ancient greek mathematicsndonalds/math184/greece.pdf · later greek mathematicians, in particular...

21
Ancient Greek Mathematics The Greek Empire • Ancient Greek civilization generally accepted to date from around 800 BC. Primarily centered on the Aegean Sea (between modern-day Greece and Turkey) containing hundreds of islands and loosely affiliated city-states. • Many wars between city-states other empires (e.g. Persians). • By 500 BC covered much of modern Greece, the Aegean and southern Italy. As a trading/sea- faring culture, built/captured city-states (colonies/trading-outposts)all around the north and east coast of the Mediterranean from Spain round the Black Sea and Anatolia (modern Turkey) to Egypt. • Alexander the Great (356–323 BC) extended empire around the eastern Mediterranean inland capturing mainland Egypt and then east to western India and Babylon where he died. • Eventually becomes part of the Roman Empire c.146 BC though Romans left Greek largely essentially intact apart from crushing several rebellions. • Greek civilization flourished even as the Rome collapsed, continuing as part of the Byzantine Empire. Map of Greek Empire c.500 BC from timemaps.com Ancient Greece is important for far more than just mathematics and one course cannot begin to do justice to it. Much of modern western thought and culture including philosophy, art logic and science has roots in Ancient Greece. While undeniably important, western culture has often over- emphasized the role of the Greeks and downplayed the contribution of other cultures to our inherited knowledge.

Upload: others

Post on 26-Jan-2020

8 views

Category:

Documents


0 download

TRANSCRIPT

  • Ancient Greek Mathematics

    The Greek Empire

    • Ancient Greek civilization generally accepted to date from around 800 BC. Primarily centeredon the Aegean Sea (between modern-day Greece and Turkey) containing hundreds of islandsand loosely affiliated city-states.

    • Many wars between city-states other empires (e.g. Persians).

    • By 500 BC covered much of modern Greece, the Aegean and southern Italy. As a trading/sea-faring culture, built/captured city-states (colonies/trading-outposts)all around the north andeast coast of the Mediterranean from Spain round the Black Sea and Anatolia (modern Turkey)to Egypt.

    • Alexander the Great (356–323 BC) extended empire around the eastern Mediterranean inlandcapturing mainland Egypt and then east to western India and Babylon where he died.

    • Eventually becomes part of the Roman Empire c.146 BC though Romans left Greek largelyessentially intact apart from crushing several rebellions.

    • Greek civilization flourished even as the Rome collapsed, continuing as part of the ByzantineEmpire.

    Map of Greek Empire c.500 BC from timemaps.com

    Ancient Greece is important for far more than just mathematics and one course cannot begin todo justice to it. Much of modern western thought and culture including philosophy, art logic andscience has roots in Ancient Greece. While undeniably important, western culture has often over-emphasized the role of the Greeks and downplayed the contribution of other cultures to our inheritedknowledge.

    http://www.timemaps.com

  • Mathematics and Philosophical Development

    • Inquiry into natural phenomena encouraged through the personification of nature (sky = man,earth = woman) which pervaded early religion.

    • By 600 BC ‘philosophers’ were attempting to describe such phenomena in terms of naturalcauses rather than being at the whim of the gods. For example, all matter was suggested to becomprised of the four elements (fire, earth, water, air).

    • Development of Mathematics linked to religion (mysticism/patterns/assumption of perfec-tion in the gods’ design), philosophy (logic) and natural philosophy (description of the naturalworld). Mathematics elevated from purely practical considerations to an extension of logic: theGreeks were unhappy with approximations even when such would be perfectly suitable forpractical use. Led to the development of axiomatics and proof.

    • Limited extant mathematics from pre-300 BC. Most famous work is Euclid’s Elements c.300 BC,indisputably the most important mathematical work in western mathematics and a primarytextbook in western education until the early 1900’s. Probably a compilation/editing of earlierworks, its importance meant other works were sacrificed and sometimes subsumed by it.

    • The word theorem (theory, theorize, etc.) comes from the Greek theoreo meaning I contemplate. Atheorem is therefore an observation based on contemplation.

    • Later mathematics included Ptolemy’s Almagest c.150 AD on Astronomy and the fore-runnerof trigonometry: basis of western astronomical theory until the 1600’s.

    Enumeration

    The ancient Greeks had two primary forms of enumeration, both developed c.800–500 BC.

    Attic Greek (Attica = Athens): Strokes were used for 1–4. The first letter of the words for 5, 10, 100,1000 and 10000 denoted the numerals. For example,

    • πeντe (pente) is the Greek word for five, whence Π denoted 5.

    • δeκα (deca) means ten, so ∆ = 10.

    • H (hekaton), X (khilias) and M (myrion/myriad) denoted 100, 1000 and 10000 respectively.

    • Combinations were used, e.g. ∆∆ΠΠ||| = 223.The construction of large numbers was very similar to the more familiar Roman numeral system.

    Ionic Greek (Ionia = middle of Anatolian coast): the alphabetdenoted numbers 1–9, 10–90 and 100–900 in the same wayas Egyptian hieratic numerals were formed. The alphabetdiffers from modern Greek due to three archaic symbols ϛ,ϙ, ϡ (stigma, qoppa, sampi).Larger numbers used a left subscript to denote thousandsand/or M (with superscripts) for 10000, as in Attic Greek.For example,

    35298 =,λ,εσϙη =γ

    M,εσϙη

    1 α 10 ι 100 ρ2 β 20 κ 200 σ3 γ 30 λ 300 τ4 δ 40 µ 400 υ5 ε 50 ν 500 φ6 ϛ 60 ξ 600 χ7 ζ 70 o 700 ψ8 η 80 π 800 ω9 θ 90 ϙ 900 ϡ

    2

  • Eventually a bar was placed over numbers to distinguish them from words (e.g. ξθ = 89). Modernpractice is to place an extra superscript (keraia) at the end of a number: thus 35298 = ͵λ͵εσϙηʹReciprocals/fractions were denoted with accents: e.g. θ́ = 19 . The use of Egyptian fractions persistedin Europe into the middle ages.

    Both systems were fine for record-keeping but terrible for calculations! Later Greek mathematicians,in particular Ptolemy, adapted the Babylonian sexagesimal system for calculation purposes thus ce-menting the use of degrees in astronomy and navigation.

    Early (pre-Euclidean) Greek Mathematics

    Euclid’s Elements forms a natural breakpoint in Greek mathematical history; almost everything thatcame before the Elements was eventually swallowed by it. Pre-Euclidean mathematics is thereforelargely a discussion of the origins of some of the ideas in Euclid.

    Thales of Miletus (c.624–546 BC)

    Often thought of as the first western scientist, Thales is also important in mathematics.

    • Olive Trader based in Miletus, a city-state in Anatolia.

    • Contact with Babylonian traders/scholars probably led to his learning some geometry and anattempt to organize his discoveries.

    • Stated some of the first abstract propositions: in particular,

    – The angles at the base of an isosceles triangle are equal.

    – Any circle is bisected by its diameter.

    – A triangle inscribed in a semi-circle is right-angled (still known as Thales’ Theorem).

    • Proofs not forthcoming. The major development was the statingof abstract general principles. Thales’ propositions concern alltriangles, circles, etc. The Babylonians and Egyptians were merelyobserved to use certain results in calculations and gave no indica-tion that they appreciated the general nature of their results.

    • Mathematical reasoning, if he conducted such, was almost cer-tainly visual. For example, by 425 BC, Socrates could describe howto halve/double the area of a square by joining the midpoints ofedges.

    • Thales arguably more important to the history of reasoning: offered arguments/ discussionsconcerning the ‘stuff’ of which the universe is made.

    3

  • Pythagoras of Samos c.572–497 BC

    • Much travelled (Egypt, Asia, Babylon, Italy) though his story was probably over-emphasisedafter his death. Eventually settled in Croton (southeast Italy) where he founded a school/cult,persisting over 100 years after his death. Mathematical results/developments came from thegroup collectively.

    • More of a mystic/philosopher than a mathematician. Core belief that number is fundamentalto nature. Motto: “All is number”. Emphasised form, pattern, proportion.

    • Pythagoreans essentially practiced a mini-religion (they were vegetarians, belived in the trans-migration of souls, etc.).

    • The following quote1 helps give a flavor of the Pythagorean way of life.

    After a testing period and after rigorous selection, the initiates of this order were al-lowed to hear the voice of the Master [Pythagoras] behind a curtain; but only aftersome years, when their souls had been further purified by music and by living inpurity in accordance with the regulations, were they allowed to see him. This pu-rification and the initiation into the mysteries of harmony and of numbers wouldenable the soul to approach [become] the Divine and thus escape the circular chainof re-births.

    Several famous results are attributable to the Pythagoreans. They were particularly interested inmusical harmony and the relationship of such to number. For instance, they related intervals inmusic to the ratios of lengths of vibrating strings:

    • Identical strings whose lengths are in the ratio 2:1 vibrate an octave apart.

    • A perfect fifth corresponds to the ratio 3:2.

    • A perfect fourth corresponds to the ratio 4:3.

    Using such intervals to tune musical instruments (in particular pianos) is still known as Pythagoreantuning.

    Theorems 21–34 in Book IX of Euclid’s Elements are Pythagorean in origin:

    Theorem (IX.21). A sum of even numbers is even.

    Theorem (IX.27). Odd less odd is even.

    The Pythagoreans studied perfect numbers: equal to the sum of their proper divisors (e.g. 6 =1 + 2 + 3). They seem to have observed the following, though it is not known if they had a proof.

    Theorem (IX.36). If 2n − 1 is prime then 2n−1(2n − 1) is perfect.

    They also considered square and triangular numbers and tried to express geometric shapes as num-bers, all in service of their belief that all matter could be formed from the combination of basic shapes.Cultish the Pythagoreans may have been, but they discovered many things and certainly had loftygoals!

    1Van der Waerden, Science Awakening pp 92–93

    4

  • Length, Number, Incommensurability and Pythagoras’ Theorem

    Our modern notion of continuity facilitates tight relationship between length and number: any objectcan be measured with respect to any fixed length. For instance, we’re happy stating that the diagonalof a square is

    √2 times the side. To the Greeks and other ancient cultures, the only numbers were

    positive integers. Appreciating the distinction between length and number is crucial to understandingseveral of the major ideas of Greek mathematics. In particular, it helps explain the primacy of Ge-ometry in their mathematics: lengths are real things that the Greeks wanted to compare using numbers.

    A core Pythagorean belief was that of commensurability: given two lengths, there exists a sub-lengthdividing exactly into both. One could then describe the relationship between lengths with a ratio.For instance, if the longer length contained three of the sub-lengths and the smaller two, this couldbe expressed with the ratio 3 : 2. In modern language,

    ∀x, y ∈ R+, ∃k, l ∈N, r ∈ R+ such that x = kr and y = lr

    This is complete nonsense for it insists that every ratio of real numbers is rational!The supposition of commensurability clearly fits with the Pythagoreans’ mystical emphasis on theperfection of number. The discovery that it was false produced something of a crisis. A (possibly)apocryphal story suggests that a disciple named Hippasus (c.500 BC) was set adrift at sea as punish-ment for revealing it. Nevertheless, it is generally accepted that the Pythagoreans provided the firstevidence of the existence of irrational numbers in the form of incommensurable lengths.

    By 340 BC, Aristotle was happy to state that incommensurable lengths exist.

    Theorem (Aristotle). If the diagonal and side of a square are commensurable, then odd numbers equal evennumbers.

    Inferred proof—original unknown. Consider Socrates’ doubled-square from earlier. Label the sides of the blue square a and thediagonal b where a and b are integers denoting multiples of thecommon sub-length (the existence of a common sub-length is thehypothesis!).

    We may assume that at least one of a or b is odd, for otherwisethere exists a larger common sub-length. However the largersquare (side b) is twice the smaller (side a) so b is even and thesquare number b2 is divisible by four. But then a is also even.

    Whichever of a, b was odd is also even: contradiction!

    b

    a a

    Note the similarity of this argument2 to the standard modern proof of the irrationality of√

    2

    2For those with musical training, a similar argument shows that the Pythagorean notion of perfect fifths in music is alsoflawed. The cycle of fifths is a musical principal stating that an ascension through twelve perfect fifths takes you throughevery note in the standard chromatic scale, finishing seven octaves above where you start. This is essentially a claim that

    27 =(

    32

    )12⇐⇒ 219 = 312: palpable nonsense!

    5

  • While there is no evidence that the Pythagoreans ever provided a correct proof of their famous The-orem, one argument possibly attributable to the Pythagoreans used the idea of commensurability.

    ‘Proof’ of Pythagoras’ Theorem. Label the right triangle a, b, cwhere c is the hypotenuse and drop the altitude to the hy-potenuse. Let d be the length of the a-side of the hypotenuse.Similar triangles tell us that

    a : d = c : a =⇒ a2 : ad = cd : ad =⇒ a2 = cd (∗)

    Thus the square on a has the same area as the rectangle belowthe d-side of the hypotenuse. Repeat the calculation on the otherside to obtain b2 = c(c− d). Now sum these for the proof.

    In the language of the Pythagoreans, the only acceptable numberswere integers, so the symbols a, b, c, d in were the integer multiplesof an assumed common sub-length. This restriction completelydestroys the generality of the proof.

    a b

    cd

    It is clear from the organization of Book I of Euclid’s Elements that one of its primary goals wasto provide a rigorous proof of Pythagoras’ Theorem which did not depend on the flawed notion ofcommensurability. With our modern understanding of real numbers, there is nothing wrong with theabove argument. We have fully adapted to the idea that length and number are freely interchangeabledue to the completeness of the real numbers: ancient mathematicians had no such knowledge.

    Zeno of Elea c.450 BC

    Zeno might be called the patron saint of devil’s advocates. His fame comes from his suggestionof several ingenious arguments/paradoxes involving the infinite and the infinitesimal. Here areperhaps the two most famous of these.

    • Achilles and Tortoise: Achilles starts a race behind a Tortoise. After a given time t1, Achillesreaches the Tortoise’s starting position, but the Tortoise has moved on. After another timeinterval t2, Achilles reaches the Tortoise’s second position: again the Tortoise has departed. Inthis manner Achilles spends an infinite sum of time intervals t1 + t2 + t3 + · · · in the chase.Zeno’s paradoxical conclusion: Achilles never catches the Tortoise.The problem was that Zeno refused to accept that the total duration could be finite, even thoughit be split into infinitely many subintervals of time. The resolution of this paradox is at the heartof the modern notion of infinite series.

    • Arrow paradox: An arrow is shot from a bow. At any given instant the arrow doesn’t move. Iftime is made up of instants, then the arrow never moves.This time Zeno is debating the the idea that a finite time period can be considered as a sum ofinfinitessimal instants. Again the problem is one of limits and infinite sums.

    Zeno’s paradoxes have stimulated philosophers for thousands of years, continuing well into the 18thcentury as the theory of calculus was fleshed out. We shall revisit this controversy later. For thepresent, it is enough to consider how radical the fundamental ideas of calculus are: to measure anarea one essentially slices it into infinitely many, infinitesimally thin strips before summing them up.Try selling this idea to someone who has never studied calculus!

    6

  • Theaetetus of Athens (417–369 BC)

    Theaetetus is probably the source for much of the most difficult part (Book X) of Euclid’s Elements.He took what is now known as the Euclidean Algorithm and applied it to lengths/magnitudes. Hereis the essential definition of (in)commensurability (which we see in Book VII).

    Definition. Suppose that a and b are (positive) magnitudes where a > b (i.e. a is longer than b).Apply the algorithm: in what follows only the quotients qk are integers, everything underlined is alength.

    a = q1b + r1 r1 < bb = q2r1 + r2 r2 < r1

    r1 = q1r2 + r3 r3 < r2Repeat the algorithm: one of two things happens.

    • The algorithm terminates with some remainder length rn dividing exactly into rn−1. In this wesay that a and b are commensurable with greatest common sub-length rn. Since rn divides in toboth original lengths, the sides can then be said to be in the ratio of the resulting divisors.

    • The algorithm never terminates and we say that a and b are incommensurable. The sequence ofquotients could then be used to describe the ratio of lengths.

    Here is an example of incommensurable magnitudes in the style of Theaetetus’. A regular pentagonABCDE is drawn. We sketch a proof that the side AB and diagonal AC are incommensurable.

    1. Prove that4BAG is isosceles.

    2. Take a = |AC| and b = |AB| = |AG|. The firstline of the algorithm now reads

    |AC| = |AG|+ |GC| = |AB|+ |GC|

    so we write a = q0b + r1 where r1 = |GC| andq1 = 1.

    3. Since |GC| = |AF|, the second line of the algo-rithm reads

    |AG| = |AF|+ |FG| = |GC|+ |FG|

    so that we again have a quotient of q2 = 1.

    A

    B

    C D

    EF

    G

    H

    J

    K

    4. Appealing to congruent isosceles triangles 4DCG ∼= 4EHF we see that |GC| = |FH| is thediagonal of the interior regular pentagon. The third line of the algorithm is therefore the sameas the first: we are back to considering the ratio of the diagonal to the side of a regular pentagon.The algorithm therefore continues forever and all quotients are 1.

    Eudoxus of Knidos c.390–337 BC

    Eudxus was arguably the most prolific of the pre-Euclidean mathmaticians. Apart from attendingand perhaps teaching at Plato’s academy, he is famous for explaining how to calculate with ratios oflengths. For example:

    7

  • Definition. A : B is greater than C : D if there are positive integers m, n such that mA > nB andmC ≤ nD.Indeed calculation (∗) on page ?? is an example of Eudoxus’ work. In this way he essentially de-scribed how to view ratios as numbers. At first glance this appears as if Eudoxus is simply telling ushow to work with rational numbers, but recall that A, B, C, D are lengths not integers. His mathematicsreally told the Greeks how to ‘get round incommensurability’ by approximating incommensurableratios with rational numbers.The majority of the mathematical work of these pre-Euclideans is preserved in Euclid, although it isoften difficult to decide which ideas are attibutable to whom.

    Education and Transmission pre-Euclid

    The earliest known Greek textbook/compilation is Elements of Geometry, written around 430 BC byHippocrates of Chios3 No copy survives, although most of its material probably made it into Book Iof Euclid.

    Around this time there were several Greek schools, mostly private and open only to men.4 Typicallyarithmetic was taught until age 14, followed by geometry and astronomy until age 18. The wordmathematics was used broadly at this time: mathemata in Greek merely means ‘things learned’ or ‘sub-jects of instruction’. As an example of how education hasn’t changed much in 2500 years, considerthe following quote from Isocrates (c.400 BC):

    If it does no other good, the teaching of mathematics keeps the young out of mischief andthe study of it helps to train one’s mind and sharpen one’s wits.

    The most famous scholars of ancient Greece were Socrates, Plato and Aristotle (each was the teacherhis successor).5 Their writings formed the core of the western philosophical tradition. Havinglearned much of his mathematics from a Pythagorean named Archytas, Plato indeed founded anacademy in Athens. The centrality of geometry to the Greek curriculum was evidenced by the fa-mous inscription above the entryway: “Let none ignorant of geometry enter here.” Indeed the wordgeometry is itself of ancient Greek origin, as are many of the words derived from it.

    Geo- γη̃ (gi) (pre-5th century BC). Could mean land, soil, earth or The Earth as a goddess Γη̃.

    -metry µέτρoν (metron). Could refer to a weight or measure, a dimension (length, width, etc.), or themetre (rhythm) in music.

    Related words: Gaia (earth-goddess), Geography (earth-write), Geology (Earth-study), Meter (bothverb and noun), Metronome (literally measure-law/division).

    Constructions and Geometry

    By the middle of the 5th century BC Greek mathematicians were solving geometric problems usingruler-and-compass constructions. The is idea of using a combination of a peg and cord to constructcircles and lines perhaps came to Greece from India or could have grown organically. Constructionswere based on three basic rules:

    3Not Hippocrates of Cos, of hippocratic oath fame!4The Pythagoreans were an outlier in accepting female students.5The birth of Socrates to the death of Aristotle covered 470–322 BC.

    8

  • 1. Given two points, one may draw a straight line segment joining them

    2. Any line segment may be indefinitely extended.

    3. One can draw a circle given its center and radius.

    Greek geometry, in large part, was based on constructing solutions to problems using sequences ofthese steps. This wasn’t just practical, it had a philosophical basis as well. A large proportion ofGreek theorems were stated as problems: e.g. to bisect a given angle. The theorem didn’t just tell youthat such could be done, the proof told you how to do it. Indeed by the late 400’s BC Greeks werealready making reference to the second and third of the three impossible constructions of antiquity:

    1. Trisecting a general angle.

    2. Doubling (the volume of) a given cube.

    3. Squaring the circle.

    It took until the 1800’s and the advent of field theory for proofs that these constructions were in-deed impossible using ruler-and-compass constructions. The belief that all geometric problems notonly could be solved, but could be solved constructively, drove many mathematicians to seek andhypothesize constructions for these problems.Euclid indeed provides constructions of many basic shapes, including how to construct an equilateraltriangle, a square and a regular pentagon in a given circle. Once these constructions are obtains, oncecan eaily double number of sides by bisection. Regular polygons with sides 3, 6, 12, 24,. . . , 4, 8,16,. . . , 5, 10, 20,. . . may then be constructed. Indeed one may also produce combinations: draw anequliateral triangle and a pentagon in the same circle to create a regular 15-gon.The Greeks were unable to construct regular n-gons with 7, 9, 11, 13, etc. sides. In 1796 Carl FriedrichGauss (then 19) proved that a regular 17-gon was constructible using algebra. Eventually in 1837 anecessary and sufficient condition for constructiblilty was proved:

    Theorem. A regular n-gon is constructible if and only if n = 2kF1 · · · Fr where F1, . . . , Fr are distinct primesof the form 2(2

    n) + 1.

    The next prime-sided constructible n-gon is the 257-gon correspoinding to n = 3.

    Here is a concrete construction of a regular pentagon in acircle:

    1. Draw perpendicular diameters AB and CD.

    2. Bisect the radius OA at M.

    3. Draw an arc centered at M with radius |CM|. Let N bethe intersection of this arc with OB.

    4. Draw an arc centered at N with radius |CN|. Let R bethe intersection of this arc with the original circle.

    5. Move the length |CR| around the circle to create a reg-ular pentagon.

    O AB

    C

    D

    MN

    R

    Of course, one still has to prove that such a construction is correct. Perhaps the easiest way for us tocheck it is to use a little algebra: suppose that the circle has radius 2, then a couple of applications

    9

  • of Pythagoras’ tells us that |CR| =√

    10− 2√

    5. Analyzing a 36°, 72°, 72° isosceles triangle and thetrigonometric multiple angle formulae will allow you to check that this is indeed the correct sidelength. Euclid first presents a less practical construction in the Elements (Book IV, Theorem 11), butthe above follows immediately from Theorem 10 of Book XIII.6

    The appearance of√

    5 in this discussion also relates to the golden ratio 1+√

    52 : this is in fact the ratio

    between the diagonal and the side of the pentagon. Pentagons and pentagrams were something ofan obsession for the Pythagoreans who viewed them as a mystic symbol.

    Alexandria, Euclid and the Elements

    Euclid worked in the Library of Alexandria, now on the north coast ofEgypt. The area was part of the empire resulting from the conquests ofAlexander the Great (after whom the city was named). The Library isthought to have been constructed around 320 BC as a means of orga-nizing the knowledge of the world and for the demonstration of Greekpower. Although the Library was destroyed or seriously damaged onseveral occasions, the knowledge was preserved and Alexandria re-mained the center of western(!) scholarship until around 500 AD.

    Map of Alexandria c.400 AD7

    The Elements is, without doubt, the most influential mathematics text in history. If was producedin Alexandria around 300 BC. It is better thought of as a complilation of the work of earlier math-ematicians (many of whom we’ve discussed) rather than an original work. Partly due to its fame

    6If a regular pentagon, hexagon and decagon are incribed in the same circle, then their sides will form a right triangle.7Note the ‘Pharos’ (Great Lighthouse), one of the wonders of the ancient world

    10

  • it was edited and added to over the centuries, eclipsing and subsuming other important works. Inparticular, Theon of Alexandria (c.400 AD) and his daughter Hypatia8 famously edited the Elementsin an attempt to improve the work and make it easier to follow. Unfortunately we have insufficientevidence to be able to extract all their edits from the original.The earliest (almost) complete copies of Euclid date from the ninth century. The pictured version,written in Greek, is at the Vatican, and does not contain some of Theon’s edits, thus showing thatmultiple versions of the text were circulating during the first millennium AD.

    Earliest Fragment c.100 AD Full copy, Vatican, 9thC

    It is impossible to overstate the influence of the Elements on mathematics and on western philosophymore widely. Most people obtaining a serious education in Europe before 1900 would have devotedmany hours to its study. Many editions and variations have been produced, four of which are shownbelow:

    Latin translation, 1572 High School textbook, 1903

    8Hypatia is arguably the first famous female mathematician, and a prolific scholar in her own right.

    11

  • Pop-up edition, 1500’s Byrne’s color edition, 1847

    Axiomatics The key feature of Euclid is its axiomatic approach. Euclid begins with a list of ax-ioms/postulates and definitions and proceeds to prove theorems deduced from these. His approachhas several mistakes and omissions (not fully corrected until c.1900), but it remains a tour de force oflogical construction. The axiomatic approach is essentially universal in modern mathematics, barelychanged from Euclid’s day. Its broader influence comes from the belief of generations of scholars thatits study would improve their logical abilities.

    Book I

    Consists of 48 theorems, the final two being Pythagoras’ and its converse. It seems likely that Euclidorganized Book I with the goal of proving this important result in a thorough manner. The bookbegins, as all of the books do, with a list of definitions, axioms and postulate. Here are Euclid’sfamous postulate from Book I.

    1. Given any two points, a straight line can be drawn between them

    2. Any line may be indefinitely extended

    3. Given a center and a radius, a circle may be drawn

    4. All right angles are equal to each other

    5. If a straight line crosses two others so that the angles on the same side make less than two rightangles, then the two lines meet on that side of the original9

    Depending on the edition, the axioms included basic statements describing equality and arithmetic,for instance

    a = b =⇒ a + c = b + c9The fifth postulate is awkwardly phrased and is set up in order to argue by contradiction. It is equivalent to the modern

    statement known as known as Playfair’s postulate: given a line and a point not on the line, there exists a unique parallel tothe line through the point. For centuries, mathematicians tried to prove that this postulate was a theorem of the others.Eventually, with the advent of hyperbolic geometry, it was shown to be necessary. Euclid’s reluctance to use the parallelpostulate until theorem 29 suggests he understood this awkwardness.

    12

  • Basic Theorems Built on the postulates with very pictorial proofs. Euclid’s approach was to vieweach theorem as a problem to be solve: he often presented a construction and then proved that theconstruction was correct. Here are four example theorems.

    • On a given line segment, an equilateral triangle may be constructed (Thm I. 1)

    • Any line segment can be bisected (Thm I. 10)

    • If two straight lines cut one another, opposite angles are equal (Thm I. 15)

    • An angle in a semicircle is right-angle (Thm III. 31). This is Thales’ Theorem: it is presented inBook III as part of a larger result, although this specific result is an easy corollary of two resultsin Book I.

    Parallel lines, their construction and uniqueness

    A large part of Book I concerns the construction, detection and uniqueness of parallel lines (i.e. lineswhich do not intersect).

    Theorem (I. 16 (Exterior angle theorem)). If one side of a tri-angle is protruded, then the exterior angle is larger than eitherof the opposite interior angles. In the language of the picture, wehave δ > α and δ > β.

    Proof. Take the midpoint M of AC and draw the bisectorBM. Extend it the same distance beyond M to E. ConnectCE. The opposite angles at M are equal and so we havecongruent triangles: 4AMB ∼= 4CME. It follows that theangle indicated at C is also α. Clearly this is less than δ.Bisect the other edge to see that β < δ.

    α

    β δ

    M

    A

    B C

    E

    α

    Theorem I. 16 essentially constructs a parallel line to AB through a given point C not on the line AB.It remains to prove that this line really is parallel.

    13

  • Theorem (I. 27). Suppose that a line falls on two other lines insuch a way that the indicated angles are equal. Then the twolines are parallel.

    Proof. Suppose the lines were not parallel. Then they mustmeet on one side. WLOG suppose they meet on the rightside at point C. But Theorem I. 16 says that the angle α atB, being external to4ABC must be greater than the angleα at A. Contradiction.

    α

    α

    α

    α

    A

    B CIt follows that the line CE constructed in the proof of Theorem I. 16 really is parallel to AB. Thefollowing theorem is an immediate corollary of the last.

    Theorem (I. 28). Suppose that a line falling on two other linesmakes the same angles. Then the two lines are parallel. α

    α

    Up to now, only the first four of Euclid’s postulates are used. The next theorem (the converse to I. 27)invokes the fifth parallel postulate to show us that this is the only way to create parallel lines.

    Theorem (I. 29). Suppose that a line falls on two parallel lines.Then the alternate angles are equal.

    Proof. We must prove that α = β. Suppose not and WLOGthat α > β. But then β + γ < α + γ and so β + γ is lessthan a straight edge. By the parallel postulate, the lines`1, `2 meet on the left side of the picture, whence `1 and `2are not parallel.

    `1

    `2

    αγ

    β

    Angles in a triangle Euclid is now in a position to prove the most famous result about triangles:that the interior angles sum to 180°. Euclid words this slightly differently. Compare the constructionwith

    Theorem (I. 32). If one side of a triangle is protruded, the exte-rior angle is equal to the two interior and opposite angles.

    Proof. 1. Construct CE parallel to BA

    2. Then ∠ABC = ∠ECD and∠ACE = ∠BAC

    3. ∴ ∠ACD = ∠ACE +∠ECD= ∠BAC +∠ABC

    A

    B C D

    E

    14

  • Pythagoras’ Theorem: I. 47 If a triangle has a right-angle, then the square on the hypotenuse equalsthe sum of the squares on the other two sides.We give a condensed version of Euclid’s argument.

    A

    B CO

    LD E

    F

    G

    H

    K

    1. Thm I. 41 states that a parallelogram is double a trian-gle with the same base and height. Therefore

    Area(4ABD) = 12

    Area(�BOLD)

    2. Similarly Area(4FBC) = 12 Area(�ABFG)

    3. Side-Angle-Side (Thm I. 4) =⇒ 4ABD ∼= 4FBC

    4. ∴ Area(�ABFG) = Area(�BOLD)

    5. Similarly Area(�ACKH) = Area(�OCEL)

    6. Put everything together for the result.

    Thm I. 48 (the last of Book I) is the converse: if the square on one side equals the sum of the othertwo, then the triangle is right-angled.

    Overview of the remaining books of Euclid

    Book II • Mostly attributable to the Pythagoreans• Geometric solutions to problems: constructing a length, even if incommensurate with the

    unit• (Thm II. 11) A straight line can be divided so that the rectangle contained by the whole

    and one of the segments is equal to the square on the remaining segment.This can be rephrased as follows: given AB = a, find H on AB, such that AH = x, wherex2 = a(a− x). Euclid is essentially solving a particular quadratic equation geometrically.Here is his solution:

    1. Construct the square ABDC on AB

    2. Bisect AC; call the midpoint E

    3. Connect EB

    4. Extend AC

    5. Lay off EF = EB on AC extended

    6. Construct the square FGHA

    Note that the ‘solution’ is x = |AH| =√

    5−12 so that AH :

    HB is the golden ratio.

    A B

    C D

    E

    F G

    H

    15

  • Book III • Mostly theorems regarding circles (e.g. Thales’ theorem) and tangency

    • Probably most of the material came from Hippocrates

    Book IV • Construction of regular 3, 4, 5, 6, 15-sided polygons in and around a circle

    • Also from Hippocrates

    Book V • Ratios/magnitudes á la Eudoxus

    • (Thm V. 11) If a : b = c : d and c : d = e : f then a : b = e : f

    Book VI • Ratios of magnitudes applied to geometry

    • (Thm VI. 4) Triangles with equal angles have corresponding sides proportional

    • (Thm VI. 8) The altitude from the right angle of a right triangle divides it into two trianglessimilar to each other and the the original

    • (Thm VI. 31) Corrected Pythagorean proof (see page ??) of Pythagoras’ using Eudoxus’proportions and Thm VI. 8.

    Book VII • Divisibility and the Euclidean algorithm

    • Probably due to the Pythagoreans

    Book VIII • Number progressions, geometric sequences

    • Possibly due to Archytas (Pythagorean and teacher of Plato) from studies in music

    Book IX • Numbers: even/odd + perfect numbers

    Book X • Discussion of commensurable and incommensurable lines

    • Long and difficult, possibly derived from Theaetetus

    Book XI • Solid geometry (lines/planes in 3D)

    • (Thm XI. 28) A parallelepiped is bisected by its diagonal plane

    Book XII • Ratios of areas and volumes (Eudoxus)

    • (Thm XII. 2) The areas of circles are in the same ratio as the squares on their diameters

    Book XIII • Construction of regular polyhedra inside a sphere and their classification

    • (Thm XIII.10) If a regular pentagon, hexagon and decagon are incribed in the same circle,then their sides will form a right triangle.

    Post-Euclidean Mathematics

    Eratosthenes of Cyrene (276–194 BC)

    • Became Librarian at Alexandria

    • Famous for the sieve for finding primes:

    1. List the integers n ≥ 2.2. Leave 2 and delete all its multiples.

    16

  • 3. Leave 3 and delete its multiples.

    4. Repeat ad infinitum: each time one reaches a number, leave it and delete its multiples.

    5. The remaining list contains all the primes.

    The sieve is shown below applied to all numbers 2 ≤ n ≤ 100

    2 3 4X 5 6X 7 8X 9X 10X 11 12X 13 14X 15X 16X 17 18X 19 20X 21X 22X23 24X 25X 26X 27X 28X 29 30X 31 32X 33X 34X 35X 36X 37 38X 39X 40X41 42X 43 44X 45X 46X 47 48X 49X 50X 51X 52X 53 54X 55X 56X 57X 58X59 60X 61 62X 63X 64X 65X 66X 67 68X 69X 70X 71 72X 73 74X 75X 76X77X 78X 79 80X 81X 82X 83 84X 85X 86X 87X 88X 89 90X 91X 92X 93X 94X

    95X 96X 97 98X 99X 100X

    • First calculation of the circumference of the Earth

    – Syene (modern-day Aswan, Egypt) is c.5000 stadiafrom Alexandria (c.1,000,000 yds)

    – At the summer solstice: sun overhead Syene, in-clined 7°12′ = 150 · 360° at Alexandria

    – Earth’s circumference therefore ≈ 250, 000 stadia ≈28, 000 miles

    – Accurate value 25, 000 miles

    rays

    sun’sA

    S

    Archimedes of Syracuse (287-212 BC)

    Archimedes is arguably the greatest of the ancient Greek mathematicians. The term greek is, again,something of a misnomer: his place of birth and residence (Syracuse) is in modern-day Sicily, thelarge island at the foot of the Italian peninsula. At the time of his birth this was part of the Greekempire: he famously helped defend Syracuse against the Romans using catapults and he died at thehands of the Romans after they captured the city. It is believed that he travelled to Alexexandria inhis youth and perhaps met and studied with eminences such as Eratosthenes.Archimedes is rare among ancient Greek mathematicians for being highly practical. He is creditedwith a large number of inventions and technical innovations, including Archimedes’ screw, an inven-tion for elevating water and still used in modern irrigation. He is also acknowledged as the founderof hydrostatics. Archmedes’ principle states that an object immersed in water loses weight equal to thatof the displaced water.

    Example A cube of side length 20cm floats such that the water-line is half-way up the cube. ByArchimedes’ principle, the weight of the cube is the same as that of a volume of water with dimen-sions 20× 20× 10 = 4000cm3 which weighs roughly 4kg.

    17

  • Levers Archimedes made a great study of levers, both for practical purposes and a as a method ofcalculation, as we’ll see in a moment. His work was put to good use, not least in the construction ofcatapults to keep away the pesky Romans! The basic idea can be summarised as follows.

    r1 r2M1

    M2

    In the picture, consider the two ratios M1 : M2 and r2 : r1, where Mi is the weight of mass i and ri isthe distance of the mass from the pivot point.

    • The lever will balance if and only if M1 : M2 = r2 : r1• The lever will rotate clockwise if M1 : M2 < r2 : r1• The lever will rotate counter-clockwise if M1 : M2 > r2 : r1

    In modern language, we would compute the torques τ1 = M1r1 and τ2 = M2r2. If these are equal,the lever balances. Instead Archimedes would have thought about this using Eudoxus’ theory ofproportions.

    The Method: is Archimedes the founder of calculus? A previously unknown work of Archimedes,called the The Method was discovered in 1899. As an amazing application of the lever principle,Archimedes makes an argument that looks remarkably like modern calculus: indeed he could rea-sonably be claimed to be the its practitioner by 1800 years! The method was outlined in a letter toEratosthenes and includes part of an argument for proving Archimedes’ favorite theorem.

    Theorem. If a cone, a hemisphere and a cylinder have the same base and height, then their volumes are in theratio 1:2:3

    Here is a somewhat modernized version of half the result.

    Place the cone below the cylinder and compare the red cross-sections. The circular cross-section of the cone has radius yand so the cross-sectional area is proportional to the squareon the radius (i.e. πy2).For the annular cross-section of the cylinder minus the hemi-sphere, Archimedes used Pythagoras’ to argue that its areais proportional to the difference of the square on the radiusof the cylinder (i.e. 12) and the square x2 = 1− y2.Since the red cross-sections are in balance with respect toa vertical lever with pivot at the center of the picture,Archimedes concludes that the cone would be in balancewith the part of the cylinder outside the hemisphere. Oth-erwise said,

    Vcone = Vcylinder −Vhemisphere

    in line with the desired ratios.

    y

    y1

    x

    18

  • Here is another argument of Archimedes’ with a suggestion of calculus. A disk comprises infinitelymany concentric circles, the circumference of each being proportional to its radius. ‘Unwind’ thesecircles and one obtains a triangle. One side is the radius of the disk, the other its circumference. ThusA = 12 r · c.

    c

    r

    The Method includes several of these calculus-like discussions. While the method is undoubtedlyefficient, Archimedes himself acknowledged that his method didn’t constitute a proof and providedalternative proofs elsewhere in his writings. The essential problem is this;

    • Can we really say that an area equals its cross-sectional lines? Or that a volume equals its cross-sectional areas? Lines have no width so if we add them up we have no area. If they have width,then infinitely many of them have infinite area.

    These are really variations of Zeno’s paradoxes regarding infinitesimals and indivisibles.Both of the above arguments would essentially be resurrected in the early 1600’s by Cavalieri andGalileo as the development of calculus gathered pace and the debate surrounding infinitesimals andindivisibles heated up. Indeed the same duality of presentation characterised the later developmentof calculus: Newton and others clearly found the infinitessimal method highly efficient, but they feltthe need to present geometric proofs to convince readers that the results weren’t mere trickery. It istempting to imagine what might have happened if Archimedes’ method had been accepted and pre-served as part of the Greek canon: would calculus have developed 1800 years earlier and how mightthis have affected technological development? Would the space-race have happened in 500AD?! It isa romantic notion certainly, but also a simplistic one. There were other factors at play in the 1600’sthat made the acceptance of infinitesimals & indivisibles more likely. Still, history is full of what-if ’s. . .

    Quadratures Archimedes also computed area by approximation. Here is an example of how hecould approximate the area of a circle.

    1. Inscribe a regular hexagon in a circle of radius 1 and compute its perimeter (= 6).

    2. Now halve each angle to inscribe a regular dodecagon in the circle. Compute its perimeter(= 12

    √2−√

    3).

    3. Repeat the halving process: Archimedes did this with 24, 48 and 96-sided figures and an in-creasing sequence of perimeters bounded above by 2π.

    4. Repeat the same calculation, this time with exscribed polygons to obtain a decreasing sequencebounded below by 2π. He argued, essentially, that these two sequences convered to the perime-ter of the circle and he could therefore obtain arbitrarily accurate approximations to its circum-ference, and thus to π. Indeed with 96-sided polygons he was able to show that 3 1071 < π < 3

    17 .

    19

  • Here is a sketch of Archimedes’ approach using modern no-tation. As an induction step, consider the diagram. Givena triangular segment of the circle with altitude d0, we halvethe angle to compute the new altitude d1. Everything followsfrom three applications of Pythagoras’:

    1 = d02 + h02

    (2h1)2 = h02 + (1− d0)2

    1 = d12 + h12

    Expanding and cancelling, we obtain

    d21 =12(1 + d0)

    1

    1

    d0

    h0

    h0

    d1 h1

    h1

    1 − d0

    Since d0 =√

    32 and h0 =

    12 , it is easy to compute the entirity of both sequences:

    d1 =12

    √2 +√

    3, d2 =12

    √2 +

    √2 +√

    3, . . . dn =12

    √2 +

    √2 + · · ·+

    √2 +√

    3

    h1 =12

    √2−√

    3, h2 =12

    √2−

    √2 +√

    3, . . . hn =12

    √2−

    √2 + · · ·+

    √2 +√

    3

    where the nth term in each case contains n copies of the digit 2 under the square-root. The circumfer-ence and area of the 6 · 2n-sided polygon inscribed in the circle are therefore

    Cn = 12 · 2nhn An = 6 · 2ndnhn = 6 · 2n−1hn−1 =12

    Cn−1

    Of course these sequences increase to 2π and π respectively. For a 96-sided polygon, Archimedeswould have had to approximate

    C4 = 12 · 24h4 = 96

    √√√√2−

    √2 +

    √2 +

    √2 +√

    3 > 6.282 =⇒ π > 3.141 > 31071

    The sequence of exscribed perimeters and areas could be likewise computed: indeed it is easy to seethat Cexn =

    1dn Cn. In modern language, with the technology of trigonometry, Archimedes is simply

    saying that a subdivision of the circle according to k-sided in- and ex-scribed polygons yields theinequality

    12

    k sin2πk

    < π < k tanπ

    k

    which produce any desired level of accuracy by elementary limit-theory10

    10 limx→0

    sin xx = 1 and the squeeze theorem.

    20

  • Late Greek Period: 200 BC–500 AD

    • Apollonius (225 BC): wrote eight-volume book Conics on the subject.

    • Greece falls under Roman rule.

    • Alexandria remained important: educated Greeks still spoke and wrote greek rather than Latin.

    • Hipparchus (140 BC) computed chords (essentially sine tables, although word was not used)for Astronomy.

    • Greek mathematics had something of a hiatus until c.100 AD: this included the time whenJulius Caesar ruled Rome (died 44 BC).

    • Heron (75 AD) proved the formula√

    s(s− a)(s− b)(s− c) for the area of triangle, where thesemi-perimeter is s = 12 (a + b + c).

    • Ptolemy (150 AD) extended the work of Hipparchus on trigonometry and wrote the astronom-ical masterwork Almagest.11

    • c.400 Theon and Hypatia produce the most widely-read edition of Euclid’s Elements as well asimproving upon several earlier mathematical topics.

    • In 395 the Roman empire split into eastern and western parts, the western rapidly declined un-der the pressures of corruption and attacks by barabrians. By 500 AD, the western empire hadcollapsed. The turmoil did not leave the former parts of the Greek empire untouched. Alexan-dria experienced riots and a bloody power-struggle (Hypatia herself was murdered by a mobin 415) and the library of Alexandria was severly damaged and possibly destroyed at this time.In 642, Alexandria was finally captured by the Islamic caliphate. Most of the material in thelibrary likely survived by being copied and preserved in various places of learning, generallyin the newly rising caliphate.

    11We shall study trigonometry and astronomy in the next section.

    21