (2p − 1). There results the equation

2pb = 2p + 2p b+ · · ·which is impossible. Thus b = 1. To show that (2p − 1) is prime, wewrite a similar equation as above

2p(2p − 1) = (2p − 1)((2p − 1) + · · ·+ 1)= (2p − 1)(2p + · · ·)

where the term · · · refers to the sum of all other the factors of (2p− 1).Now cancel (2p − 1). This gives

2p = (2p + · · ·)If there are any other factors of (2p − 1), this equation is impossible.Thus, (2p − 1) is prime, and the proof is complete.

4 The Primal Challenge

The search for large primes goes on. Prime numbers are so fundamentaland so interesting that mathematicians, amateur and professional, havebeen studying their properties ever since. Of course, to determine if agiven number n is prime, it is necessary only to check for divisibility bya prime up to

√n. (Why?) However, finding large primes in this way

is nonetheless impractical17 In this short section, we depart history and17The current record for largest prime has more than a million digits. The square root of

any test prime then has more than 500,000 digits. Testing a million digit number against allsuch primes less than this is certainly impossible.


take a short detour to detail some of the modern methods employed inthe search. Though this is a departure from ancient Greek mathematics,the contrast and similarity between then and now is remarkable. Justthe fact of finding perfect numbers using the previous propositions hasspawned a cottage industry of determining those numbers p for which2p−1 is prime. We call a prime number aMersenne Prime if it has theform 2p − 1 for some positive integer p. Named after the friar MarinMersenne (1588 - 1648), an active mathematician and contemporaryof Fermat, Mersenne primes are among the largest primes known today.So far 38 have been found, though it is unknown if there are othersbetween the 36th and 38th. It is not known if there are an infinity ofMersenne primes. From Euclid�s theorem above, we also know exactly38 perfect numbers. It is relatively routine to show that if 2p − 1 isprime, then so also is p.18 Thus the known primes, say to more thanten digits, can be used to search for primes of millions of digits.

Below you will find complete list of Mersenne primes as of January,1998. A special method, called the Lucas-Lehmer test has been devel-oped to check the primality the Mersenne numbers.18If p = rs, then 2p − 1 = 2rs − 1 = (2r)s − 1 = (2r − 1)((2r)s−1 + (2r)s−2 · · ·+ 1)


Number Prime Digits Mp Year Discoverer(exponent)

1 2 1 1 � Ancient2 3 1 2 � Ancient3 5 2 3 � Ancient4 7 3 4 � Ancient5 13 4 8 1456 anonymous6 17 6 10 1588 Cataldi7 19 6 12 1588 Cataldi8 31 10 19 1772 Euler9 61 19 37 1883 Pervushin

10 89 27 54 1911 Powers11 107 33 65 1914 Powers12 127 39 77 1876 Lucas13 521 157 314 1952 Robinson14 607 183 366 1952 Robinson15 1279 386 770 1952 Robinson16 2203 664 1327 1952 Robinson17 2281 687 1373 1952 Robinson18 3217 969 1937 1957 Riesel19 4253 1281 2561 1961 Hurwitz20 4423 1332 2663 1961 Hurwitz21 9689 2917 5834 1963 Gillies22 9941 2993 5985 1963 Gillies23 11213 3376 6751 1963 Gillies24 19937 6002 12003 1971 Tuckerman25 21701 6533 13066 1978 Noll - Nickel26 23209 6987 13973 1979 Noll27 44497 13395 26790 1979 Nelson - Slowinski28 86243 25962 51924 1982 Slowinski29 110503 33265 66530 1988 Colquitt - Welsh30 132049 39751 79502 1983 Slowinski31 216091 65050 130100 1985 Slowinski32 756839 227832 455663 1992 Slowinski & Gage33 859433 258716 517430 1994 Slowinski & Gage34 1257787 378632 757263 1996 Slowinski & Gage35 1398269 420921 841842 1996 Armengaud, Woltman,?? 2976221 895932 1791864 1997 Spence, Woltman,?? 3021377 909526 1819050 1998 Clarkson, Woltman, Kurowski?? 26972593 2098960 1999 Hajratwala, Kurowski?? 213466917 4053946 2001 Cameron, Kurowski


What about odd perfect numbers? As we have seen Euler char-acterized all even perfect numbers. But nothing is known about oddperfect numbers except these few facts:

� If n is an odd perfect number, then it must have the form

n = q2 · p2k+1,where p is prime, q is an odd integer and k is a nonnegative integer.

� It has at least 8 different prime factors and at least 29 prime factors.

� It has at least 300 decimal digits.

Truly a challenge, finding an odd perfect number, or proving there arenone will resolve the one of the last open problems considered by theGreeks.

5 Figurate Numbers.

Numbers geometrically constructed had a particular importance to thePythagoreans.

Triangular numbers. These numbers are 1, 3, 6, 10, ... . Thegeneral form is the familiar

1 + 2 + 3 + . . .+ n =n(n+ 1)

2.

Triangular Numbers


Square numbers These numbers are clearly the squares of the integers1, 4, 9, 16, and so on. Represented by a square of dots, they prove(?)the well known formula

1 + 3 + 5 + . . .+ (2n− 1) = n2.

1 2 3 4 5 61

3

5

7

9

11

Square Numbers

The gnomon is basically an architect�s template that marks off�similar� shapes. Originally introduced to Greece by Anaximander,it was a Babylonian astronomical instrument for the measurement oftime. It was made of an upright stick which cast shadows on a planeor hemispherical surface. It was also used as an instrument to measureright angles, like a modern carpenter�s square. Note the gnomon hasbeen placed so that at each step, the next odd number of dots is placed.The pentagonal and hexagonal numbers are shown in the below.

Pentagonal Numbers Hexagonal Numbers

Figurate Numbers of any kind can be calculated. Note that the se-


quences have sums given by

1 + 4 + 7 + . . .+ (3n− 2) = 3

2n2 − 1

2n

and1 + 5 + 9 + . . .+ (4n− 3) = 2n2 − n.

Similarly, polygonal numbers of all orders are designated; thisprocess can be extended to three dimensional space, where there resultsthe polyhedral numbers. Philolaus is reported to have said:

All things which can be known have number; for it is notpossible that without number anything can be either con-ceived or known.

6 Pythagorean Geometry

6.1 Pythagorean Triples and The Pythagorean Theorem

Whether Pythagoras learned about the 3, 4, 5 right triangle while hestudied in Egypt or not, he was certainly aware of it. This fact thoughcould not but strengthen his conviction that all is number. It wouldalso have led to his attempt to find other forms, i.e. triples. How mighthe have done this?

One place to start would be with the square numbers, and arrangethat three consecutive numbers be a Pythagorean triple! Consider forany odd number m,

m2 + (m2 − 12

)2 = (m2 + 1

2)2

which is the same as

m2 +m4

4− m

2

2+1

4=m4

4+m2

2+1

4

orm2 = m2


Now use the gnomon. Begin by placing the gnomon around n2.The next number is 2n+ 1, which we suppose to be a square.

2n+ 1 = m2,

which impliesn =

1

2(m2 − 1),

and thereforen+ 1 =

1

2(m2 + 1).

It follows that

m2 +m4

4− m

2

2+1

4=m4

4+m2

2+1

4

This idea evolved over the years and took other forms. The essential factis that the Pythagoreans were clearly aware of the Pythagorean theorem

Did Pythagoras or the Pythagoreans actually prove the Pythagorean the-orem? (See the statement below.) Later writers that attribute the proofto him add the tale that he sacrificed an ox to celebrate the discovery.Yet, it may have been Pythagoras�s religious mysticism may have pre-vented such an act. What is certain is that Pythagorean triples wereknown a millennium before Pythagoras lived, and it is likely that theEgyptian, Babylonian, Chinese, and India cultures all had some �proto-proof�, i.e. justification, for its truth. The proof question remains.

No doubt, the earliest �proofs� were arguments that would notsatisfy the level of rigor of later times. Proofs were refined and retunedrepeatedly until the current form was achieved. Mathematics is full ofarguments of various theorems that satisfied the rigor of the day andwere later replaced by more and more rigorous versions.19 However,probably the Pythagoreans attempted to give a proof which was upto the rigor of the time. Since the Pythagoreans valued the idea ofproportion, it is plausible that the Pythagoreans gave a proof based onproportion similar to Euclid�s proof of Theorem 31 in Book VI of TheElements. The late Pythagoreans (e400 BCE) however probably didsupply a rigorous proof of this most famous of theorems.19One of the most striking examples of this is the Fundamental Theorem of Algebra, which

asserts the existence of at least one root to any polynomial. Many proofs, even one by Euler,passed the test of rigor at the time, but it was Carl Friedrich Gauss (1775 - 1855) that gaveus the Þrst proof that measures up to modern standards of rigor.


There are numerous proofs, more than 300 by one count, in theliterature today, and some of them are easy to follow. We present threeof them. The first is a simple appearing proof that establishes thetheorem by visual diagram. To �rigorize� this theorem takes more thanjust the picture. It requires knowledge about the similarity of figures,and the Pythagoreans had only a limited theory of similarity.

(a+ b)2 = c2 + 4(1

2ab)

a2 + 2ab+ b2 = c2 + 2ab

a2 + b2 = c2

b

a

b a

b

a

ba

c

cc

c

This proof is based upon Books I andII of Euclid�s Elements, and is sup-posed to come from the figure to theright. Euclid allows the decomposi-tion of the square into the two boxesand two rectangles. The rectanglesare cut into the four triangles shownin the figure.

b

a

ba

b

a

ba

Then the triangle are reassembled into the first figure.

The next proof is based on similarity and proportion and is aspecial case of Theorem 31 in Book VI of The Elements. Consider thefigure below.


A

B CD

If ABC is a right triangle, with right angle at A, and AD is perpen-dicular to BC, then the triangles DBA and DAC are similar to ABC.Applying the proportionality of sides we have

|BA|2 = |BD| |BC||AC|2 = |CD| |BC|

It follows that|BA|2 + |AC|2 = |BC|2

Finally we state and prove what is now called the Pythagorean Theoremas it appears in Euclid The Elements.

Theorem I-47. In right-angled triangles, the square upon the hy-potenuse is equal to the sum of the squares upon the legs.

A C

B

D

E

L

M N

G

Pythagorean Theorem

Proof requirements: SAS congruence, Triangle area = /2 = base = height

hbb

h


This diagram is identical to the original figure used in the Euclid�sproof theorem. The figure was known to Islamic mathematicians as theFigure of the Bride.

Sketch of Proof. Note that triangles4ADC and4ADE are congruentand hence have equal area. Now slide the vertex C of 4ADC to B.Slide also the vertex B of 4ADE to L. Each of these transformationsdo not change the area. Therefore, by doubling, it follows that the areaof the rectangle ALME is equal to the area of the square upon the sideAB. Use a similar argument to show that the area of the square uponthe side BC equals the area of the rectangle LCNM .

This stamp was issued by Greece. Itdepicts the Pythagorean theorem.

6.2 The Golden Section

From Kepler we have these words

�Geometry has two great treasures: one is the Theoremof Pythagoras; the other, the division of a line into extremeand mean ratio. The first we may compare to a measure ofgold; the second we may name a precious jewel.�


A line AC divided into extreme and mean ratio is defined to meanthat it is divided into two parts, AP and PC so that AP:AC=PC:AP,where AP is the longer part.

A Q P C

Golden Section

AP : AC = PC : AP

Let AP = x and AC = a. Then the golden section isx

a=a− xx

,

and this gives the quadratic equation

x2 + ax− a2.The solution is

x =−1±√5

2a.

The golden section20 is the positive root:

x =

√5− 12

∼ .62

The point Q in the diagram above is positioned at a distance fromA so that |AQ| = |PC|. As such the segment AP is divided into meanand extreme ratio by Q. Can you prove this? Of course, this idea canbe applied recursively, to successive refinements of the segment all intosuch sections.

In the figure to the rightQ1, Q2, Q3, . . . are selected sothat |AQ1| = |QP |, |AQ2| =|Q1Q|,|AQ3| = |Q2Q1|, . . . respectively.

A Q P C

Golden Section

| AP | : | AC | = | PC | : | AP |

Q1

Q3

Q2

20...now called the Golden ratio. Curiously, this number has recurred throughout the devel-opment of mathematics. We will see it again and again.


The points Q1, Q2, , Q3, . . . divide the segments AQ, AQ1, | >AQ2, . . . into extreme and mean ratio, respectively.

The Pythagorean Pentagram

And this was all connected with the construction of a pentagon. Firstwe need to construct the golden section. The geometric construction,the only kind accepted21, is illustrated below.

Assume the square ABCE has side length a. Bisecting DC at E con-struct the diagonal AE, and extend the segment ED to EF, so thatEF=AE. Construct the square DFGH. The line AHD is divided intoextrema and mean ratio.

A B

CD EF

G H

Golden Section

Verification:

|AE|2 = |AD|2 + |DE|2 = a2 + (a/2)2 = 5

4a2.

Thus,

|DH| = (√5

2− 12)a =

√5− 12

a.

The key to the compass and ruler construction of the pentagon isthe construction of the isosceles triangle with angles 36o, 72o, and 72o.We begin this construction from the line AC in the figure below.21In actual fact, the Greek �Þxation� on geometric methods to the exclusion of algebraic

methods can be attributed to the inßuence of Eudoxus


α

A

B

CD

EP

Q

Pentagon

α

β

180 − β + 2α = 180β = 72

A P Q C

Divide a line AC into the �section� with respect to both endpoints.So PC:AC=AP:PC; also AQ:AC=QC:AQ. Draw an arc with center Aand radius AQ. Also, draw an arc with center C with radius PC.Define B to be the intersection of these arcs. This makes the trianglesAQB and CBP congruent. The triangles BPQ and AQB are similar,and therefore PQ : QB = QP : AB. Thus the angle 6 PBQ =6 QABAB = AQ.

Define α := 6 PAB and β := 6 QPB. Then 180o−β−2α = 180o.This implies α = 1

2β, and hence (2 + 1

2)β = 180. Solving for β we,

get β = 72o. Since 4 PBQ is isoceles, the angle 6 QBP = 32o. Nowcomplete the line BE=AC and the line BD=AC and connect edges AE,ED and DC. Apply similarity of triangles to show that all edges havethe same length. This completes the proof.

6.3 Regular Polygons

The only regular polygons known to the Greeks were the equilaterialtriangle and the pentagon. It was not until about 1800 that C. F. Guassadded to the list of constructable regular polyons by showing that thereare three more, of 17, 257, and 65,537 sides respectively. Precisely, heshowed that the constructable regular polygons must have

2mp1p2 . . . pr


sides where the p1, . . . , pr are distinct Fermat primes. A Fermat primeis a prime having the form

22n

+ 1.

In about 1630, the Frenchman Pierre de Fermat (1601 - 1665) con-jectured that all numbers of this kind are prime. But now we knowdifferently.


Pierre Fermat (1601-1665), was a courtattorney in Toulouse (France). He was anavid mathematician and even participated inthe fashion of the day which was to recon-struct the masterpieces of Greek mathemat-ics. He generally refused to publish, butcommunicated his results by letter.

Are there any other Fermat primes? Here is all that is known to date.It is not known if any other of the Fermat numbers are prime.

p 22p+ 1 Factors Discoverer

0 3 3 ancient1 5 5 ancient2 17 17 ancient3 257 257 ancient4 65537 65537 ancient5 4,294,957,297 641, 6,700,417 Euler, 17326 21 274177,672804213107217 39 digits composite8 78 digits composite9 617 digits composite Lenstra, et.al., 199010 709 digits unknown11 1409 digits composite Brent and Morain, 198812-20 composite

By the theorem of Gauss, there are constructions of regular poly-gons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,

2mp1p2 . . . pr

sides where the p1, . . . , pr are distinct Fermat primes.


6.4 More Pythagorean Geometry

Contributions22 by the Pythagoreans include

� Various theorems about triangles, parallel lines, polygons, circles,spheres and regular polyhedra. In fact, the sentence in Proclusabout the discovery of the irrationals also attributes to Pythago-ras the discovery of the five regular solids (called then the �cosmicfigures�). These solids, the tetrahedron (4 sides, triangles), cube (6sides, squares, octahedron (8 sides, triangles), dodecahedron (12sides, pentagons), and icosahedron (20 sides, hexagons) were pos-sibly known to Pythagoras, but it is unlikely he or the Pythagoreanscould give rigorous constructions of them. The first four were as-sociated with the four elements, earth, fire, air, and water, andbecause of this they may not have been aware of the icosahedron.Usually, the name Theaetetus is associated with them as the math-ematician who proved there are only five, and moreover, who gaverigorous constructions.

Tetrahedron Cube Octahedron

Dodecahedron Icosahedron

� Work on a class of problems in the applications of areas. (e.g. toconstruct a polygon of given area and similar to another polygon.)

� The geometric solutions of quadratics. For example, given a linesegment, construct on part of it or on the line segment extended aparallelogram equal to a given rectilinear figure in area and falling

22These facts generally assume a knowledge of the Pythagorean Theorem, as we know it.The level of rigor has not yet achieved what it would become by the time of Euclid


short or exceeding by a parallelogram similar to a given one. (In

modern terms, solveb

cx2 + ax = d.)

6.5 Other Pythagorean Geometry

We know from from Eudemus that the Pythagoreans discovered theresult that the sum of the angles of any triangle is the sum of two rightangles. However, if Thales really did prove that every triangle inscribedin a right triangle is a right triangle,he surely would have noted the resultfor right triangles. This follows directlyfrom observing that the base angles ofthe isosceles traingles formed from thecenter as in the figure just to the right.The proof for any triangle followsdirectly. However, Eudemus notes

A

B

CO

a different proof. This proof requires the �alternating interior angles�theorem. That is:Theorem. (Euclid, The Elements BookI, Proposition 29.) A straight linefalling on parallel straight lines makethe alternate angles equal to oneanother, the exterior angle equal to theinterior and opposite angle, and theinterior angles on the same side equalto two right angles.

A

B

C

D E

From this result and the figure just above, note that the angles/ABD = /CAB and /CBE = /ACB. The result follows.

The quadrature of certain lunes (crescent shaped regions) wasperformed by Hippocrates of Chios. He is also credited with thearrangement of theorems in an order so that one may be proved from aprevious one (as we see in Euclid).


BA

LuneC

D

We wish to determine the area of the lune ABCD, where the largesegment ABD is similar to the smaller segment (with base on one legof the right isosceles triangle 4ABC). Because segments are to eachother as the squares upon their bases, we have the

Proposition:The area of the large lune ABCD is the area of the triangle4ABC.

This proposition was among the first that determined the area of a curvi-linear figure in terms of a rectilinear figure. Quadratures were obtainedfor other lunes, as well. There resulted great hope and encouragementthat the circle could be squared. This was not to be.

7 The Pythagorean Theory of Proportion

Besides discovering the five regular solids, Pythagoras also discoveredthe theory of proportion. Pythagoras had probably learned in Babylonthe three basic means, the arithmetic, the geometric, and the subcon-trary (later to be called the harmonic).

Beginning with a > b > c and denoting b as the �mean of aand c, they are:

1a− bb− c =

a

aarithmetic a+ c = 2b


2a− bb− c =

a

bgeometric ac = b2

3a− bb− c =

a

charmonic

1

a+1

c=2

b

The most basic fact about these proportions or means is that if a > c,then a > b > c. In fact, Pythagoras or more probably the Pythagore-ans added seven more proportions. Here is the complete list from thecombined efforts of Pappus and Nicomachus.

Formula Equivalent

4a− bb− c =

c

a

a2 + c2

a+ c= b

5a− bb− c =

c

ba = b+ c− c

2

b

6a− bb− c =

b

ac = a+ b− a

2

b

7a− cb− c =

a

cc2 = 2ac− ab

8a− ca− b =

a

ca2 + c2 = a(b+ c)

9a− cb− c =

b

cb2 + c2 = c(a+ b)

10a− ca− b =

b

cac− c2 = ab− b2

11a− ca− b =

a

ba2 = 2ab− bc

The most basic fact about these proportions or means is that if a > c,then a > b > c. (The exception is 10, where b must be selecteddepending on the relative magnitudes of a and c, and in one of thecases b = c.) What is very well known is the following relationshipbetween the first three means. Denote by ba, bg, and bh the arithmetic,geometric, and harmonic means respectively. Then

ba > bg > bh (1)

The proofs are basic. In all of the statements below equality occurs ifand only if a = c. First we know that since (α − γ)2 ≥ 0, it followsthat α2 + γ2 ≥ 2αγ. Apply this to α =

√a and β =

√b to conclude


that a + c > 2√ac, or ba ≥ bg. Next, we note that bh = 2

ac

a+ bor

b2g = bhba. Thus ba ≥ bg ≥ bh.

What is not quite as well known is that the fourth mean, some-times called the subcontrary to the harmonic mean is larger than allthe others except the seventh and the ninth, where there is no greaterthan or less than comparison over the full range of a and c. The proofthat this mean is greater than ba is again straight forward. We easilysee that

b =a2 + c2

a+ c=

(a+ c)2 − 2aca+ c

= 2ba −b2gba≥ ba

by (1). The other proofs are omitted.

Notice that the first six of the proportions above are all of a

specific generic type, namely having the forma− bb− c = · · ·. It turns

out that each of the means (the solution for b) are comparable. Thecase with the remaining five proportions is very much different. Fewcomparisons are evident, and none of the proportions are much in usetoday. The chart of comparison of all the means below shows a plus(minus) if the mean corresponding to the left column is greater (less)than that of the top row. If there is no comparison in the greater or lessthan sense, the word �No� is inserted.


i/j 1 2 3 4 5 6 7 8 9 10 111 + + - - - No No No No +2 - + - - - No No - No No3 - - - - - - No - No No4 + + + + + No + No + +5 + + + - + No + No + +6 + + + - - No No No No +7 No No + No No No No - No No8 No No No - - No No No + +9 No + + No No No + No No No10 No No No - - No No - No No11 - No No - - - No - No No

Comparing Pythagorean Proportions

Linking qualitative or subjective terms with mathematical propor-tions, the Pythagoreans called the proportion

ba : bg = bg : bh

the perfect proportion. The proportion

a : ba = bh : c

was called the musical proportion.

8 The Discovery of Incommensurables

Irrationals have variously been attributed to Pythagoras or to the Pythagore-ans as has their study. Here, again, the record is poor, with much ofit in the account by Proclus in the 4th century CE. The discovery issometimes given to Hippasus of Metapontum (5th cent BCE). Oneaccount gives that the Pythagoreans were at sea at the time and whenHippasus produced (or made public) an element which denied virtuallyall of Pythagorean doctrine, he was thrown overboard. However, laterevidence indicates that Theaetetus23 of Athens (c. 415 - c. 369 BCE)23the teacher of Plato


discovered the irrationality of√3,√5, . . . ,

√17, and the dates suggest

that the Pythagoreans could not have been in possession of any sort of�theory� of irrationals. More likely, the Pythagoreans had noticed theirexistence. Note that the discovery itself must have sent a shock to thefoundations of their philosophy as revealed through their dictum All isNumber, and some considerable recovery time can easily be surmised.

Theorem.√2 is incommensurable with 1.

Proof. Suppose that√2 = a

b, with no common factors. Then

2 =a2

b2

ora2 = 2b2.

Thus24 2 | a2, and hence 2 | a. So, a = 2c and it follows that

2c2 = b2,

whence by the same reasoning yields that 2 | b. This is a contradiction.

Is this the actual proof known to the Pythagoreans? Note: Unlikethe Babylonians or Egyptians, the Pythagoreans recognized that thisclass of numbers was wholly different from the rationals.

�Properly speaking, we may date the very beginnings of �theo-retical� mathematics to the first proof of irrationality, for in �practical�(or applied) mathematics there can exist no irrational numbers.�25 Herea problem arose that is analogous to the one whose solution initiatedtheoretical natural science: it was necessary to ascertain something that24The expression m | n where m and n are integers means that m divides n without

remainder.25I. M. Iaglom, Matematiceskie struktury i matematiceskoie modelirovanie. [Mathematical

Structures and Mathematical Modeling] (Moscow: Nauka, 1980), p. 24.


it was absolutely impossible to observe (in this case, the incommensu-rability of a square�s diagonal with its side).

The discovery of incommensurability was attended by the intro-duction of indirect proof and, apparently in this connection, by thedevelopment of the definitional system of mathematics.26 In general,the proof of irrationality promoted a stricter approach to geometry, for itshowed that the evident and the trustworthy do not necessarily coincide.

9 Other Pythagorean Contributions.

The Pythagoreans made many contributions that cannot be described indetail here. We note a few of them without commentary.

First of all, connecting the concepts of proportionality and relativeprime numbers, the theorem of Archytas of Tarantum (c. 428 - c. 327BCE) is not entirely obvious. It states that there is no mean proportionalbetween successive integers. Stated this way, the result is less familiarthan using modern terms.

Theorem. (Archytas) For any integer n, there are no integral solutionsa to

A

a=a

Bwhere A and B are in the ratio n : n+ 1.

Proof. The proof in Euclid is a little cumbersome, but in modernnotation it translates into this: Let C and D be the smallest numbersin the same ratio as A and B. That is C and D are relatively prime.Let D = C + E Then

C

D=

C

C + E=

n

n+ 1

which implies that Cn+ C = Cn+ En. Canceling the terms Cn, wesee that E divides C. Therefore C and D are not relatively prime, acontradiction.26A. Szabo �Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden?�, Acta

Antiqua, 4 (1956), p. 130.


The Pythagoreans also demonstrated solutions to special types oflinear systems. For instance, the bloom of Thymaridas (c. 350 BCE)was a rule for solving the following system.

x+ x1 + x2 + . . .+ xn = s

x+ x1 = a1

x+ x2 = a2

. . .

x+ xn = an

This solution is easily determined as

x =(a1 + . . .+ an)− s

n− 2It was used to solve linear systems as well as to solve indeterminatelinear equations.

The Pythagoreans also brought to Greece the earth-centered cos-mology that became the accepted model until the time of Copernicusmore that two millenia later. Without doubt, this knowledge originatedin Egypt and Babylon. Later on, we will discuss this topic and itsmathematics in more detail.


References

1. Russell, Bertrand, A History of Western Philosophy, Simon andSchuster Touchstone Books, New York, 1945.

Anaxagoras of Clazomenae1(c. 500- c 428 BC)

As with many mathematicians of the 5th century BCE, littleis known of Anaxagoras’ life. He was born in Clazomenae, nowin Turkey. He was fully committed to science, and neglected hisconsiderable wealth toward its end. Most probably, he fit withinthe Ionian school2 as he like his predecessors was concerned withexplaining phenomena in terms of matter or physical forces.

In about 480 he moved lived and thrived in Athens, enjoyingthe friendship of Pericles (c. 495 - 429 BCE, Athens). Pericles, anAthenian statesman largely responsible for the development, in thelater 5th century BC, of Athenian democracy and the Athenian em-pire, making Athens the political and cultural focus of Greece. Thearrival of the Sophist philosophers in Athens happened during hismiddle life, and he took full advantage of the society of Zeno. Itis from the Sophists that Anaxagoras is said to have learned im-passivity toward trouble and insult and skepticism toward divinephenomena. When Pericles suffered decline in influence, Anaxago-ras lost influence as well, and was attacked for his “impious” views.In fact, the attack on Anaxagoras may well have been an indirectattack on Pericles. Imprisoned in about 450 BCE and then later re-leased probably due to the intervention of Pericles, he was compelledto leave Athens to retire to Lampsacus for the remainder of his life.In the end, according to other sources he was condemned to deathfor advocating the Persian cause, at the ripe age of seventy-two.

The particular charge for his imprisonment was for assertingthat the sun was not a diety but a huge red-hot stone as large asall of Peloponnesus and the moon borrowed its light from the sun.That is, moonlight is reflected light. This epoch making discoverypermitted the first accurate description and explanation of solar andlunar eclipses. His full explanation was not without error, but theidea took root and became the accepted theory very soon thereafter.

Anaxagoras’ theory of the cosmos was his most original con-tribution. He viewed that the cosmos was formed by mind in twostages: first, by a revolving and mixing process that still continues;and, second, by the development of living things. As to the first,

1 c°G. Donald Allen, 19992Thales, Anaximander, Anaximenes, Diogenes of Apollonia, et al.

Anaxagoras 2

he theorized that the formation of the world began with a vortexset up in a portion of mixed mass in which all things were together.This rotating motion began at the center and gradually spread inwider and wider circles. From this, the elements of the world be-gan to separate into ‘aether’ and air. From the air was distilled theclouds, water, earth and stones. Using ideas of centrifugal force,he postulated the process of condensation of the air into the solidearth. These ideas were later used by Kant and Laplace to describethe formation of the solar system. Admittedly, these cosmologicaltheories that are abstract, unverifiable, but vaguely plausible. Thisis how new science begins, with guesses and plausibility argumentslong before experiments.

On the constitution of matter he differed from Greek thinkersof the time who had tried to explain the physical universe by an as-sumption of a single fundamental element (cf. Anaximander). Some,including Parmenides, asserted that such an assumption could notaccount for movement and change, and, others posited more ele-ments (e.g. air earth, water, fire) as necessary for material explana-tion. Anaxagoras posited an infinite number. Unlike his predeces-sors Anaxagoras included materials found in living bodies, such asleaf, bark, flesh, and bone.

He wrote the book On Nature, the first widely circulated bookon scientific subjects. Cost: 1 drachma.

Though Anaxagoras was mostly a natural philosopher ratherthan a mathematician, while in prison, he attempted to square thecircle3 using only a straight edge and compass. More precisely,sources state that he “wrote” on this subject. The term “wrote”was at times used to indicate “studied” or “investigated”. Notethat it was probably known that regular polygons, of an arbitrarilylarge number of sides, inscribed in a circle could be squared usingonly the straight edge and compass. From this preliminary work,most likely, the general “squaring the circle” conjecture was made.

3One of the oldest problems in mathematics, the problem of squaring the circle was finallyresolved in the negative by the German mathematician C. L. F. Lindemann (1852 - 1939) inabout 1881. Precisely stated, this problem poses the question of constructing a square havingthe same area as a given circle. Though solutions were given by the ancients using a variety ofmethods, the goal was always to determine a solution using only a straight edge and compass.This problem has a remarkable history and has been one of those mathematical problemsthat attracted professional and amateur mathematicians alike. Even today, mathematicsdepartments all over the world each year receive new straight edge and compass constructions.Of course all are incorrect.

Anaxagoras 3

Can you perform this construction? There are two steps. Thefirst is to construct a rectangle having the same area as a triangle.For this use the triangle base with half the altitude. The second isto square the rectangle. To do this suppose the dimensions of therectangle are a and b, with a > b. The area is ab, and the squarewith this area has side length

√ab. Form a right angle and construct

the length of one side to be a−b2. From the endpoint construct an

arc of radius a+b2. Extend the second side of the right angle to meet

this arc. By the Pythagorean theorem, the length of the second side

isq(a+b2)2 − (a−b

2)2 =

√ab. See the diagram below.

Altitude

Base

Triangle Rectangle( )/2a-b

( )/2a+bab

Rectangle Square

xNow repeat the process for all the triangles constructed from theinscribed polygon (all with a vertex at the origin). Add them upand create the square.

The attempt conveys a remarkable amount of information. Itshows clearly that that Greeks were more than just casually in-terested in non-practical problems, and that they had a very cleardistinction between the exact and the approximate. It demonstratesa differentiation of acceptable and not acceptable methods, and thismay have preceeded or have developed along side acceptable logicfor proofs. The price paid in this particular situation is that theGreeks were prepossessed by straight edge and compass construc-tions for much of their history. Thus, precluded from developmentwas the vast range of functions known to us and the general conceptof function, as well. In all of the ancient Greek literature, func-tions were always specific examples, whether the conic sections ortrigonometric functions, or other dynamically formed curves.

Yet, Anaxagoras lived in the age where the great problems ofantiquity were formed:

1. Doubling the cube. (Delian problem)

Anaxagoras 4

2. Trisecting the angle.

3. Squaring the circle.

Anaxagoras represents bold, rational inquiry. He representedthe Greek trademark: “the desire to know”. His principle interestwas in philosophy, where his main belief was that “reason rules theworld.”

The Heroic AgeIf Greek civilization seems familiar to the modern mind it is becausethe Greeks loved reason in both practice and form. It was a neces-sary aspect of cultural life, a necessary part of political persuation,and a replacement for appeals to tradition and unchallenged religion.Science and theology, at times sister subjects, were simultaneouslyliberated together with political thought and became the principleparts of the unique adventure of the Greek mind. This period hasbeen called the heroic age, for many reasons, but partiularly for usbecause the Greeks attempted to solve these very difficult problems.That they were difficult is evidenced from that fact that it wouldbe two millennia before their resolution was complete – all in thenegative. The problems would tempt, perplex, and ultimately resistthe efforts of the very best mathematicians of every age until the 19th

century, with the circle squaring problem resolved in 1881. Attemptsto solve these problems and other ancient problems would drive thedevelopmental directions of mathematics until modern times.

Greek Numbers and Arithmetic!

! Introduction

The earliest numerical notation used by the Greeks was the Attic system.It employed the vertical stroke for a one, and symbols for“5”, “10”,“100”, “1000”, and “10,000”. Though there was some steamlining ofits use, these symbols were used in a similar way to the Egyptian system,being that symbols were used repeatedly as needed and the system wasnon positional. By the Alexandrian Age, the Greek Attic system ofenumeration was being replaced by the Ionian or alphabetic numerals.This is the system we discuss.

The (Ionian) Greek system of enumeration was a little more sophis-ticated than the Egyptian though it was non-positional. Like the Atticand Egyptian systems it was also decimal. Its distinguishing feature isthat it was alphabetical and required the use of more than 27 differentsymbols for numbers plus a couple of other symbols for meaning. Thismade the system somewhat cumbersome to use. However, calculationlends itself to a great deal of skill within almost any system, the Greeksystem being no exception.

" Greek Enumeration and Basic Number Formation

First, we note that the number symbols were the same as the letters ofthe Greek alphabet.

symbol value symbol value symbol value! 1 " 10 # 100$ 2 % 20 & 200' 3 ( 30 ) 300* 4 + 40 , 400- 5 . 50 / 500

6 0 60 1 6002 7 3 70 4 7005 8 6 80 7 8008 9 90 900

where three additional characters, the (digamma), the (koppa),1 c°2000, G. Donald Allen

Greek Numbers and Arithmetic 2

and the (sampi) are used. Hence,&62 = 287

Larger Numbers

Larger numbers were also available. The thousands, 1000 to 9000,were represented by placing adiacritical mark ! before a unit. Thus

!'&62 = 3287

In other sources we see the diacritical mark placed as a subscript beforethe unit. Thus

!'&62 = 3287

The uses of a M was used to represent numbers from 10,000 onup. Thus

9 ! = 50: 000 9"#&62 = 120: 287

Alternatively, depending on the history one reads9- = 50: 000 9!% ¢ &62 = 120: 287

Archimedes, in his book The Sand Reckoner, calculated the numberof grains of sand to fill the universe". This required him to developan extention the power of Greek enumeration to include very largenumbers.

Fractions

In the area of fractions, context was crucial for correctly reading afraction. A diacritical mark was placed after the denominator of the(unit) fraction. So,

$ ! =1

2;<= +$ ! =

1

42

but this latter example could also mean 40!" .More complex fractions could be written as well, with context again

being important. The numerator was written with an overbar. Thus,

.! 6* =51

84

Numerous, similar, representations also have been used, with in-creasing sophistication over time. Indeed, Diophantus uses a fractionalform identical to ours but with the numerator and denominator in re-versed positions.

2The reader may ask, ”What universe?” It was the universe of Aristarchus, the so called ‘ancientCopernicus’ because he proposed a sun centered universe with the earth and other planet orbiting it. Moreon this later.

Greek Numbers and Arithmetic 3

# Calculation

The arithmetic operations are complex in that so many symbols areused. However, as you can imagine, addition amounts to grouping andthen carrying. For example 5 + 7 = - + 2 = %$ = 12, not terribly unlikewhat we do. Multiplication was carried out using the distributive law.For example:

&62 ¢ $ = (& + 6 + 2)£ $= (200 + 80 + 7)£ 2= 400 + 160 + 14 = 574

= /3*

Remarkably, division was performed in essentially the same way as wedo it today.

Eudoxus of Cnidus1

Eudoxus (c. 400 BCE), son of Aeschines, is ranked among the great-est of the ancient mathematicians, surpassed perhaps only by Archimedes| but more on Archimedes later. The few facts known concerninghis life are derived substantially fron the histories written by Dio-genes LaÄertius in the 3rd century BCE.

² Eudoxus was born in Cnidus, on the Black Sea. Eudoxuslearned mathematics and medicine at a school that rivaled thatof Hippocrates of Cos. A well-to-do physician, very much im-pressed by his ability, paid his way to Athens so that he couldstudy at Plato's Academy (est. 387), He also spent 16 months inEgypt during the reign of Nectanebo I (380-363). At Heliopo-lis, now a Cairo suburb, Eudoxus learned the priestly wisdom,which included astronomy.

² He studied mathematics with Archytus (a Pythagorean) in Tar-entum.

² He studied medicine with Philistium on Sicily.

² At the age of 23 he went to Plato's academy in Athens to studyphilosophy and rhetoric.

² He established a school having many pupils at Cyzicus on thesea of Marmora (Marmara) in what is now Balikhisar, Turkey.(Balikhisar, in a position of strategic commercial importance,was likely founded as a colony of Miletus in 756 BCE.)

² In 365 B. C. he returned to Athens with his pupils. He becamea colleague of Plato.

² At the age of 53 he died in Cnidos, highly honored as a law-giver/legilator.

² He was the leading mathematician and astronomer of his day.

Eudoxus was the most reknown astronomer and mathematician ofhis day. In astronomy devised an ingenious planetary system basedon spheres.

1 c°2000, G. Donald Allen

Eudoxus of Cnidus 2

? The spherical earth is at rest at the center.

? Around this center, 27 concentric spheres rotate.

? The exterior one caries the ¯xed stars,

? The others account for the sun, moon, and ¯ve planets.

? Each planet requires four spheres, the sun and moon, three each.

Consider the moon.

² The outer sphere rotates in one day as the sphere of the starsand with axis perpendicular to the zodiac circle. One period is24 hours.

² The next middle sphere rotates on an circle at an angle to theplane of zodiac circle, and from east to west. One period is223 lunations. From this sphere the \recession of the nodes" isrealized.

² The inner sphere rotates about an axis inclined to the axis ofthe second at an angle equal to the highest latitude attainedby the moon, and from west to east. The draconitic month,the period of this sphere, is 27 days, 5 hours, 5 minutes. Themoon is ¯xed on the great circle at this angle.

Eudoxus of Cnidus 3

Homocentric spheres for the moon

EarthMoon

Zodiac Circle

? The description of the motion of the planets is more clever still.

? This model was improved by Callippus of Cyzicus (°. 370 BCE), a student of Eudoxus, who added spheres to improve the theory,especially for Mercury and Venus and by Aristotle who added to this\retrograde" spheres. Aristotle's cosmos, modelled like an onion,consisted of a series of some 55 spheres nested about a center, theEarth. However, these emendations never accounted for variation ofluminosity, which had been observed and bounded elongations (e.g.Venus is never observed to be more than about 48o and Mercurynever more than about 24o from the Sun).

? Eudoxus also described the constellations and the rising and set-ting of the ¯xed stars.

? Within 50 years the whole theory had to be abandoned.

Eudoxus's contributions to mathematics include:

Eudoxus of Cnidus 4

² A theory of proportion; this allowed the study of irrationals(incommensurables)2.

² The concept of magnitude, as not a number but stood for suchas line segments, angles, areas, etc, and which could vary con-tinuously. Magnitudes were opposed to numbers, which couldchange discontinuously. This avoided giving numerical valuesto lengths, areas, etc. Consequently great advances in geometrywere made3.

² The method of exhaustion.² Establishing rigorous methods for ¯nding areas and volumes ofcurvilinear ¯gures (e.g. cones and spheres).

² A profound in°uence in the establishment of deductive organi-zation of proof on the basis of explicit axioms.

There is little question that Eudoxus added to the body of geometricknowledge. Details are scant, but probably his main contributionscan be found in Euclid, Books V, VI, and XII.

The Theory of Proportion of Eudoxus is found as De¯nition 5 ofEuclid, Book V.

Magnitudes are said to be in the same ratio, the ¯rst tothe second and the third to the fourth, when, if any equi-multiples whatever be taken of the ¯rst and third, and anyequimultiples whatever of the second and fourth, the for-mer equimultiples alike exceed, are alike equal to, or alikefall short of, the latter equimultiples respectively taken incorresponding order.

In modern terms: a=b = c=d if and only if, for all integers m and n,whenever ma < nb then mc < nd, and so on for > and =.

2Incommensurables had only been recently discovered, reportedly by Hippasus of Metapon-tum

3at the expense of algebra. All mathematicians were driven to geometry. The consequence

persists. Even today, we still speak of x2 and x3 as x squared and x cubed.

Eudoxus of Cnidus 5

This is tantamount to an in¯nite process. But it was needed to dealwith incommensurables.

The Method of Exhaustion unquestionably helped resolve numberof loose ends then extant. It contained as Proposition 1 of Book X.

Two unequal magnitudes being set out, if from the greater there issubtracted a magnitude greater than its half, and from that which isleft a magnitude greater than its half, and if this process be repeatedcontinually, there will be left some magnitude which will be less thanthe lesser magnitude set out.

Let a > ² > 0 be given4. Let a > s1 > a=2, and a1 = a ¡ s1. Leta1 > s2 > a1=2 and a2 = a1 ¡ s2. Continue this process, generatingthe sequence a1; a2; ; : : :, we eventually have an < ².

How does this di®er from our limit concept today?

With this result, Eudoxus was able to establish following:

Proposition 2. (Book XII) Circles are to one another asthe squares on the diameters.

This was proved on the basis of the previous proposition.

Proposition 1. (Book XII) Similar polygons inscribed incircles are to one another as the squares on the diameters.

4Note the assumption that ² > 0 is super°uous in the language of the time as there wereonly positive numbers. So, the assumption is implicit

Eudoxus of Cnidus 6

To prove the Proposition 2, polygonal ¯gures, of inde¯nitely in-creasing numbers of sides, are both inscribed and circumscribed inthe circle. Assuming Proposition 2 does not hold will lead to thecontradiction that the result must be false for the polygons also.

Proof of Proposition 2. Let a and A, d and D be the repectivelydiameters of the circles. Suppose that

a

A>d2

D2:

Then there is an a0 < a so that

a0

A=d2

D2:

Set ² = a¡ a0. Let pn ( resp. Pn) be the inscribed regular polygonsof n sides in circle a (resp A). Then

a¡ p2n < 1

2(a¡ pn):

By the method of exhaustion it follows that for large enough n

a¡ pn < a¡ a0 = ²which implies that

pn > a0:

We know thatpnPn=d2

D2=a0

A:

Eudoxus of Cnidus 7

Thus, since pn > a0, it follows that Pn > A. But this is impossible,

and we have a contradiction.

To complete the proof, it must now be shown that

a

A<d2

D2

is also impossible.

This is a double reductio ad absurdum argument, a requirement ofthis method.

Eudoxus also demonstrated that the ratios of the volumes of twospheres is as the ratio of the cubes5 of their radii.

On pyramids Eudoxus proved

Proposition 5. (Book XII) Pyramids which are of the same heightand have triangular bases are to each other as their bases.

Other propositions are more famous6:

Proposition. The volume of every pyramid is one third of the prismof on the same base and with the same height.

Proposition. The volume of every cone is one third of the cylinderon the same base and with the same height.

Curiously, the proof is by the method of slabs, familiar to all collegefreshmen after completing their ¯rst calculus course.

5This is a modern word; the original word as given by Archimedes is triple.6Try to prove either of these without calculus!! Yet, the Egyptians seems to have been

aware of the formulas. Levels of understanding progress.

EUCLID, fl. 300 BCE1

The name Euclid is known to almost every high school student asthe author of The Elements, the long studied treatise on geometry andnumber theory. No other book except the Bible has been so widelytranslated and circulated. From the time it was written it was regardedas an extraordinary work and was studied by all mathematicians, eventhe greatest mathematician of antiquity — Archimedes, and so it hasbeen through the 23 centuries that have followed. It is unquestionablythe best mathematics text ever written and is likely to remain so intothe distant future.

This miniature found in a manuscript of the Roman surveyors in Wolfenbuttel,6th century CE is purportedly an image of Euclid.

1 Euclid, the mathematician

Little is known about Euclid, fl. 300BC, the author of The Elements.He taught and wrote in Egypt at the Museum and Library at Alexandria,


Euclid 2

which was founded in about 300 BCE by Ptolemy I Soter, who 2

Almost everything about him comes from Proclus’ Commentary,4th cent AD. He writes that Euclid collected Eudoxus’ theorems, per-fected many of Theaetetus’, and completed fragmentary works left byothers. His synthesis of these materials was so masterful that scarcelyany mathematician today is unfamiliar with this work.

Euclid is said to have said to the first Ptolemy who inquired ifthere was a shorter way to learn geometry than the Elements:

...there is no royal road to geometry

Another anecdote relates that a student after learning the very firstproposition in geometry, wanted to know what he would gain by know-ing such proposition, whereupon Euclid called his slave and said, ”Givehim threepence since he must needs make gain by what he learns.”

There are also remarks in the Islamic literature that attributes namesto Euclid’s father and grandfather, that gives his birthplace as Tyre, andprovides a very few other details about Euclid, including the admonitionplaced on the doors of many Greek schools forbidding anyone fromentering who has not first learned the elements of Euclid.

Of the character of Euclid there is only a remark by Pappus thatEuclid was unassuming, not boasting of his work and honest and fairto the contributions of others. These comments seem to have come asa pointed contrast to Apollonius3

? He , who we will discuss later. This, 700 years after Euclid’sdeath, can scarcely be considered authoritative. Indeed, by this timeEuclid was more legend than person.

2 Sources of The Elements

Before Euclid there was geometry. The latest compiler before Euclidwas Theudius, whose textbook was used in the Academy. It was was

2Ptolemy I was a Macedonian general in the army of Alexander the Great. He became ruler of Egyptin 323 BCE upon Alexander’s death and reigned to 285/283 BCE.

3Apollonius was known as the “great geometer” because of his work on conics. He seems to have felthimself a rival of Archimedes, twenty five years his senior. His accomplishments in proving tangencieswithout coordinates is singularly remarkable, and he is considered one of the greatest of the ancients of theHelenistic period.

Euclid 3

probably the one used by Aristotle. But soon after The Elements ap-peared, all others were forgotten. If the greatness of a masterpiece canbe measured by the number of people that study it, The Elements mustrank second of all written works, with only the The Bible precedingit. Judging by the number of references, it must have been a classicalmost from the time of publication. The most accomplished mathe-maticians of antiquity studied The Elements, and several of them wrotecommentaries on it. Among them are Heron, Proclus, Pappus, Theonof Alexandria, and Simplicius. Some authors added books (chapters)and other improved or modified the theorems or proofs. In fact, con-siderable effort has been expended to determine what the original workcontained. This is difficult in that it was written about 2300 yearsago, and no copies are extant. Only a few potsherds dating from 225BC contain notes about some propositions, Many new editions wereissued. The most significant was prepared by Theon of Alexandria, 4thcentury, CE. Theon’s scholarly recension was for centuries the basis ofall known translations. Another version was found in the Vatican byPeyrard (early 19th century) with the customary attributions to Theonabsent. From this, it was possible to determine an earlier, root versionof The Elements closer to the original. However, it was not until theDanish scholar J. L. Heiberg in 1883-1888, working with the Peyrardmanuscript and the best of the Theonine manuscripts together withcommentaries by Heron and others, that a new and definitive text wasconstructed. This version is widely regarded as closest of all to theoriginal, both in organization and constitution.

When the Greek world crumbled in the 5th century, it was the Islam-ics that inherited the remains. At first disdaining any regard for ancientwork and indeed destroying what they found, substantially on religiousbases, they later embraced the Greek learning through as many ancienttexts as could be recovered. They actively sought out the remainingGreek editions, even by making lavish purchases, and translated themto Arabic. We will discuss Islamic mathematical contributions to ourmathematical heritage in more detail later. For now it suffices to saythat it was the Arabic translations that provided the primary sourcematerials for the Latin translations that were to emanate from MoorishSpain in the 12th and 13th centuries.

Three Arabic translations were made during the Islamic period ofenlightenment. One was produced by al-Hajjaj ibn Yusuf ibn Matar,first for the Abbassid caliph Harun ar-Rashid (ruled 786-809) and again

Euclid 4

for the caliph al-Ma‘Mun (ruled 813-833); The second was made byHunayn ibn Ishaq (ruled 808-873), in Baghdad. His translation wasrevised by Thabit ibn Qurrah4 The third was made by Nasir ad-Dinat-Tusi in the 13th century.

Of the Latin translations, the first of these was produced by theEnglishman Adelard of Bath (1075 - 1164) in about 1120. Adelardobtained a copy of an Arabic version in Spain, where he travelledwhile disguised as a Muslim student. There is, however, some evidencethat The Elements was known in England even two centuries earlier.Adelard’s translation, which was an abriged version with commentary,was followed by a version offered by the Italian Gherard of Cremona(1114 - 1187) who was said to have translated the ‘15 books’ of TheElements. Certainly this was one of the numerous editions This versionwas written in Spain. Because it contains a number of Greek words suchas rhombus where Adelard’s version contains the Arabic translations,it is likely independent of Adelard’s version. Moreover, Gherard nodoubt used Greek sources as well. Gherard’s manuscript was thoughtlost but was discovered in 1904 in France. It is a clearer translation thatAdelard’s, without abbreviations and without editing, being a word forword translation containing the revised and critical edition of Thabit’sversion. A third translation from the Arabic was produced by JohannesCampanus of Novara (1205 - 1296) that came in the late 13th century.The Campanus translation is similar to the Adelard version but it isclearer and the order of theorem and proof is as now, with the prooffollowing the proposition statement.

The first direct translation from the Greek without the Arabic in-termediate versions was made by Bartolomeo Zamberti in 1505. Theeditio princeps of the Greek text was published at Basel in 1533 by Si-mon Grynaeus. The first edition of the complete works of Euclid wasthe Oxford edition of 1703, in Greek and Latin, by David Gregory. Alltexts, including the one we quote from, are now superceded by EuclidisOpera Omnia (8 volumes and a supplement, 1883-1916), which wereedited by J.L. Heiberg and H. Menge.

The earliest known copy of The Elements dates from 888AD andis currently located in Oxford.

4Abu’l-Hasan Thabit ibn Qurra (826 - 901) was court astronomer in Baghdad, though he was a nativeof Harran. Thabit generalized Pythagoras’s theorem to an arbitrary triangle. He was regarded as Arabicequivalent of Pappus, the commentator on higher mathematics. He was also founder of the school thattranslated works by Euclid, Archimedes, Ptolemy, and Eutocius. Without his efforts many more of theancient books would have been lost.

Euclid 5

Note. There is an important web site at www.perseus.tufts.eduwhich details many facets of the ancient Greek world. It also containsthe statements of the propositions in The Elements.

3 Euclid’s Other Works

Five works by Euclid have survived to our day:

1. The Elements

2. Data — a companion volume to the first six books of the Ele-ments written for beginners. It includes geometric methods for thesolution of quadratics.

3. Division of Figures — a collection of thirty-six propositions con-cerning the division of plane configurations. It survived only byArabic translations.

4. Phaenomena — on spherical geometry, it is similar to the workby Autolycus

5. Optics — an early work on perspective including optics, catoptrics,and dioptrics.

All these are in the TAMU library.

Three works by Euclid have not survived:

1. Porisms — possibly an ancient version of analytic geometry.

2. Surface Loci — ?

3. Pseudaria — ?

4 The Elements

The Elements was one of the first books printed with the Gutenbergpress, though not by Gutenberg personally. It was first published inVenice by Erhard Ratdolt. This book had 21

2inch margins in which

Euclid 6

were placed the figures. It was the first mathematical book of impor-tance printed. The reason this or other mathematical texts were pub-lished so late was the technical difficulty of printing the figures. Thereis a remarkable similarity with the contemporary difficulty of producingmathematical typography for Web-based course. That is substantiallythe reason why these materials are in Acrobat pdf format. Our sourcefor the results in The Elements are from the Sir Thomas L. Heath trans-lation into English of Heiberg’s Greek version. The general style ofThe Elements contrasts dramatically with a modern mathematics text-book. Indeed, these days only research monographs have a similar style.Namely, there is no examples, no motivations, no calculation, no wittyremarks, no introduction, no preamble. The expensive method of manu-script reproduction, hand transcriptions, probably dictated this economyof scale. However, original commentary and the like may have beenlost through the many new editions and translations.

4.1 The Elements — Structure: Thirteen Books

It comes as a surprise to many that The Elements contains so muchmathematics, including number theory and aspects of series and limits.The Elements can be topically divided into four sections.

² Books I-VI — Plane geometry

² Books VII-IX — Theory of Numbers

² Book X — Incommensurables

² Book XI-XIII — Solid Geometry

Each of the books was organized in the following order.

² Definitions

² Axioms or common notions — general statements obvious to all

² Postulates — particular to the subject at hand

² Theorems

Present here is the considerable influence of Aristotle, who outlinedthe logical requirements of an argument. The axioms were general

Euclid 7

statements, so primitive and so true that there could be no hope ofany sort of proof. A typical example: If equals be added to equals,the wholes are equal. This axiom, used repeatedly in almost everyarea of mathematics is completely fundamental. Axioms have bearingthroughout all of reason. Postulates are the primitive basis of the subjectat hand, and in The Elements form the set of constructs that are possible.In Book I there are five postulates. Here is one: To describe a circlewith any center and distance. This means the Euclid states withoutproof that a circle of any diameter and radius may be constructed. Thispostulate, just barely more that defining what a circle is, allows circlesto be constructed as needed. Of course, the theorems constitute themain content of the material at hand. This organization, which is thestandard today, is remarkable in that it was developed concurrently withthe materials themselves. It is reasonable to conclude that the theoremsof The Elements assumed through many forms and were proved manyways before Euclid locked them into his timeless masterpiece.

4.2 The Elements — Book I

² Definitions — 23

1. A point is that which has no part

2. A line is breadthless length.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straightlines on itself.

8. A plane angle is the inclination to one another of two lines in aplane which meet one another and do not lie in a straight line.

9. And when the lines containing the angle are straight, the angle iscalled rectilinear.

...

Euclid 8

15. A circle is a plane figure contained by one line such that all thestraight lines meeting it from one point among those lying withinthe figure are equal to one another.

16. And the point is called the center of the circle...

17. A diameter of the circle is any straight line drawn through thecenter and terminated in both directions by the circumference ofthe circle, and such a straight line also bisects the circle.

...

23. Parallel straight lines are straight lines which, being in the sameplane and being produced indefinitely in both directions do notmeet one another in either direction.

² Postulates — 5

1. To draw a straight line from any point to any point.2. To produce a finite straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. That, if a straight line falling on two straight lines make the

interior angles on the same side less than to right angles, thetwo straight lines, if produced indefinitely, meet on that sideon which are the angles less than the to right angles.

Euclid 9

m

2

1

n

² Axioms — 5

1. Things which are equal to the same thing are also equal toone another.

2. If equals be added to equals, the wholes are equal.3. If equals be subtracted from equals, the remainders are equal.4. Things which coincide with one another are equal to one an-

other.5. The whole is greater that the part.

Some Logic

² A syllogism: “a syllogism in discourse in which, certain thingsbeing stated, something other than what is stated follows of neces-sity from their being so.” Example: If all monkeys are primatesand all primates are mammals, then it follows that all monkeysare mammals.

² modus ponens: If p, then q. p. Therefore q.

² modus tolens: If p, then q. Not q. Therefore, not p.

The 48 propositions of Book I comprise much of the standard oneyear high school geometry course. The most famous of all them isProposition I-47, the Pythagorean Theorem, which was discussed inthe chapter on Pythagoras. Here we shall consider a few of the resultswith their proofs as samples of the work.

Euclid 10

Proposition I-1. On a given finite straight line to construct an equilateraltriangle.

A B

C

Proof. To prove this construct circles at A and B of radius AB. Arguethat the intersection point C is equidistant from A and B, and since itlies on the circles, the distance is AB.

Note that in Proposition I-1, Euclid can appeal only to the definitionsand postulates. But he doesn’t use the Aristotelian syllogisms, ratherhe uses modus ponens. Note also that there is a subtle assumption ofthe continuous nature of the plane made in the visual assumption thatthe circles intersect. Flaws of this type went essentially unresolved upuntil modern times.

4.3 The Elements — Book I

Proposition I-4. (SAS) If two triangles have two sides equal to twosides respectively, and have the angles contained by the equal sidesalso equal, then the two triangles are congruent.

Note: The modern term congruent is used here, replacing Euclid’sassertion that “each part of one triangle is equal to the correspondingpart of the other.” Euclid assumes that rigid translation or rotation leavesfigures invariant and this is the final step, though never take, of everycongruence proof. The one figure can be placed upon the other, withall sides and angles in correspondence, means they are congruent.

Euclid 11

Proposition I-5. In isosceles triangles the angles at the base are equalto one another, and, if the equal straight lines are produced further, theangles under the base will be equal to one another.

A

B C

D E

F G

Proof. Extend AB to D and AC to E. Mark off equal distancesBF and CG on their respective segments. Now argue that since AFand AG are equal and AC and AB are equal and the triangles ACFand ABG share the included angle at A, they must be congruent. Thismeans than the sides FC and GB are equal. Hence, triangles FCB andGCB are (SAS) congruent. Therefore, the angles / FBC and /GCBare equal, from which the conclusion follows.

Proposition I-6. If in a triangle two angles are equal to one another,then the opposite sides are also equal.

A

B C

D

Proof. We are given that /ABC = /ACB. Assume AB 6= AC.Assume AB > AC. Make D so that DC = AB. Now argue thattriangles ABC and DBC are congruent. Thus 4DBC, the part isequal to 4ABC, the whole. This cannot be.

Proposition I-29. A straight line intersecting two parallel straight linemakes the alternate angles equal to one another, the exterior angle equal

Euclid 12

to the interior and opposite angle, and the interior angles on the sameside equal to two right angles.

A B

C D

E

F

G

H

Proof. Assume 6 AGH > 6 GHD. Then the sum of 6 AGH and6 BGH is greater than the sum of 6 BGH and 6 GHD. But the firstsum is two right angles. (Proposition I-13.) Thus the second sum isless than two right angles and thus the line are not parallel.

Proposition I-35. Parallelograms which are on the same base and in thesame parallels are equal to one another.

A D E F

B C

G

Proof. The proof follow directly once the triangles BAE and ‘CDFare shown to be congruent. And this step is argued via SAS congruence.

With I-35 established, it is shown in Proposition I-37 that triangleswhich are on the same base and in the same parallels are equal toone another, and in Proposition I-41 that if a triangle and parallel havethe same base and are between the same parallels, then the triangle ishalf the parallelogram (in area). These together with Proposition I-46on the constructibility of a square on any segment are the main tools inthe proof of the Pythagorean theorem. The formal statement is

Proposition I-47. In right-angled triangles the square on the side sub-tending the right angle is equal to the squares on the sides containingthe right angle.

Euclid 13

See more details and diagrams in the chapter on Pythagoras and thePythagoreans.

4.4 The Elements — Book II

Book II, with 14 Theorems, is differs from Book I in that it deals withrectangles and squares. It can be termed geometric algebra. Thereis some debate among Euclid scholars as to whether it was extracteddirectly from Babylonian mathematics. In any event, it is definitelymore difficult to read than Book I material.

Definition. Any rectangle is said to be contained by the two straightlines forming the right angle. Euclid never multiplies the length and

width to obtain area. There is no such process. He does multiplynumbers (integers) times length.

Proposition II-1. If there are two straight lines, and one of them is cutinto any number of segments whatever, the rectangle contained by thetwo straight lines is equal to the sum of the rectangles contained by theuncut straight line and each of the segments.

l

w

a b c

lw = l(a+b+c) = la+lb+lc

It should be apparent that this is the distributive law for multiplicationthrough addition. Yet, it is expressed purely in terms of geometry.

Euclid 14

A

G HLK

F

B D E C

Proof. Let A and BC be the two lines. Make the random cuts at Dand E. Let BF be drawn perpendicular to BC and cut at G so thatBG is the same as A. Complete the diagram as shown. Then BH isequal to BK, DL, EH Now argue that the whole is the sum of theparts.

Proposition II-2. If a straight line be cut at random, the rectanglecontained by the whole and both of the segments is equal to the squareon the whole.

A BC

D F E

Euclid 15

Proposition II-4. If a straight line is cut at random, the square on thewhole is equal to the squares on the segments and twice the rectanglecontained by the segments.

22

a

a

b

b

( ) 2a+b = a + b + ab2

Note the simplicity of visualization and understanding for the binomialtheorem for n = 2. Many propositions give geometric solutions toquadratic equations.

Proposition II-5. If a straight line is cut into equal and unequal seg-ments, the rectangle contained by the unequal segments of the wholetogether with the square on the straight line between the points of sec-tion is equal to the square on the half.

Euclid 16

A BC Db/2

b

LK H M

G FE

x

This proposition translates into the quadratic equation

(b¡ x)x+ (b=2¡ x)2 = (b=2)2:

Proposition II-14. To construct a square equal to a given rectilinearfigure.

A B

CD

E

F

G

H

a

c

P

Proof. Assume a > c. Solve x2 = ac. Construct at the midpointof AB, and produce the line EG of length (a+ c)=2. Therefore lengthof the segment FG is (a ¡ c)=2. Extend the line CD to P and con-struct the line GH of length (a + c)=2 (H is on this line.). By thePythagorean theorem the length of the line FH has square given byµ

a+ c

2

¶2¡µa¡ c2

¶2= ac

Euclid 17

4.5 The Elements — Book III

Book III concerns circles, begins with 11 definitions about circles. Forexample, the definition of the equality of circles is given ( circles areequal if they have the same diameter). Tangency is interesting in thatit relies considerably on visual intuition:

Definition 2. A straight line is said to touch a circle which, meetingthe circle and being produced, does not cut the circle.

Definition 3. A segment of a circle is the figure contained by a straightline and a circumference of a circle.

Other concepts are segments, angles of segments, and similarity ofsegments of circles are given. Euclid begins with the basics.

Proposition III-1. To find the center of a given circle.

Proposition III-2. If on the circumference of a circle two points be takeat random, the straight line joining the points will fall within the circle.

Proposition III-5. If two circles cut (touch) one another, they will nothave the same center.

The inverse problem: III-9. If a point be taken within a circle, andmore than two equal straight lines fall from the point on the circle, thepoint taken is the center of the circle.

The Elements — Book III

III-11. If two circles touch one another internally, and their centers betaken, the straight line joining their centers, if it be also produced, willfall on the point of contact.

Euclid 18

III-16. The straight line drawn at right angles to the diameter of a circlefrom its extremity will fall outside the circle, and into the space betweenthe straight line and the circumference another straight line cannot beinterposed; ... .

Proposition III-31. (Thales Theorem) In a circle the angle in the semi-circle is a right angle, that in a greater segment less than a right angle,and that in a less segment greater than a right angle; and further theangle of the greater segment is greater than a right angle, and the angleof the less segment less than a right angle.

4.6 The Elements — Book IV

— 16 theorems Construction of regular polygons was a preoccupationof the Greeks. Clearly equilateral triangles and squares can be con-structed, that is, inscribed in a circle. Bisection allows any numberof doublings, e.g. hexagons and octagons. The inscribed pentagon isa more challenging construction. This book is devoted to the circum-scribing and inscribing regular and irregular polygons into circles. Asususal, Euclid begins with appropriate definitions. For example, a verygeneral notion of inscribed figure is given.

Definition 1. A rectilineal figure is said to be inscribed in a rectilinealfigure when the respective angles of the inscribed figure lie on therespective sides of that in which it is inscribed.

Definition 2. Similarly a figure is said to be circumscribed abouta figure when the respective sides of the circumscribed figure passthrough the respective angles of that about which it is circumscribed.

Definitions 3 and 4 give the meaning of inscribed in and circumscribedabout a circle; in the former case the angles are required to lie onthe circumference in contrast with the sides. In all there are sevendefinitions. Even the most basic result is considered by Euclid as wesee in the opening proposition.

Proposition IV-1. Into a given circle to fit a straight line equal to agiven straight line which is not greater than the diameter of the circle.

For example,

Proposition IV-5. About a given triangle to circumscribe a circle.

Euclid 19

Proposition IV-10. To construct an isosceles triangle having each ofthe angles at the base double of the remaining one.

Proposition IV-10 is the key to proving the celebrated

Proposition IV-11. In a given circle to inscribe an equilateral andequiangular pentagon.

α

A

B

CD

EP

Q

Pentagon

α

β

180 − β + 2α = 180β = 72

A P Q C

!"#"$ More regular figures.

The next regular figure to be inscribed in a circle was the 17-gon. Butthis is not in The Elements . Requiring more than 2100 years to findit, the key was understanding which polygon it should be. For this thespark of a young genius in the form of no less a mathematician thanCarl Frederich Gauss (1777 - 1855) was needed. He discovered the17-gon in 1796, at age 18.

In fact, when he was a student at Gottingen, he began work on hismajor publication Disquisitiones Arithmeticae, one of the great classicsof the mathematical literature. Toward the end of this work, he includedthis result about the 17-gon but more!!! He proved that the only regularpolygons that can be inscribed in a circle have

N = 2mp1p2 : : : pr

sides, where m is a integer and the p0s are Fermat primes.

Euclid 20

? Fermat numbers are of the form

22n

+ 1:

where n is an integer. For the first few integers they are prime and arecalled Fermat primes.

We have the following table of polygons that can be inscribed in acircle:

n 22n+ 1 discoverer

0 3 ancients1 5 ancients2 17 Gauss3 65 Gauss4 65,537 Gauss

For many years, it was an open question as to whether all such num-bers, 22n + 1, primes? In about 1730 another young genius, LeonhardEuler (1707 - 1783) factored the next one as 225+1 = 4; 294; 967; 296 =641 ¢ 6; 700; 417 The Fermat numbers were not all primes. Indeed, noothers are known as primes. A contemporary of Gauss, FernidandEisenstein (1823-1852) conjectured the following subset of the Fermatnumbers consists only of primes:

22 + 1; 222

+ 1; 2222

+ 1; 22222

+ 1; : : :

This conjecture has not been verified. The first three are the Fermatprimes, 5, 17, 65,537. The next number has 19,729 digits. Even thoughprime numbers are now known with millions of digits, this number,with not even 20,000 digits, is almost intractable. It is not a Mersennenumber (i.e. a number of the form 2p ¡ 1, where p is a prime), so theLucas-Lehmer test does not apply5. This limits the tests that can beapplied. The most primitive test, that of attempting to divide all primeswith 10,000 or fewer digits would require vastly more than the storagecapacity of all the computers on earth to hold them and far more thanthe computational power of them all to perform the calculations.6 So,another special test must be determined if the primality of such numbersis to be tested.

5See the chapter on Pythagoras and the Pythagoreans6Just to convince you of this, simply suppose that everyone on earth has a computer with 1000 gigabites

of storage and that the governments has 100 times that collective amount. This gives less than 1014+10 =1024 bytes of storage. Compare this with just the number of primes, not even the storage requirements, inthe required range range, which exceeds 109995 bytes. Furthermore, if each of these computers operated at1 teraflop (1 trillion floating point operations per second), only about 1037 computations could be carriedout in the next century.

Euclid 21

4.7 The Elements — Book V — 25 theorems

Book V treats ratio and proportion. Euclid begins with 18 definitionsabout magnitudes beginning with a part, multiple, ratio, be in the sameratio, and many others. Definition 1. A magnitude is a part of amagnitude, the less of the greater, when it measures the greater.

This means that it divides the greater with no remainder.

Definition 4. Magnitudes are said to have a ratio to one another whichare capable, when multiplied, exceeding on another.

This is essentially the Archimedian Axiom: If a < b, then there isan integer n such that na > b.

In the modern theory of partially ordered spaces, a special role isplayed by those spaces which have the so-called Archimedian Property.Consider Definition 5 on same ratios devised by Eudoxus to reckon withincommensurables.

Definition 5. Magnitudes are said to be in the same ratio, the first to thesecond and the third to the fourth, when, if any equimultiples whateverbe taken of the first and third, and any equimultiples whatever of thesecond and fourth, the former equimultiples alike exceed, are alike equalto, or alike fall short of, the latter equimultiples respectively taken incorresponding order.

In modern notation, we say the magnitudes, a; b; c; d are in thesame ratio a : b = c : d if for all positive integers m and n

ma > mc then nb > nd;

and similarly for < and =. Subtly, this definition requires an infinityof tests to verify two sets of numbers are in the same ratio.

Proposition V-1. If there be any number of magnitudes whatever whichare, respectively, equimultiples of any magnitudes equal in multitude,then, whatever multiple one of the magnitudes is of one, that multiplealso will all be of all.

In modern notation, let the magnitudes be a1; a2; ¢ ¢ ¢ ; an and let mbe the multiple. Then,

ma1 +ma2 + ¢ ¢ ¢+man = m(a1 + a2 + ¢ ¢ ¢+ an):

Proposition V-8. Of unequal magnitudes, the greater has to the same

Euclid 22

a greater ratio than the less has; and the same has to the less a greaterratio than it has to the greater.

In modern term, let a > b, and c is given. Then

a=c > b=c;

andc=b > c=a:

The Elements — Book VI — 33 theorems

Book VI is on similarity of figures. It begins with three definitions.

Definition 1. Similar rectilineal figures are such as have their anglesseverally equal and the sides about the equal angles proportional.

Definition 2. A straight line is said to have been cut in extreme andmean ratio when, as the whole line is to the greater segment, so is thegreater to the less.

Definition 3. The height of any figure is the perpendicular drawnfrom the vertex to the base.

Euclid 23

4.8 The Elements — Book VI

Proposition VI-1. Triangles and parallelograms which are under thesame height are to one another as their bases.

Proposition VI-5. If two triangles have their sides proportional, thetriangles will be equiangular and will have those angles equal whichthe corresponding sides subtend.

Proposition VI-30. To cut a given finite straight line in extreme andmean ratio.

A Q P C

Golden Section

AP : AC = PC : AP

Euclid 24

The picture says....

(a+ b)2 = c2 + 4(1

2ab)

a2 + 2ab+ b2 = c2 + 2ab

a2 + b2 = c2

Of course, you must prove all the similarity rigorously.

4.9 The Elements — Book VII — 39 theorems

Book VII is the first book of three on number theory. Euclid beginswith definitions of unit, number, parts of, multiple of, odd number, evennumber, prime and composite numbers, etc.

Definition 11. A prime number is that which is measured by the unitalone.

Definition 12. Numbers prime to one another are those which aremeasured by the unit alone as a common measure.

Euclid 25

Proposition VII-21. Numbers prime to one another are the least of thosewhich have the same ratio with them.

Proposition VII-23. If two numbers be prime to one another, the numberwhich measures the one of them will be prime to the remaining number.

Proposition VII-26. If two numbers be prime to two numbers, both toeach, their products also will be prime to one another.

Proposition VII-31. Any composite number is measured by some primenumber.

Proposition VII-32. Any number either is prime or is measured by someprime number.

4.10 The Elements — Book VIII — 27 theorems

Book VIII focuses on what we now call geometric progressions, butwere called continued proportions by the ancients. Much of this isno doubt due to Archytas of Tarentum, a Pythagorean. Numbers are incontinued proportion if

a1 : a2 = a2 : a3 = : : : :

We would write this as

a1 = a; a2 = ar; a3 = ar2; a4 = ar

3; : : : :

which is of course the same thing.

Proposition VII-1. If there be as many numbers as we please in contin-ued proportion, and the extremes of them be prime to one another, thenumbers are the least of those which have the same ratio with them.

Consider 5:3 and 8:6 and 10:6 and 16:12.

Proposition VIII-8. If between two numbers there are numbers in con-tinued proportion with them, then, however any numbers are betweenthem in continued proportion, so many will also be in continued pro-portion between numbers which are in the same ratio as the originalnumbers.

Euclid 26

Euclid concerns himself in several other propositions of Book VIIIwith determining the conditions for inserting mean proportional num-bers between given numbers of various types. For example,

Proposition VIII-20. If one mean proportional number falls betweentwo numbers, the numbers will be similar plane numbers.

In modern parlance, suppose a : x = x : b, then x2 = ab:

4.11 The Elements — Book IX — 36 theorems

The final book on number theory, Book IX, contains more familiar typenumber theory results.

Proposition IX-20. Prime numbers are more than any assigned multitudeof prime numbers.

Proof. Let p1; : : : ; pn be all the primes. DefineN = p1p2 ¢ ¢ ¢ pn+1.Then, since N must be composite, one of the primes, say p1 j N . Butthis is absurd!

1 Proposition IX-35. If as many numbers as we please are incontinued proportion, and there is subtracted from the second and thelast numbers equal to the first, then, as the excess of the second is tothe first, so will the excess of the last be to all those before it.

We are saying let the numbers be a; ar; ar2; : : : ; arn, The the dif-ferences are a(r ¡ 1) and a(rn ¡ 1). Then, the theorem asserts that

a=a(r ¡ 1) = (a+ ar + ¢ ¢ ¢+ arn¡1)=a(rn ¡ 1):

Euclid 27

Proposition 20

Prime numbers are more than anyassigned multitude of prime numbers.

A, B, C.

Proposition 36If as many numbers as we please beginningfrom an unit be set out continuously If doubleproportion, until the sum of all becomes prime,and if the sum multiplied into the last makesome number, the product will be perfect.

For let as many numbers as we please, A, B, C, D, beginning from an unit be set out in double proportion,until the sum of all becomes prime,

let E be equal to the sum, and let E by multiplying D make FG;I say that FG is perfect.

For, however many A, B C, D are in multitude, let so many E, RK, L, M be taken in double proportionbeginning from E;

therefore, ex qequali as A is toD, so is E to M. [vii. 14]Therefore the product of E, D is equal to theproduct of A, M [vii. 19]

and the product of E, D 1) is FG;

4.12 The Elements — Book X — 115 theorems

Many historians consider this the most important of the thirteen books.It is the longest and probably the best organized. The purpose is theclassification of the incommensurables. The fact that the anathema tothe Pythagoreans, the incommensurable is placed in Book X, the num-ber of greatest significance to them, may be more than a coincidence.Perhaps a slight toward the Pythagoreans; perhaps a sense of humor —if not, the irony is almost as remarkable.

Definition 1. Those magnitudes are said to be commensurable whichare measured by the same measure, and those incommensurable whichcannot have any common measure.

Note in the following definition how Euclid distinguishes magnitudesand lengths/areas.

Definition 1. Straight lines are commensurable in square when the

Euclid 28

squares on them are measured by the same area, and incommensurablein square when the squares on them cannot possibly have any area asa common measure.

The first proposition is fundamental. It is Eudoxus’ method ofexhaustion.

Proposition X-I. Two unequal magnitudes being given, if from thegreater there is subtracted a magnitude greater than its half, and fromthat which is left a magnitude greater than its half, and if this processis repeated continually, there will be left some magnitude less that thelesser of the given magnitudes.

This proposition allows an approximating process of arbitrary length.

Proposition X-36. If two rational straight lines commensurable in squareonly be added together, the whole is irrational.

4.13 The Elements — Book X1-XIII

The final three chapters of The Elements are on solid geometry and theuse of a limiting process in the resolution of area and volume problems.For example,

Proposition XII-2. Circles are to one another as the squares on thediameters.

? You will note there is no “formula” expressed.

Proposition XII-7. An pyramid is a third part of the prism which hasthe same base with it an equal height.

Proposition XII-18. Spheres are to one another in the triplicate ratio oftheir respective diameters.

Archimedes of Syracuse1

Archimedes of Syracuse (287 - 212 BCE), the most famous andprobably the best mathematician of antiquity, made so many discoveriesin mathematics and physics that it is difficult to point to any of themas his greatest.

He was born in Syracuse, the principal city-state of Sicily, the sonof the astronomer Phidias. He spent considerable time in Alexandria,where he studied with Euclid’s successors. It is there he met Conon ofSamos (fl. 245 BCE) and Eratosthenes of Cyrene (c. 276 - 195 BCE),both leading mathematicians of their day. However, he resided mostof his whole life in Syracuse, an intimate friend of the court of KingHieron II.

He was an accomplished engineer, indeed he is said to have dis-dained mechanical invention, who loved pure mathematics. With oneexception, his only extant works are on pure mathematics. His methodsof proof and discovery, though, were based substantially upon mechani-cal principles as revealed in his treatiseMethod Concerning MechanicalTheorems.

In fact, he seems to have disdained the source of his fame duringhis day, ingenious mechanical inventions, on which he left no written


Archimedes 2

description. Said Plutarch, ”he possessed so high a spirit, so profounda soul, and such treasures of scientific knowledge that, thought theseinventions had obtained for him the renown of more than human sagac-ity, he yet would not deign to leave behind him any written work onsuch subjects, ....”

Stories from Plutarch, Livy, and others describe machines inventedby Archimedes for the defense of Syracuse. These include the cata-pult and the compound pulley. Also described is his instruments ap-plying “burning-mirrors.” His fascination with geometry is beautifullydescribed by Plutarch.2

Often times Archimedes’ servants got him against his willto the baths, to wash and anoint him, and yet being there, hewould ever be drawing out of the geometrical figures, evenin the very embers of the chimney. And while they wereanointing of him with oils and sweet savors, with his figureshe drew lines upon his naked body, so far was he taken fromhimself, and brought into ecstasy or trance, with the delighthe had in the study of geometry.

During the siege of Syracuse in the Second Punic War, inventionsby Archimedes such as a catapult equally serviceable at a variety ofranges, caused great fear to the Roman attackers. Another invention,the compound pulley, was so powerfully built as to lift Roman shipsfrom the sea and drop them back into it. However, the story that he usedan array of mirrors, burning-mirrors, to destroy Roman ships is probablyapocryphal. So much fear did these machines instill in the Romans thatgeneral Marcus Claudius Marcellus, the Roman commander, gave upon frontal assault and placed his hopes in a long siege. When at lastSyracuse did fall in about 212 BCE, Archimedes was killed during thecapture of Syracuse by the Romans Plutarch recounts this story of hiskilling: As fate would have it, Archimedes was intent on working outsome problem by a diagram, and having fixed both his mind and eyesupon the subject of his speculation, he did not notice the entry of theRomans nor that the city was taken. In this transport of study a soldierunexpectedly came up to him and commanded that he accompany him.When he declined to do this before he had finished his problem, theenraged soldier drew his sword and ran him through. Marcellus was

2Plutarch(c. 46 - 119 CE), was a Greek biographer and author whose works influenced the evolutionof the essay, the biography, even into our own times.

Archimedes 3

greatly saddened by this and arranged for Archimedes’ burial.

1 Archimedes’ Works

It was to Conon that Archimedes frequently communicated his resultsbefore they were published. There were no journals, as such, in thattime. Major works were developed into books. There are nine extantbooks of Archimedes, that have come to us. Substantially in the formof advanced monographs, they are not works for students nor for thedilettante, as each requires serious study. Almost certainly, they werenot as widely copied or studied as other works such as The Elements .

But how do we know about the works? Where did they come from?For most of the second millennium, the earliest sources of Archimedesworks date from the Latin translations of Greek works made by Williamof Moerbeke (1215 - 1286). He used two Greek manuscripts. Both havedisappeared, the first before 1311 and the second disappears about the16th century. No earlier versions were known until about 1899, whenan Archimedes palimpsest was listed among hundreds of other volumesin a library in Istanbul. In 1906, the great Greek mathematical scholarwas able to begin his examination of it.

A palimpsest is a document which has been copied over by anothertext. Two reasons are offered for doing this. First, parchment wasexpensive and reusing it was an economical measure. Second, at thetime it was considered virtuous to copy over pagan texts. In the case athand, the Archimedes palimpsest was covered over by a religious text.Moreover, the original sheets were folded in half, the resulting book of174 pages having a sown binding.

What Heiberg found were four books already known but which hadbeen copied in the 10th century by a monk living in a Constantinoplemonastery. This version was independent of the two manuscripts usedby William of Moerbeke. However, a new book was found. It was theMethod concerning Mechanical Theorems. This book, though known tohave been written, had not been found to that time. Its importance liesin that in this volume, Archimedes described his method of discoveryof many of his other theorems.

The story of the Archimedes palimpsest over that past century isinteresting with suggestions of theft and manuscript alteration. Having

Archimedes 4

disappeared in 1922, it reappeared in 1998 as an auction item displayedby Christie’s in New York. It sold at auction for two million dollars inOctober of 1998 to an anonymous buyer. This buyer has agreed to makethe manuscript available for scholarly research. For further details, theinterested reader should consulthttp://www-history.mcs.st-and.ac.uk/history/HistTopics/Greek sources 1.html

The works themselves are

² On Plane Equilibria, Volume I

² Quadrature of a Parabola² On Plane Equilibria, Volume II

² On the Sphere and Cylinder, Volumes I and II² On Spirals² On Conoids and Spheroids² The Sand-Reckoner² On Floating Bodies, Volumes I and II

² On Measurement of the Circle² Method Concerning Mechanical Theorems

Another volume Stomachion, is known in fragments only. Yet anothervolume, a collection of Lemmas Liber Assumptorum comes down tous from the Arabic. In its present form, it could not been writtenby Archimedes as his name is referenced in it, though the results arelikely due to Archimedes. Overall, we may say that he worked inthe Geometry of Measurement in distinction to the Geometry of Formadvanced by his younger colleague/competitor Apollonius (260 - 185BCE). His methods anticipated the integral calculus 2,000 years beforeNewton and Leibniz. In the following subsections, we describe someof the results, recognizing the impossibility of rendering anything nearan adequate description of the overwhelming depth and volume of hisworks.

Archimedes 5

1.1 Measurement of the Circle

Among Archimedes’ most famous works is Measurement of the Circle,in which he determined the exact value of ¼ to be between the values 310

71

and 317. This result is still used today, and most certainly every reader of

these notes has used 227= 31

7to approximate ¼. He obtained this result

by circumscribing and inscribing a circle with regular polygons havingup to 96 sides. However, the proof requires two fundamental relationsabout the perimeters and areas of these inscribed and circumscribedregular polygons.

The computation. With respect to a circle of radius r, let

b1 = an inscribed hexagon with perimeterp1 and area a1

B1 = an circumscribed hexagon with perimeterP1 and area A1

Further, let b2; : : : ; bn denote the regular inscribed 6¢2 : : : 6¢2n polygons,similarly, B2 : : : Bn for the circumscribed polygons. The followingformulae give the relations between the perimeters and areas of these6 ¢ 2n polygons.

Pn+1 =2pnPnpn + Pn

pn+1 =ppnPn+1

an+1 =qanAn An+1 =

2an+1Anan+1 +An

Using n-gons up to 96 sides he derives the followingProposition 3. The ratio of the circumference of any circle to its diam-eter is less than 31

7and greater than 310

71.

1.2 On the Sphere and Cylinder

In Volume I of On the Sphere and Cylinder Archimedes proved, amongmany other geometrical results, that the volume of a sphere is two-thirdsthe volume of a circumscribed cylinder. In modern notation, we have

Archimedes 6

the familiar formula.

Vsphere =2

3V circumscribed cylinder

This he considered his most significant accomplishments, requestingthat a representation of a cylinder circumscribing a sphere be inscribedon his tomb. He established other fundamental results including

Proposition 33. The surface of any sphere is equal to four times thegreatest circle on it.

Similarly, but for cones, we have

Proposition 34. Any sphere is four times the cone which has as itsbase equal to the greatest circle in the sphere and its height equal to theradius of the sphere.

From this of course follows Archimedes relation above. In Volume II,Archimedes proves a number of results such as

Proposition 1. Given a cone or a cylinder, to find a sphere equal to thecone or to the cylinder.

Proposition 3. To cut a given sphere by a plane so that the surfaces ofthe segments may have to one another a given ratio.

Proposition 9. Of all segments of spheres which have equal surfacesthe hemisphere is the greatest in volume.

1.3 On Conoids and Spheroids

In On Conoids and Spheroids, he determined volumes of segments ofsolids formed by the revolutions of a conic, such as a parabola, about anaxis. In modern terms these are problem of integration. For example,we have

Proposition 21. Any segment of aparaboloid of revolution is half aslarge again as the cone or segmentof a cone which has the same baseand the same axis.

x

y

Archimedes 7

Though easy to verify using calculus, this result requires a carefuland lengthly proof using only the standard method of the day, i.e. doublereductio ad absurdum.

1.4 On Floating Bodies

In On Floating Bodies Archimedes literally invented the whole studyof hydrostatics. In one particular result he was able to compute themaximum angle that a (paraboloid) ship could list before it capsized— and he did it without calculus! This result, a tour de force ofcomputation, is not nearly as well known as the story which describesArchimedes crying “Eureka” after discovering whether a newly madecrown was truly pure gold.

The case of the fraudulent gold crown. King Hieron II commis-sioned the manufacture of a gold crown. Suspecting the goldsmith mayhave substituted silver for gold, he asked Archimedes to determine itsauthenticity. He was not allowed to disturb the crown in any way. Whatfollows is a quote from Vitruvius.3

The solution which occurred when he stepped into his bath andcaused it to overflow was to put a weight of gold equal to the crownand know to be pure, into a bowl which was filled with water to thebrim. Then the gold would be removed and the king’s crown put in,in its place. An alloy of lighter silver would increase the bulk of thecrown and cause the bowl to overflow.

There are some technical exceptions to this method. A better solu-tion applies Archimedes’ Law of Buoyancy and his Law of the Lever:

Suspend the wreath from one end of a scale and balance it withan equal mass of gold suspended from the other end. Immerse thebalanced apparatus into a container of water. If the scale remainsin balance then the wreath and the gold have the same volume, andso the wreath has the same density as pure gold. But if the scaletilts in the direction of the gold, then the wreath has a greater volume

3Vitruvius’s comments can be found in his work De architectura (about 27 BCE) a comprehensivetreatise on architecture consisting of 10 books.

Archimedes 8

than the gold. For more details, consult the Archimedes home page,http://www.mcs.drexel.edu./~crorres/Archimedes/index.html.

1.5 Sand-Reckoner

The Sand-Reckoner is a small treatise that is addressed to Gelon, sonof Hieron. Written for the layman, it nevertheless contains some highlyoriginal mathematics. One object of the book was to repair the inad-equacies of the Greek numerical notation system by showing how toexpress a huge number, in particular the number of grains of sand thatit would take to fill the whole of the universe. Apparently independentof the Babylonian base 60 system, Archimedes devised a place-valuesystem of notation, with a base of 100,000,000. He constructed num-bers up to 8 £ 1017. The work also gives the most detailed survivingdescription of the heliocentric system of Aristarchus of Samos, the An-cient Copernican.

1.6 On the Equilibrium of Planes

In a treatise of two volumes Archimedes discovered fundamental theo-rems concerning the center of gravity of plane figures and solids. Hismost famous theorem gives the weight of a body immersed in a liquid,called Archimedes’ principal.

1.7 Quadrature of a Parabola

In the Quadrature of a Parabola, Archimedes proved using the Methodof Exhaustion that

area segment ABC =4

3¢ 4ABC

where the triangle and parabolic segment have the same base and height.The standard technique of proof, the method of exhaustion, was used.

Archimedes 9

A

B

C

Segment of aparabola and in-scribed triangle.Note: theslope at is the sameas theline

.

B

AC

1.8 Spirals

In The Spiral Archimedes squared the circle using the spiral.

He does this by proving that, in length, OQ = arcPS. Note, PQ istangent to the spiral at P and 6 POQ is a right angle.

Archimedes 10

² He also determined the area of one revolution (0 ∙ µ ∙ 2¼ ) ofr = aµ to be4

Area =1

3[¼(a2¼)2]

That is, the area enclosed by the spiral arc of one revolution is onethird of the area of the circle with center at the origin and of radius atthe terminus of the spiral arc.

4In polar coordinates the area of the curve r = f(µ) is given by 12

R 2¼0

f2(µ) dµ. For the function

f(µ) = aµ, we have 12

R 2¼0(aµ)2 dµ = 4

3a2¼3.

Archimedes 11

He also showed how to trisect angles using the spiral. Suppose theparticular angle to be trisected is /AOB. Construct circles with centerO that intersect the spiral. Construct the line segment OD and markthe point C. Trisect the segment CD and construct circles with centerO with the respective radii at the trisection points. Since the spiralsweeps out the radius in exact proportion to the respective angle, thenew circles will intersect the spiral at equal angles from the lines OAand OB. The angle between them will be the same, as well. Thus theangles /AOB is trisected.

In another argument using acompass and ruler, he trisected anangle. Using the diagram to theright, we trisect the angle /DOA.First, extend the diameter to B insuch a way that jBCj = j0Cj.This is the part that requires the ruler. Now measure the angles. Notethat ° = 2¯. From this it follows that µ = 4¯¡¼: Finally observe that®+ µ + ¯ = ¼. Substitute µ = 4¯ ¡ ¼ and solve for ¯ to get ¯ = 1

3®.

1.9 The Method

In Method Concerning Mechanical Theorems Archimedes reveals howhe discovered some of his theorems. The method is basically a “geo-metric method of the lever.” He balances lines as one might balanceweights. This work was found relatively recently, having being redis-covered only in 1906.

Archimedes 12

2 Inventions

Archimedes’ mechanical skill together with his theoretical knowledgeenabled him to construct many ingenious machines. Archimedes spentsome time in Egypt, where he invented a device now known as Archimedes’screw. This is a pump, originally used for irrigation and for drainingmines. It is still used in many parts of the world. The image below isbut one example.5

From Pappus we have learned that in connection with his discoveryof the solution to the problem of moving a given weight by a given force,that Archimedes upon applying the law of the lever6 is to have said,“Give me a place to stand on, and I can move the earth.” Another storyrelated to this was the challenge to Archimedes by King Hieron to givea practical demonstration of this law. Thereupon, Archimedes, usingonly a compound pulley, steadily and smoothly pulled a ship from thesea onto dry dock. According to Proclus, Hieron was so impressed byArchimedes that he declared, “from that day forth Archimedes was tobe believed in everything that he might say.”

He is also said to have invented a sphere to imitate the motions ofthe sun, moon, and five planets known at that time. Cicero, who mayhave actually seen it, reported that it described details of the periodicnature of the rotations and even showed eclipses of the sun. How itoperated is conjectural, but water power is often attributed.

5An interesting website maintained by Drexel University mathematics Professor Chris Rorres, locatedat http://www.mcs.drexel.edu/~crorres/Archimedes/Screw/Applications.html, shows many illustrations ofArchimedian screws from the past and present. The illustration above can be found at this site.

6Archimedes was a master at the law of the lever and related mechanical principles, this is true.However, it is certain that this law must have been known much earlier in antiquity. In particular, theEgyptians must have applied some form of it in the construction of the pyramids.

Archimedes 13

3 Influence

The magnitude and originality of Archimedes’ achievement is monu-mental. However, his influence on ancient mathematics was limited.Many reasons could be attributed, one being that mathematics in theGreek world was in something of an eclipse of his mathematics. An-other is the hegemony of the Romans, who had little interest in theoret-ical works, particularly mathematics. Though some of his results, suchas approximations to ¼ by 22

7, became commonplace, his deeper results

of hydrostatics and quadrature were never continued in any importantway — as far as is known. This seems true, despite his publicationof The Method, in which he hoped to show others the basis of histechniques.

Nearly a millennium was to follow, when in the 8th and 9th cen-turies there were some substantial Arabic contributions that seem to beinspired by Arabic translations of Archimedes works.

The greatest influence of his work came much later in 16th and 17th

centuries with the printing of texts derived from the Greek. Knowledgeof these works was reflected in the work of the greatest mathematiciansand physicists of the day, Galileo (1564 - 1642) and Kepler (1571- 1630). Later, more mathematically sound editions such as DavidRivault’s edition and Latin translation (1615) of the complete workswas profoundly influential on mathematicians of a stature no less thanRené Descartes (1595 - 1650) and Pierre de Fermat (1601 - 1665).The ancient works including Archimedes cast a pale across these timeschallenging mathematicians of the day to understand and advance theancient results. It is widely regarded that the greatest advances of the16th century would have be delayed without them. Had The Methodbeen discovered earlier than the late 19th century, modern mathematicsmay have taken an entirely different course concluding, of course, thesame essential results but with mechanical underpinnings instead ofgeometrical ones.

The Hellenistic Period1

In this brief section, we show what type of problems mathemati-cians solved, and the extent of Greek mathematics before the collapseof the mathematical world in about 400 CE.

1 Aristarchus of Samos (CA. 310-230 BCE)

Aristarchus of Samos was considered by the ancients, particularly Vit-ruvius2, who considered him to be one of the few great men knowl-edgeable in all sciences, especially astronomy and mathematics. Notethat his dates are only surmised. For example, he lived before the SandReckoner of Archimedes, a work published before 216 BCE. He is alsoknown to have observed the summer solstice in about 280 BCE.

Perhaps, foremost among them is that he formulated the Coperni-can hypothesis He was the first to formulate the Copernican hypothesesand is sometimes called the Ancient Copernican. His heliocentric the-ory was never accepted in his day and remained dormant until it wasproposed again by Copernicus. His contributions to science and math-ematics are many. He discovered an improved sundial, with a concavehemispherical circle. In supporting his heliocentric theory, he counteredthe non parallax objection by asserting that the stars to be so far dis-tant that parallax was not measurable. For this he needed the followingmeasure of distance to the stars.

distance(earth, stars) » distance(radius of the sun)

He also determined a more accurate estimate of the solar year. Hewrote as well on light and colors.

He wrote On the Sizes and Distances of the Sun and Moon, inwhich by the way the heliocentric ideas are not mentioned or in anyevent are not present in the copied version extant. He assumes the

1 c°2000, G. Donald Allen2Vitruvius’s comments can be found in his work De architectura (about 27 BCE) which was a com-

prehensive treatise on architecture. Divided into 10 books, it deals with city planning, building materials,temple construction, the use of the Greek orders, public buildings such as theaters, private buildings, floor-ing, hydraulics, clocks, mensuration and astronomy, and engines. Though a Roman living probably duringthe time of Augustus, his outlook is essentially Hellenistic.

The Hellenistic Period 2

following two propositions that are equivalents of the following modernstatements.

1. If 0 < ® <¼

2, then the ratio

sin®

®decreases, and the

ratiotan®

®increases as ® increases from 0 to

¼

2.

2. If 0 < ¯ < ® <¼

2then

sin®

sin¯<®

¯<tan®

tan¯

(now known as Aristarchus’ theorem)

He sets out to prove a number of results including, (1) When themoon is half full, the angle between the lines of sight to the sun andthe moon is less than a right angle by 1/30 of a quadrant. From this heconcluded that the distance from the earth to the sun is more than 18but less than 20 times the distance from the earth to the moon. (Actual» 400). Without trigonometry he was aware of and used the fact that

1

20< sin 3± <

1

18:

This result gave a great improvement over all previous estimates.

(2) The angular diameters of the sun and the moon at the center ofthe earth are equal. (3) The diameter of the sun has to the diameter ofthe earth a ratio greater than 19:3 but less than 43:6. (Actual: »109:1)

The proofs of all are geometrical in nature — as we would expect.He also made other trigonometric estimates — without trigonometry.


2 Archimedes, (287 - 212 BCE)

Looming far above almost all mathematicians of this period is Archimedes.His contributions are many and his mathematics clearly the deepest ofall. A separate chapter is devoted to his life and works.

3 Apollonius of Perga (CA 262 BCE – 190 BCE)

? Apollonius was born in Perga in Pamphilia (now Turkey), but waspossibly educated in Alexandria where he spent some time teaching.Very little is known of his life. He seems to have felt himself a rival ofArchimedes. In any event he worked on similar problems. However, incontrast to the older master we may say that Apollonius worked on theGeometry of Form as distinguished from the Geometry of Measurement.He was known as the “great geometer” mainly because of his work onconics.

Apollonius was 25 years younger than Archimedes, and they to-gether with Euclid stood well above all other mathematicians of thefirst century of this period. Because of them, this period is sometimescalled the “golden age” of Greek mathematics.

Apollonius wrote many books. All but one are lost. Among thosewe know he wrote are:

1. Quick Delivery

2. Cutting-off of a Ratio

3. Cutting-off of an Area

4. Tangencies

5. Vergings (Inclinations)

6. Plane Loci

In his book Quick Delivery (lost), he gives the approximation to ¼as 3.1416. We do not know his method.

His only known work is On Conics in 8 Books of which only 4survive in Greek. The 8th book is lost completely and the books V -


VII exist in Arabic. The fourth and later books were dedicated to KingAttalus I (247 - 197 BCE), we have an approximate date of publication.The first two books were dedicated to Eudemus of Pergamum. It be-came immediately the authoritative treatise on conics, being referencedby many mathematicians. In about 500 CE, Eutocius wrote a commen-tary on them. It is this edition, altered somewhat in the ninth centuryin Constantinople, that our best manuscripts are derived. The editioprinceps of the Greek text is due substantially to Edmund Halley3 in1710. Originally, Gregory works on the Books I - IV, but died whenthe work was in progress. Features: Using the double oblique cone heconstructs the conics

parabola, ellipse, hyperbola,

whose names he fixed for all time.

He made use of the idea of Symptoms which were similar to equa-tions. There results an analytic-like geometry – but without coordinates!These come from what can be called a “symptoma” (from the Greek¾º0¹¼¿!¹®), a constant relation between certain magnitudes whichvary according to the position of an arbitrary point on the curve. Thesymptoma are very much like a modern equation. Even Archimedesused these relations for parabolas, ellipses and hyperbolas in his works.

3We will see in a later chapter that it was Edmund Halley (1656 - 1743) that was instrumental inpersuading Isaac Newton to publish his monumental Principia Mathematica.


We consider two typical results found in On Conics.

Proposition I-33. If AC is constructed, where jAEj = jEDj, then ACis tangent to the parabola.

Proof. Assume AC cuts the parabola at K. Then CK lies within theparabola. Pick F on that segment and construct a ? to AD at B andG on the curve. Then

jGBj2jCDj2 >

jBF j2jCDj2 =

jABj2jADj2 : (¤)

Since G and C lie on the curve, the symptom shows that jBGj2 =pjEBj and jCDj2 = pjEDj implies jBGj2

jCDj2 =jBEjjDEj . This implies from

(*) that

jBEjjDEj >

jABj2jADj2 ; and so

4jBEj ¢ jEAj4jDEj ¢ jEAj >

jABj2jADj2 :

Thus 4jBEj¢jEAjjABj2 > 4jDEj¢jEAj

jADj2 . Since jAEj = jDEj it follows thatjADj2 = 4jDEj¢jEAj and since jBEj > jEAj it follows that 4jBEj jEAj >jABj2. (Why?) Since AB = EA+ EB, this is a contradiction.


Proposition I-34. (ellipse) Choose A so that

jAHjjAGj =

jBHjjBGj

Then AC is tangent to the ellipse at C.

It is apparent that these Äsymptoma used by Apollonius and byothers before him were a definite precursor to analytic geometry. Abouttwo millennia would pass before coordinate geometry became commonplace.


The Apollonius model for the sun used epicycles. The sun rotates abouta circle whose center revolves about the earth. The directions of rotationare opposed. The ratio

ED

DS:= eccentricity

There was also the eccenter model, designed to explain the seasonsbetter.

He improved upon the numbering system of Archimedes by usingthe base 104.

In the works of Apollonius and Archimedes, Greek mathematicsreached its zenith. Without his predecessors, the foremost being Euclid,Apollonius never would have reached this height. Together, they domi-nated geometry for two thousand years. With the works of Apollonius,mathematics was now well beyond the reaches of the dedicated amateur.Only a professional would be able to advance this theory further.


4 Hipparchus (fl. 140 BCE)

Hipparchus of Nicaea was a scientist of the first rank. So carefullyaccurate were his observations and calculations that he is known inantiquity as the “lover of truth.” He worked in nearly every field ofastronomy and his reckonings were canon for 17 centuries.

Only one work of his remains, a commentary on Phainomena ofEudoxus and Aratus of Soli. We know him, however, from Ptolemy’sThe Almagest. Indeed the “Ptolemaic Theory” should be called Hip-parchian. His mathematical studies of astronomical models required acomputation of a table of sines. He constructed a table of chords forastronomical use. Here a chord was given by

Crd ® = 2R sin ®=2

in modern notation.

Hipparchus knew the half angle formula as well. He could computethe chord of every angle from 71

2

± to 180± Using Babylonian observa-tions, he improved the lunar, solar and sidereal years. He reckoned thesolar year at 3651

4days, less 4 minutes, 48 seconds — an error of 6

minutes from current calculations. He computed the lunar month at 29days, 12 hours, 44 minutes, 21

2seconds — less than one second off.

He also computed the synodic periods of the planets with astonishingaccuracy. He estimated the earth-moon distance at 250,000 miles, lessthan 5 percent off.

Hipparchus almost concluded the orbit of the earth about the sun


to be elliptic through his theory of “eccentrics” to account for orbitalirregularities.

In about 129 BCE, he made a catalog of 1080 known fixed stars interms of celestial longitude and latitude. Comparison of his chart withthat of Timochares from 166 years earlier, he made his most brilliantobservation. Noting a 20 shift in the apparent position of the stars hepredicted the precession of the equinoxes4 — the advance, day by day,of the moment when the equinoctial points come to the meridian. Hecalculated the precession to be 36 seconds/year — 14 seconds slowerthan the current estimate of 50 seconds.

5 Claudius Ptolemy(100-178 AD)

Ptolemy was an astronomer/mathematician He wrote The Geography—a compilation of places in the known world along with their geographiccoordinates. His most enduring work is the Mathematical Collection,later called The Almagest, from the Arabic ‘al-magisti’ meaning “thegreatest.” It consists of 13 books, which contains

1. table of chords 1=2± – 180± 1=2± intervals — R = 60, insexagesimal.

2. He solve triangles, planes and spherical triangles.

3. The book was very algorithmic – this was a text/tutorial not aresearch monograph.

4. We also find the well known Ptolemy’s Theorem.

4equal nights


Theorem. jACj jBDj = jADj jBCj+ jABj jDCj

The Almagest was the standard reference work for astronomers untilthe time of Copernicus.

6 Heron of Alexandria 1st century A.D.

Very little is known of Heron’s life. However we do have his bookMetrica which was more of a handbook for mathematics than whatwould now be called a research monograph. In it we find the famousHeron’s formula. For any triangle of sides a; b; and c, and withperimeter s = a+ b+ c, the areas is given by

A = (s(s¡ a)(s¡ b)(s¡ c))1=2:

He also gives formulas for the area of regular polygons of n sides,each of length a:

A3 ¼ 13

30a2

A5 ¼ 5

3a2

A7 ¼ 43

12a2

Heron gives an example of finding the cube of a non-cube number.The number is A = 100. Here are his instructions,


Take the nearest cube numbers to 100 both above andbelow. These are 125 and 64. Now compute the differences

125¡ 100 = 25

100¡ 64 = 36

Multiply 5 by 36 to get 180 and then add 100 to get 280. Takethe fraction 180

280= 9

14and add to the smaller cube obtaining

4 914This is as nearly as possible the cube root (“cubic side”)

of 100 units.

Just how Heron arrived at this method, and indeed, just what the methodis we can only conjecture. However, in Heath5 we find the conjectureof Wertheim. Allowing the (positive) differences to be d1 and d2, andthe cube roots of the higher and lower cube to be a1 and a2, an approx-imation can be taken as

a1 +a2d

a2d1 + a1d2

Such was the state of the art for determining cube roots in the firstcentury AD. In fact, with just elementary methods, it can be shown thatthis approximation is quite good, with accuracy exceeding that achievedby application of the differential method.6

7 Nicomachus of Gerasa(fl. 100 CE)

Nicomachus was probably a neo-Pythagorean as he wrote on numbersand music. The period from 30BCE to 641 CE is sometimes called theSecond Alexandrian School.

He studied in Alexandria. Only two of his books are extant:

1. Introduction to Arithmetic

2. Introduction to Harmonics

A third book on geometry is lost.5Thomas Heath, A History of Greek Mathematics, II, Dover, New York, page 341.6The differential method: Approximate f(x) by f(x) ¼ f !(a)(x¡ a) + f(a) where both f(a) and

f !(a) are easily computed.


In the first of the works Introduction to Arithmetic given in two bookswe have in Book I a classification of integers no without proofs Forexample we have

even: even £ even (powers of two)even £ odd (doubles of odd)odd £ even (all others)

odd: primescomposites

perfect: c alculates 6, 28, 496, 8128 ala Euclid

Also, Nicomachus provides a classification of ratios of numbers.Assume a=b is completely reduced form of A=B. (i.e. a and b arerelative prime)

relation ratio of A to B is aa = nb multipleb = na submultiplea = b+ 1 super particulara+ 1 = b sub super particulara = nb+ 1 multiple super particular

a = nb+ k 1 < k < b multiple super particularetc.

In Book II Nicomachus discusses plane and solid numbers but againwith no proofs. (Were the proofs removed in the many translations?)He considers the very Pythagorean:

triangular numbers

square numbers

pentagonal numbers

hexagonal numbers

heptagonal numbers


He notes an interesting result about cubes:

1 = 1

3 + 5 = 8

7 + 9 + 11 = 27

13 + 15 + 17 + 19 = 64

¢ ¢ ¢ =This should be compared with the summation of odd numbers to achievesquares. (Recall, the square numbers of Pythagoras.)

The other known work, Introduction to Arithmetic was a “hand-book” designed for students, primarily. It was written at a much lowerlevel than Euclid’s Elements but was studied intensively in Europe andthe Arabic World throughout the early Middle Ages.

8 Other Great Geometers

8.1 Hypsicles of Alexandria

(fl. 175 BCE) added a fourteenth book to the Elements on regular solids.In short, it concerns the comparison of the volumes of the icosahedronand the dodecahedron inscribed in the same sphere.

? In another work, Risings, we find for the first time in Greekmathematics the right angle divided in Babylonian manner into 90 de-grees. He does not use exact trigonometry calculations, but only a roughapproximation. For example, he uses as data the times of rising for thesigns from Aries to Virgo as an arithmetic progression.

? He also studied the polygonal numbers, the nth one of which isgiven by

1

2n[2 + (n¡ 1)(a¡ 2)]:

8.2 Diocles of Carystus, fl. 180 BCE

Diocles of Carystus invented the cissoid, or ivy shaped curve. It wasused for the duplication problem. In modern terms it’s equation can be


written asy2(a¡ x) = x3:

In fact, what we know of Diocles comes from references by Euto-cius (6th cent. A.D.) to Apollonius’ Conics in his version of Diocles,Burning Mirrors. Since the Cissoid was references by Geminus, whosucceeded Apollonius, we have dates on Diocles of 190-180 B.C, thereferences to Apollonius an addition by Eutocius.

? Diocles also solved an open problem posed by Archimedes, thatof dividing a sphere into two parts whose volumes have a prescribedratio. (Compare this with the result of Archimedes: To cut a givensphere by a plane so that the surfaces of the segments may have to oneanother a given ratio.

He was able to solve certain cubics by intersecting an ellipse and ahyperbola. He studied refraction and reflection in the book On BurningMirrors. However, this book may actually be a combination of threeshort works, one on burning mirrors, one on doubling the cube byintersecting parabolas, and the third on the problem of Archimedes.

8.3 Nicomedes,(fl. 260 BCE)

Nicomedes wrote Introduction Arithmetica. He discovered the con-choid and used it for angle trisection and finding two mean proportion-als. We say the x and y are two mean proportionals between a and bif a : x = x : y = y : b. Of course this amounts to solving the forthe intersection of the parabola ay = x2 and y2 = bx. The conchoid isgiven by

r = a sec µ § d:The little we know of these results come from Pappus (c. 350 CE) whodescribed them and Eutocius (6th century CE) who indicated Nicomedes’great, even boastful, pride in his discovery. Pappus further associatesNicomedes with Dinostratus and others as having applied the quadra-trix, invented by Hippias (c. 460 BCE), for squaring the circle.

8.4 Eratosthenes of Cyrene, (c. 276 - c. 195 BCE)

Eratosthenes achieved distinction in many fields and ranked second onlyto the best in each. His admirers call him the second Plato and some


called him beta, indicating that he was the second of the wise men ofantiquity. By the age of 40, his distinction was so great that Ptolemy IIImade him head of the Alexandrian Library. He wrote a volume of verseand a history of comedy, as well. He wrote mathematical monographsand devised mechanical means of finding mean proportions in continuedproportion between two straight lines. He also invented the sieve fordetermining primes. This sieve is based on a simple concept:

Lay off all the numbers, then mark of all the multiples of 2, then 3,then 5, and so on. A prime is determined when a number is not markedout. So, 3 remains uncovered after the multiples of two are marked out;5 remains uncovered after the multiples of two and three are markedout. Although it is not possible to determine truly large primes in thisfashion, the sieve was used to determine early tables of primes. 7

In a remarkable achievement he attempted the measurement of theearth’s circumference, and hence diameter. Using a deep well in Syene(nowadays Aswan) and an Obelisk in Alexandria, he measured the anglecast by the sun at noonday in midsummer at both places. He measuredthe sun to be vertical in Syene and making an angle equal to 1/50 of acircle at Alexandria he measured the circumference of the earth to be25,000 miles. Remember, this measurement of the radius of the earthwas made in » 250 B.C. Here’s the diagram:

In modern terms, we have distance = radius£angle. Using theangle to be 1/50 of a circle, we obtain

7An interesting Java applet demonstrating this may be found at Peter Alfeld’s web site:http://www.math.utah.edu/ alfeld/Eratosthenes.html


50£ 5000 = 250; 000 stades= 25; 000 miles

He also calculated the distance between the tropics as 1183of the circum-

ference, which make the obliquity of the ecliptic 230 510 2000, an errorof 1/2 of one percent.

Having measured the earth he set out to describe it, compilingaccounts of all voyagers. He advocated regarding each person not asa Greek or Babylonian, but as an individual with individual merits —certainly a modern-sounding tenet.

It is said that in his old age he became blind and committed suicideby starvation.

8.5 Perseus (fl. 75 BCE)

Perseus discovered spiric sections, curves obtained by cutting solids,not cones. In modern terms a spire is given by

(x2 + y2 + z2 + c2 ¡ a2)2 = 4c2(z2 + x2)

8.6 Zenodorus (fl. 180 BCE)

About Zenodorus we know only by his reference to a result of Archimedesand his style of writing that he must have succeeded Archimedes butnot by very much. In his book On Isoperimetric Problems, Zenodorusstudied isoperimetric figures (same perimeter-different shape), a subjectwith a long history in Greek mathematics. This subject is extremelydifficult and during the 18th century evolved into the Calculus of Vari-ations. Indeed, even the great Isaac Newton was challenged to solve adifficult problem in this area, the so-called Brachistochone problem.8This result, of course, was well beyond the technical level of Greekmathematics. Other modern problems include the “soap bubble” prob-lem and others. However, Zenodorus (and later Pappus) offered somecredibly difficult theorems. Here are a few typical results.

Proposition. Of all regular polygons of equal perimeter, that is greatestin areas has the most angles.

8Given two points in space, one higher than the other. If a wire is attached to the points what shapeshould it have for a frictionless bead to slide from the highest point to the lowest in least time.


Proposition. A circle is greater than any regular polygon of equal con-tour.

Proposition. Of all polygons of the same number of sides and equalperimeter the equilateral and equiangular polygon is the greatest in area.

August 27, 2000

Ancient Algebra1

1 Origins.

Our word ‘Algebra’ is derived from the Arabic expression

al-jabr wa’l muqabala

which occurs in the title of the first Arabic text on algebra written byAl-Khwarizmi in the 9th century. We have the words:

Al jabr — restoration or completion(add equals to equals toeliminate negative terms)

Al-muqabala — reduction or balancing(cancelling terms that occur onboth sides of an equation)

So, to Al-Khwarizmi, ‘Algebra’ is the art of reducing and solvingequations. In modern algebra, the emphasis has shifted to structure.Its roots started with the work of Galois on the possibility of solvingequations by means of radicals (1830).

2 Three Kinds of Algebra

In ancient times there were three kinds of algebra.

A. Mixed Algebra. This is Babylonnian type in which line segmentsand areas, etc. are added together and set equal to numbers.

B. Numerical Algebra. Herein only rational numbers m=n are admit-ted as coefficients and solutions of equations. (cf. Diophantus)


Ancient Algebra 2

C. Geometric Algebra. Here line segments, areas, and volumes arekept strictly apart, e.g. when lengths are multiplied, area results. Thisis found in Greek mathematics, we know, but also in Chinese andIndian mathematics. Solutions to quadratics and linear equations areaccomplished geometrically. In this connection, modern mathematicianssuch as Decartes, solved geometric problems by first converting themto algebraic ones and then the solutions back to geometric terms.

Diophantus, ca. 2401

1 Introduction

So little is known of Diophantus, that the dates of his life are givenin the two century range 150 AD - 350 AD, likely »250 AD. He isbelieved to have lived to be about 84 years. According to traditionhis age is determined from the \conundrum", dating from the ¯fth-sixth century:

God granted him to be a boy for the sixth part of hislife, and adding a twelfth part to this, He clothed his cheekswith down; He lit him the light of wedlock after a seventhpart, and five years after his marriage He granted him a son.Alas! late-born wretched child; after attaining the measureof half his father’s life, chill Fate took him. After consolinghis grief by this science of numbers for four years he endedhis life.

2 Works

Diophantus was proli¯c. He wrote

Arithmetica (13 Books { only 6 are now Extant)

On Polygonal Numbers of which only fragments now exist

Porisms (may have originally been part of Arithmetica, as inthe latter they are referenced as though they are there

In ancient and even in more recent times, commentaries wouldfrequently be written on notable books. Indeed, one measure of thebook's value to the professional community is the number of com-mentaries written on it. For Arithmetica commentaries were writtenby


Diophantus 2

² Hypatia, daughter of Theon of Alexandria, who commentedonly on the ¯rst 6 books.

² Psellus 11th century,² Georgius Pachymeres (1240-1310)² M. Plamides (1260-1310)² several Arab mathematicians

Translations of Arithmetica:

² Regionontanus { 1463 ¯rst to call attention to Diophantus.

² Rafael Bombelli { 1570 translated a manuscript found in theVatican. It was not published but was included in his own bookAlgebra (1572).

² Wilhelm Holzman (aka Xylander) { produced an excellentLatin translation in 1575.

² Bachet { 1621, published the present standard edition. Thesecond edition was carelessly produced, but does contain the epoch-making notes of Fermat { the editor was S. Fermat.

² Simon Stevin » 1585, French version of Books I{IV based onXylander.

² Otto Schultz { 1822G. Wertheria » 1890 g German versions based on Bachet.

3 Arithmetica

Before considering several of the types of results found in Arithmeticait is worth taking a few moments to consider a little backgroundabout algebra and the types of problems Diophantus solves. In the¯rst case, because geometrical reasoning is not particularly penalizedby rhetoric and was the dominant mathematical form of the day,algebra was compelled to follow suit. However, the problems solvedby Diophantus are every bit as tricky today as they were two milleniaago.

Diophantus 3

To be speci¯c, we may ask what was the form of algebra in thesevery early days? There are roughly three classi¯cations of algebracentered on the form of presentation.

1. Rhetorical algebra » complete prose.2. Syncopated Algebra » use of some abbreviations and symbols{ Diophantus and latter to early 17th century

3. Symbolic Algebra » complete symbolic notation { no prose.

The last form did not appear until relatively recent times. Symbol-ism was not in anything near wide usage until the early seventeenthcentury. Pure symbolism in mathematical expression is unques-tionably a contributing factor that has resulted in the explosion ofmathematics dating from that time, which if anything is expandingtoday at an ever increasing pace2. Diophantus wrote with limitedsymbols, but it must be surmised that after a length of time he musthave developed a deep intuition and problem familiarity that per-mitted him to use many abbreviations (semi-symbolism) that neverappeared in the published works.

First of all the problems of Arithmetica are algebraic. Diophan-tus was not the ¯rst to consider such problems nor was he the orig-inator of his principle technique, that of false position. Indeed, theEgyptians used the method of false position to solve relatively simplealgebraic equations. Recall the solution of the problem x+ 1

7x = 19

was solved ¯rst by assuming that x = 7 and then correcting thefalse solution by multiplication 7£ 19

8. Other, similar, problems were

solved by this method as well. We have also seen the Babylonianmathematicians solve simple linear systems using a false assump-tion.

In Heron's Metrica several indeterminate problems are posed.Indeterminate problems are of a type where there may be severalsolutions and the student is asked to ¯nd one or all of them. In mostof these problems the student is asked to ¯nd an integer or rationalsolutions. Here is an example of one such problem.

2Equally, one might argue that modern computers and computer algebra systems, with their immensenumeric and symbolic computational power may contribute to another explosive wave of mathematicaldevelopment, one that has only just begun. To be sure, the powerful mathematics is being done todaywithout any technology whatever and will continue throughout next millennium. However, the opportunitiesthat modern technology afford may inspire new directions of research impossible without them, just assymbols permitted four centuries ago.

Diophantus 4

Example. Find two rectangles for which the perimeter of the ¯rstis three times that of the second and the area of the second is threetimes that of the ¯rst. Mathematically, we are asked to ¯nd integersa; ; b; c; and d for which

a+ b = 3cd

c+ d = 3ab

Example. Find a triangle having rational sides with area 5. Alter-natively, ¯nd a right triangle with rational sides such that the sumof its area and perimeter is a given value.

Problems of these types can be prodigiously challenging to solve,even with modern symbolism. Moreover, the use of a purely rhetor-ical system can only make them more di±cult. A good number ofmathematicians, high on our lists of the truly great, are rememberedpartly if not exclusively due to their genius at solving indeterminateproblems and proving theorems about them.

3.1 Symbolism of Diophantus

? The beginning of symbolism: The unknown (x { to us)

x =½³ S ³ 0 invarious editionsy 0S± ®½

x2 := ¢¨

x3 := K¨

x4 := ¢¨¢

x5 := ¢K¨

x6 := K¨K

³x := 1=x

¢¨x

:= 1=x2

There is no symbol for +. Essentially, the plus operation was thedefault. No symbol between variables implies the plus operation.We also have

¤j := minus±M := units

Diophantus 5

An example:

K¨®¢¨i° ³"±M ¯ = x3 + 13x3 + 5x+ 2

More examples:

K¨® ³´ ¤j¢¨"±M ® =x3 ¡ 5x2 + 8x¡ 1

¢¨i"¤j ±M ¸µ = 15x2 ¡ 39

Note that in the ¯rst example above Diophantus collects the neg-ative terms so that what was written corresponds literally to x3 +8x ¡ (5x2 + 1). Diophantus introduced su±cient symbolism to be-come well aware of the laws of exponents, which is relatively simpleto perceive from modern notation.

3.2 The methods and ground rules of Diophantus

The types of solutions.

² No solutions are accepted other than positive rational numbers.² Excluded are negative numbers and surds3, and imaginary num-bers. For examples, Diophantus would describe 4 = 4x + 20as absurd because the solution x = ¡4. Neither would thesolution of x2+1 = 0 be permitted, as the roots are imaginary.

The types of equations:

² (A) Determinate equations { single variable.² (B) Indeterminate equations { two or more unknowns. Herethere is a weakness in notation.

3A number that is can be obtained from rational numbers by a finite number of additions, multiplications,divisions, and root extractions is called a surd. An irrational number of the form !

pa in which a is rational

is called a pure surd of index n. For n = 2 the surd is quadratic. Surds that are not pure are called mixed.Geometrically, surds are all constructable numbers on the basis of a compass and straight edge.

Diophantus 6

3.3 Some examples

General form for intermediate equations of the second degree:

Ax2 +Bx+ C = y2 { single equationx+ y = mx2 + y2 = n

{ two variables

(1) The single equation: Ax2 +Bx+ C = y2

Case (i) A = C = 0A = 0

Bx = y2

Bx+ C = y2

¾ Soln. Takey2 = m2 andsolve

Case (ii) C = 0 Ax2 +Bx = y2Soln. Takey = m

nx

Case (iii) B = 0 Ax2 + C = y2

(®) A = a2 : a2x2 + c = (ax§m)2 ! x = §C¡m2

2ma

(¯) C = c2 : Ax+ c2 = (mx§ c)2 ! x = § 2mcA¡m2

Case (iv) Ax2 +Bx+ C = y2

(2) Double equations:

mx2 + ®x+ a = u2

nx2 + ¯x+ b = w2

² Simplest case:

®x+ a = u2

¯x+ b = w2

To add the same number to two given numbers so as to makeeach a square.

Diophantus gives two complex solutions, the second assuminga = b = n2.

Diophantus 7

? Other examples from Book II:

x2 + y = u2; y2 + x = v2

(Assume y = 2mx+m2, and one equation is satis¯ed.)

x2 ¡ y = u2; y2 ¡ x = v2x2 + (x+ y) = u2; y2 + (x+ y) = v2

(x+ y)2 + x = u2; (x+ y)2 + y = v2

y2 ¡ z = u2; z2 ¡ x = v2; x2 ¡ y = w2

Solve.

x+ a = u2

x+ b = v2

Solution.a¡ b = u2 ¡ v2 = (u¡ v)(u+ v)

Select

u¡ v = a¡ bu+ v = 1:

Solve for u; v. Hope x comes out to be plus. Else factor di®erently.

Example. Solve

x+ 3 = u2

x+ 2 = v2

1 = u2 ¡ v2 = (u¡ v)(u+ v) = 14£ 4:

Take

u¡ v = 1=4u+ v = 4

2u = 17=4! u = 17=8

2v = 15=4! v = 15=8

! x = 97=64:

Diophantus 8

Note: The factorization 12£ 2 above yields a negative x.

? From Book III come the quadratic systems.

(x+ y + z)2 ¡ x2 = u2; (x+ y + z)2 ¡ y2 = v2;(x+ y + z)2 ¡ z2 = w2

and

x+ y + z = t2; y + z ¡ x = u2;z + x¡ y = v2; x+ y ¡ z = w2

and

yz + x2 = u2; zx+ y2 = v2;

xy + z2 = w2

? From Book IV come the examples.

x2 + y2 + z2 = (x2 ¡ y2) + (y2 ¡ z2) + (x2 ¡ z2)x2 + y2 + z2 + w2 ¡ (x+ y + z + w) = a

and the quadratic type system

x2 + y = u2 x+ y = u

x2 + y = u x+ y = u2

? From Book IV we also have the cubic type systems

x2y = u; xy = u3

and

x3 + y2 = u3; z2 + y2 = v2

x3 + y3 = x+ y

Example. Solve x3 + y = (x+ y)3.

Diophantus 9

Solution. (Method of False Position.) Assume x = 2y. Thus

8y3 + y = (3y)3 = 27y3

y = 19y3

19y2 = 1

But 19 is not a square!! Retracing steps, note that 19 = 33 ¡ 23,and 3 comes from the assumption x = 2y. Hence, we need to ¯ndtwo consecutive numbers such that the di®erence of their cubes isa square. That is, we will take x = zy, where z is chosen so that(z + 1)3 ¡ z3 is a perfect square. Thus,

(z + 1)3 ¡ z3 = 3z2 + 3z + 1 ´ (1¡ 2z)21 + 4z2 ¡ 4z

Or

z2 ¡ 7z = 0z(z ¡ 7) = 0:

Soz = 7:

Take x = 7y. It follows that 343y3+y = (8y)3, or 169y2 = 1. Hencey = 1=13 and x = 7=13.

3.4 Other Problems

The Method of Limits It is desired to ¯nd a power xn between twogiven numbers a and b.

To solve this problem Diophantus multiplies a and b bypowers 2n; 3n; : : : until some nth power, cn lies betweenapn and bpn. Then he sets x = c=p as it is easily seen thatxn = cn=pn lies between a and b.

More Method of Limits. Divide a number into a sum of squareseach one of which satis¯es some property.

Example 1. Divide 13 into the sum of two squares, each of which isgreater than 6.

Diophantus 10

Example 2. Divide 10 into the sum of three squares, each of whichis greater than 3.

The work of Diophantus has attracted mathematicians for thelast two millenia. No diminution of e®ort has occurred. Indeed,solving polynomial equations for integer solutions is now a majorarea of mathematics usually included within analytic number the-ory, with countless applications. It includes some of the deepestand most di±cult mathematics being done today. For example, theso-called \Fermat's last theorem" is among the many Diophantineequations whose solutions were extraordinarily di±cult to decide.

4 Modern Diophantine Equations.

Definitions. A polynomial Diophantine equation is an equation ofthe form

(1) P (x1; x2; : : : ; xm) = 0

where m 2 Z+ and P 2 Z[x1; x2; : : : ; xm], i.e. P is a polynomialwith integer coe±cients. The xi, i = 1; : : : ;m are assumed to beintegers. If P (a1; a2; : : : ; am) = 0 for ai 2 Z, i = 1; : : : ;m, we saythat (a1 : : : am) is a solution of (1).

Examples.

y3 = x2 + 999 Solution: (1,10)

x21 + x22 + x

23 = 7 no solutions

There are two types of questions normally asked: descriptiveand quantitative. Consider the Diophantine equation

x21 + x22 + x

23 + x

24 = n

The descriptive question asks for a solution in integers for a givenand ¯xed n. The quantitative question asks for the number of solu-tions.

Here's a typical theorem in the theory of Diophantine Equa-tions.

Diophantus 11

Theorem. If a; b; c; d and e are not zero and not all of the same sign,there exist integral solutions, not all zero, of

ax2 + by2 + cz2 + du2 + ev2 = 0:

However, the same theorem for

ax2 + by2 + cz2 + du2 = 0

must have additional conditions. ( Namely, two of a; b; c; d must notbe even and

1

4abcd = 5(mod 8)

.

Theorem. If k is any integer, then

k2 = 0 or 1(mod4):

Corollary. For any two integers u and v,

u2 + v2 = 0; 1; or 2(mod4):

Therefore, no integer congruent to 3(mod 4) can be written as thesum of two squares.

Theorem. (Lagrange) Every positive integer can be written as thesum of four squares. (Zero is admissible as one of the squares.)

Theorem. If a; b; c have no common factor > 1, all integral solutionsof ax+ by + cz = 0 are given by

x = bk ¡ cn y = cs¡ ak z = an¡ bs;where s; n, and k are integers.

A famous example. Consider the polynomial

P (x; y; z) = xn + yn ¡ zn:

Diophantus 12

If n = 2, solutions of P (x; y; z) = 0 are Pythagorean triples forwhich it can be shown that, if primitive, it must have the form

x = m2 ¡ n2 y = 2mn; z = m2 + n2:

If n ¸ 3 Fermat conjectured and in 1995 Andrew Wiles provedthat there can be no integer solutions.

Recall the polynomial:

P (x1; x2; : : : ; xn) = 0:

A. Hilbert (10th problem). \Is there an algorithm for determiningwhether or not a given Diophantine equation (polynomial) has asolution?" (1902)

Answer. No. J. Robinson (1952) Mitijasevi¶c (1970).

B. If we know there is a solution, then we can ¯nd it by applying allm-tuples and testing them.

C. To ¯nd algorithms for restricted classes of Diophantine equations

yes, n = 1 Greeks n = 3 A. Bakerfor n = 2 Gauss

D. Is there an upper bound on the number of solutions? Or arethere an in¯nite number?

Compare: x2 + y2 = z2 and x3 + y3 = z3

Note. The value of finding only some solutions of a fixed Diophantineequation is usually rather small.

Diophantine approximation The approximation of irrationals by ra-tionals is one problems characteristic to the ¯eld of Diophantineapproximation. For example, as is well known any irrational ® isapproximable by in¯nitely many rationals h

k. Thus

j®¡ hkj < ²

Diophantus 13

has in¯nitely many solutions for every ². But how small can this bein terms of the rational approximants? Can we have

j®¡ hkj < 1

k²

which means that jk®¡ hj < ². Can we have

j®¡ hkj < 1

k2²

which means that jk® ¡ hj < 1k². The answer is contained in the

following theorem.

Theorem. For any irrational ® there exist in¯nitely many rationalshksuch that

j®¡ hkj < 1p

5k²:

No number greater thanp5 can replace the

p5 above.

For algebraic numbers, there are more general versions of thistheorem pertaining to the zeros of polynomial with integer coe±-cients.

Another type of result is this:

Theorem. For any irrational ® the numbers ®; 2®; 3®; : : : are uni-formly distributed modulo 1.

Recall that a sequence ®1; ®2; ®3; : : : is uniformly distributed overan interval I if for every subinterval J , the number of elements ofthe sequence ®1; ®2; : : : ; ®n that are in J , denoted by n(J), satis¯es

limn!1

n(J)

n=jJ jjIj

where jJ j is the length of J .

August 27, 2000

Pappus of Alexandria, (fl. c. 300-c. 350)!

1 Introduction

Very little is known of Pappus' life. Moreover, very little is knownof what his actual contributions were or even exactly when he lived.We do know that he recorded in one of his commentaries on theAlmagest2 that he observed a solar eclipse on October 18, 320. He isregarded, though, as the last great mathematician of the HelenisticAge. At this time higher geometry was in complete abeyance untilPappus. From his descriptions, we may surmise that either theclassical works were lost or forgotten. His self-described task is to`restore' geometry to a place of signi¯cance.

2 Pappus’ Work

Toward this end wrote The Collection or The Synagogue, an extanttreatise on geometry which we discuss here and several commen-taries, now all lost except for some fragments in Greek or Arabic.One of the commentaries, we note from Proclus, was on The Ele-ments. Basically, The Collection is a treatise on Geometry, whichincluded everything of interest to him. Whatever explanations orsupplements to the works of the great geometers seemed to himnecessary, he formulated them as lemmas.

The ¯rst published translation into Latin was made by Com-mandinus in 1589. Others including Eisenmann, John Wallis addedto the translations. Friedrich Hultsch gave the de¯nitive Greek textwith Latin translation in 1876-8.

Features:

² It is very broad, designed to revive classical geometry.1 c°2000, G. Donald Allen2Claudius Ptolemy(100-178 AD) wrote the Mathematical Collection, later called The Almagest, from

the Arabic ‘al-magisti’ meaning “the greatest.”

Pappus 2

² It is a guide or handbook to be read with the Elements andother original works.

² Alternative methods of proof are often given.² The work shows a thorough grasp of all the subjects treated,independence of judgment, mastery of technique; the style isterse but clear. Pappus is an accomplished and versatile math-ematician.

² The range of names of predecessors is immense. In some cases,our only knowledge of some mathematicians is due to his ci-tation. Among many others he mentions Aristaeus the elder,Carpus of Antioch, Conon of Samos, Demetrius of Alexandria,Geminus, Menelaus, and of course the masters.

Summary of Contents:

² Book I and ¯rst 13 (of 26) propositions of Book II. Book IIwas concerned with very large numbers { powers of myriads(i.e. 10,000).

² Book III begins with a summary of ¯nding two mean propor-tionals (a : x = x : y = a : y) between two straight lines. Inso doing, he gives the solutions of Eratosthenes, Nicomedes,and Heron. He adds solutions of his own that resembles closelythat of Eutocius of Sporus. He also de¯nes plane problems,solid problems, and linear problems, of which the two meanproportionals problem is of the latter type. Pappus

{ Distinguishes (1) plane problems, solvable with straightedge and compass

{ Distinguishes (2) solid problems, requiring the conics forsolution, e.g. solving certain cubics.

{ Distinguishes (3) linear problems, problems invoking spi-rals, quadratrices, and other higher curves

{ Gives a constructive theory of means. That is, given anytwo of the numbers a; b; c and the type of mean (arithmetic,geometric, or harmonic), he constructs the third.

Pappus 3

{ Describes the solution of the three famous problems of an-tiquity, asserts these are not plane problems » 19th cen-tury.

{ Treats the trisection problem, giving another solution in-volving a hyperbola and a circle.

{ Inscribes the ¯ve regular solids in the sphere.

In this book Pappus goes to some length to distinguish theo-rems from problems. Citing the apparent fact that his prede-cessors combined them as one, he separates those statementscalling for a construction as problems, and statements that callupon hypotheses to draw implications as theorems. It is not in-cumbent on the problem originator to know whether or not theconstruction can be made. It is the solver's task to determineappropriate conditions for solution.

Pappus goes to some length in his study of the three claasicmeans of anticuity, the arithmetic, the geometric, and the har-monic. Recall the chapter on Pythagoras

a : c = a¡ b : b¡ c Harmonica : a = a¡ b : b¡ c Arithmetica : b¡ a¡ b : b¡ c Geometric

where b is the designated mean of a and c. He o®ers geometricsolutions of each. Precisely, for any two of the quantities heconstructs the third.

² Book IV covers a variety of geometrical propositions. Foremostit contains an extention of theorem of Pythagorus for parallel-ograms constructed on the legs of any triangle. This result hasitself a variety of generalization, and seems to reveal the essenceof the Pythagorean theorem itself.

Pappus 4

Generalization of Pythagorean’s Theorem If ABC is a triangleand on AB;AC any parallelograms are drawn as ABDE andACFG, and if DE and FG are extended to H and HA bejoined to K. Then BCNL is a parallelogram and

area ABDE + area ACFG = area BCNL

.

N

A

B C

D

E

F

G

H

L M

KProof. The proof is similar to the original proof of the Pythagorean the-orem as found in The Elements. First, BL and NC are de¯nedto be parallel to HK. BLHA is a parallelogram and CAHMis a parallelogram. Hence BCNL is a parallelogram.

By \sliding" DE to HL, it is easy to see that

area BDEA = area BLHA;

and by sliding HA to MK it follows that

area BKML = area BLHA:

Thusarea BDEA = area BKML:

Similarly,area ACFG = area KCNM:

Putting these conclusions together gives the theorem

area ABDE + area ACFG = area BCNL:

Pappus 5

Note. Both parallelograms need not be drawn outside ABC.

Also in Book IV we ¯nd material about the Archimedian spiral,including methods of ¯nding area of one turn | di®ers fromArchimedes.

He also constructs the conchoid of Nicomedes. In addition,he constructs the quadratix in two di®erent ways, (1) usinga cylindrical helix, and (2) using a right cylinder, the base ofwhich is an Archimedian spiral.

He considers the three problems of antiquity, alluding to themas \solid" problems.3 He o®ers two solutions of the trisectionproblem, both involving the use of hyperbolas.

² Book V Here we see in the introduction his comments on thesagacity of bees. This statement on the bees celebrates thehexagonal shape of their honeycombs.

[The bees], believing themselves, no doubt, to be en-trusted with the task of bringing from the gods to themore cultured part of mankind a share of ambrosia inthis form,: : : do not think it proper to pour it carelesslyinto earth or wood or any other unseemly and irregularmaterial, but, collecting the fairest parts of the sweetestflowers growing on the earth, from them they preparefor the reception of the honey the vessels called honey-combs, [with cells] all equal, similar and adjacent, andhexagonal in form.That they have contrived this in accordance with a

certain geometrical forethought we may thus infer. Theywould necessarily think that the figures must all be ad-jacent one to another and have their sides common, inorder that nothing else might fall into the interstices andso defile their work. Now there are only three rectilinealfigures which would satisfy the condition, I mean regularfigures which are equilateral and equiangular, inasmuchas irregular figures would be displeasing to the bees: : : .[These being] the triangle, the square and the hexagon,

3An obvious conclusion here is that by this time the ancients generally believed that no classical compassand straight edge constructive solution of these problems was possible.

Pappus 6

the bees in their wisdom chose for their work that whichhas the most angles, perceiving that it would hold morehoney than either of the two others.Bees, then, know just this fact which is useful to them,

that the hexagon is greater than the square and the tri-angle and will hold more honey for the same expenditureof material in constructing each.

In his recounting of various propositions of Archimedes On theSphere and Cylinder he gives geometric proofs of the two famil-iar trigonometric relations of which the most familiar is

sin(x+ y) + sin(x¡ y) = 2 sinx cos y

He also reproduces the work of Zeodorus on isoperimetric ¯g-ures. He includes the following result.Proposition. Of all circular segments having the same circum-ference the semicircle is the greatest.

On solids we ¯nd the thirteen semi-regular solids discoved byArchimedes. We also see a number of isoperimetric results suchasProposition. The sphere is greater than any of the regular solidswhich has its surface equal to that of the sphere.The proof is

similar to that of Zenodorus. He also shows

Proposition. Of regular solids4 with surfaces equal, that isgreater which has more faces.

² Book VI determines the center of an ellipse as a perspective ofa circle. It is also astronomical in nature. It has been called the\Little Astronomy". It covers optics { re°ection and refraction.

² Book VII, the `Treasury of Analysis' is very important becauseit surveys a great number of works on geometric analysis ofloci, nearly all of which are lost. Features:

{ The Book begins with a de¯nition of analysis and synthesis.4Recall, the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron are the five regular

solids. And there are no more, as established during the time of Plato by Theaetetus

Pappus 7

¤ Analysis, then takes that which is sought as if it wereadmitted and passes from it through its successive con-sequences to something which is admitted as the resultof systhesis. Unconditional controvertability required.

¤ In Synthesis, reversing the process, we take as alreadydone that which was last arrived at in the analysis and,by arranging in their natural order as consequenceswhat before were antecedents, and successively con-necting them one with the other, we arrive ¯nally atthe construction of what was sought.

{ A list of the books forming the `treasury' is included, to-gether with a short description of their contents.

{ As an independent contribution Pappus formulated thevolume of a solid of revolution, the result we now call theThe Pappus { Guldin Theorem. P. Guldin (1577-1643)

{ Most of the remaining of the treatise is collections of lem-mas that will assist the reader's understanding of the orig-inal works.

Pappus also discusses the three and four lines theorem of Apol-lonius.

² Succinctly, given three lines: Find the locus of points forwhich the product of the distances from two lines is the square ofthe distance of the third. (The solution is an ellipse.)

² Given four lines: Find the locus of points for which theproduct of the distances from two lines is the product of the distanceof the other two.

Pappus 8

Pappus’ Theorem.

Volume of revolution = (area bounded by the curve)£ (distance traveled by the center of gravity)

Center ofGravity

a b

y

Volume of revolution V =R ba ¼f

2(x)dx

Area bounded by the curve:

A =Z b

af(x)dx

The ¹y center of gravity:

¹y =12

R ba yf(x)dxR ba f(x)dx

=

R ba f

2(x)dxR ba f(x)dx

Pappus:V = 2¼¹yA

Pappus’, on the Pappus-Guldin Theorem

‘Figures generated by a complete revolution of a plane figure aboutan axis are in a ratio compounded (1) of the ratio of the areas of thefigures, and (2) of the ratio of the straight lines similarly drawn to (i.e.drawn to meet at the same angles) the axes of rotation from the respec-tive centres of gravity. Figures generated by incomplete revolutionsare in the ratio compounded (1) of the ratio of the areas of the figuresand (2) of the ratio of the arcs described by the centres of gravity

Pappus 9

of the respective figures, the latter ratio being itself compounded (a)of the ratio of the straight lines similarly drawn (from the respectivecentres of gravity to the axes of rotation) and (b) of the ratio of theangles contained (i.e. described) about the axes of revolution by theextremities of the said straight lines (i.e. the centres of gravity).’

Pappus’ theorem surface area.

Pappus 10

3 End Game | The End of the Greek School

Following Pappus was no mathematician with abilities of the greatmasters. The school at Alexandria was diminished with only anoccasional bright star yet to shine. The world was turning to Chris-tianity, which at that time in no way resembled what it would be-come centuries later. To the Christians, the ancient schools werepagan, and paganism must be destroyed.

3.1 Theon of Alexandria

Theon of Alexandria (c. 390) lived toward the end of the periodwhen Alexandria was a center of mathematics. His most valuablecontributions are his commentaries on various of the masterpieces,now growing old in libraries. One commentary was on Prolemy'sSyntaxis in eleven books. In it we learn of the Greek use of sexages-imal fractions, arithmetic, and root extraction. Theon also wrotecommentaries on Euclid's Optics.

More signi¯cantly, Theon wrote commentaries on Euclids's El-ements. It appears that his e®ort was not directed toward the pro-duction of an authoritative and accurate addition. (Remember, 700years have past since it was written.) Rather, Theon seems to havebeen intent on making what he regarded as improvements. Accord-ing to Heath5, he made alterations where he thought mistakes ap-peared, he altered some passages too hastily, he made emendationsto improve the linguistic form of Euclid, he added explanations tothe original, by adding or altering propositions as needed, and headded intermediate steps to Euclid's proofs to assist student under-standing. In summary, his intent was to make the monumental workmore accessible. We know all of this only because of the discoveryof the earlier non-Theonine edition in the Vatican.

5Thomas Heath, A History of Greek Mathematics, II, Dover, New York, 1921 page 527. This is areprinting of the original published by the Clarendon Press, Oxford in 1920.

Pappus 11

3.2 Hypatia

(c. 370-418) By 397 Christianity became the state religion of theRoman empire and paganism \pagamni" was banned. The Alexan-drian school was considered a center of pagan learning and becameat risk. Hypatia, daughter of Theon of Alexandria, became a leaderof the neoplatonic school6 and was so eloquent and persuasive inher beliefs that she was feared a threat to Christianity. In conse-quence, she was slain in 418 by a fanatical mob led by the Nitrianmonks when she refused to repudiate her beliefs. Some accountsargue that this may have resulted because of a dispute between theRoman prefect Orestus and the patriarch bishop Cyrillus (later St.Cyril. Regardless of the cause of her death, the surrounding eventsand political hostility resulted in the departure of many scholarsfrom Alexandria.

Her mathematical contributions are not well known and indeedare all lost. It has been surmised by statement of Suidas thatshe wrote commentaries on Diophantus' Arithmetica, Apollonius'On Conics, and possibly on Ptolemy astronomical works. AfterHypatia, Alexandrian mathematics came to an end, though thereis evidence that little remained at this late date. For further re-sources on Hypatia, see http://www-groups.dcs.st-and.ac.uk/ his-tory/Mathematicians/Hypatia.html.

3.3 Eutocius of Ascalon

Eutocius (c. 480 - c. 540), likely a pupil of Ammonius in Alexandria7,probably became head of the Alexandrian school after Ammonius.There is no record of any original work by Eutocius. However, his

6One unfortunate tenet of neoplatonism, the last school of Greek philosophy, was its declaration ofideological war against the Christians. Created by the great philosopher Plotinus, and carefully editedand promoted by Porphyry (c. 234 - c. 305), neoplatonism featured an extreme spiritualism and a greatersympathy with the less sharply defined hierarchies of the Platonists. Porphyry, who incorporated Aristotle’slogic into neoplatonism, also attacked Christian doctrines on both a philosophical and exegetical basis.Antagonism developed. Interestingly, Porphyry, Iamblichus (c. 250 - c. 330) , and much later Proclus (410 -485) were all prominant neoplatonists and more importantly (for us) were capable mathematicians. Porphyrywrote a commentary on on the Elements Iamblichus wrote a commentary on Nicomachus’ Introductioaritmetica, and Proclus we will study in the next section. Interestingly, until modern times, mathematicianswere often also philosophers or clerics. One of the deepest philosopher-mathematicians was René du PerronDescartes (1596 - 1650).

7who inturn was a student of Proclus

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commentaries contain much historical information which might oth-erwise have been completely lost. In particular, he wrote commen-taries of Archimedes' On the sphere and cylinder that inspired somebrief interest in his great mathematical works resulting in transla-tions into a more familiar dialect and more suitable for students. Asimilar result held for his commentaries of Apollonius' On Conics.

3.4 Athens

Ironically, the center of Greek mathematics returned to Greece forthe ¯rst time in nearly 1000 years. The Academia of Plato, whichhad access to it own ample ¯nancial means, maintained itself for alonger time. Proclus Diadochus (411 - 485), though he studied fora brief time in Alexandria, later moved to Athens where he wrote amost important work, namely his commentaries on Elements, Book I.Because of the wealth of information he included, it is now one of ourmain sources of information on the history of geometry. Evidently,Proclus had access to a library of considerable resources includingthe History of Geometry by Eudemus (°. 335) and other great works.Proclus, a neoplatonist, was venerated even in his own time as beinga man of great learning.

After came Isidore of Alexandria and Damascius of Damascus.They were heads of the school. There was also Simplicius, whowrote commentaries on Aristotle. But in 529, on the order of theEmperor Justinian, the school of Athens, the last rampart of thepagan world, was closed.

The last center of Greek culture was Constantinople. Here livedIsidore of Milete and Anthemius of Tralles, both architects andmathematicians. It was probably Isidore who added the so-called15th Book of the Elements, which contains propositions on regularpolyhedra. At least, the propositions were probably his. After theselast °utterings, the history of Greek mathematics died.

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4 The Decline of Greek Mathematics

Why did mathematics decline so dramatically from the Golden Age?No doubt an entire chapter could be devoted to this topic, evenbooks with carefully crafted answers could be o®ered up. However,among the main points that could be argued are these:

² There were always only a few that could a®ord to spend theirlives pursuing mathematics. Mathematics, in particular geome-try, was clearly now at a level that demanded professional prac-titioners. It would have to spring in new directions to gatheraround it a loyal cadre of dedicated amateurs that could sustainthe few professionals and feed their numbers.

² The teaching tradition diminished partly due to the politicalstrife around the eastern Mediterranean.

² Roman in°uence (never inclined to mathematics) was impor-tant.

² Arab hegemony { destruction of the library of Alexandria { andof the seat of learning.

² Christian intolerance and the unfortunate classi¯cation of math-ematics as a \pagan art".

allen, donald - the origin of greek mathematics

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