The History of Mathematics Early Greek

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1 The History of Mathematics Early Greek
Michael Flicker OLLI Summer 2014

2 Pre Greek Mesopotamia 3500 BCE – 2000 BCE Babylonia 2000 BCE – 600 BCE
Egypt 2000 BCE – 1500 BCE India & China

3 Uruk 3350 – 3200 BCE Five thousand clay tablets, reused as building rubble in the central temple precinct of the city of Uruk, constitute the world’s assemblage of written records. 10% were written by trainee administrators as they learned to write. Most of those exercises are standardized lists of nouns used within the book-keeping system, but one tablet (W 19408,76) contains two exercises on calculating the areas of fields. It is the worlds oldest piece of recorded mathematics.

4

5 Oldest Datable Mathematical Table c
Oldest Datable Mathematical Table c BCE Shuruppag North of Uruk (VAT12593)

6 VAT 12593 First securely datable mathematical table in world history. Table comes from the Sumerian city of Shuruppag to the north of Uruk c 2600 BCE

7 Earliest Known Mathematical Diagram IM 58045 2350 – 2250 BCE
Exactly what the associated problem was is unclear.

8 √2 Yale Collection 7289 (1800 – 1600 BCE)
30 1; 24; 51; 10 42; 25; 35 Observation x 1; 24; 51; 10 = 42; 25; 35 If side of square is 1, tablet says diagonal is 1; 24; 51; 10 = √2 = … 1; 24; 51; 11 = How did they calculate the approximation to √2? 30 x 1; 23; 51; 10 = 42; 25; 35

9 Babylonian Geometry Plimpton 322 Tablet (c 1800 BCE)
One or more columns have broken off the left-hand side of the tablet. Their content have been the subject of much speculation

10 Primary Egyptian Sources
Rhind Mathematical Papyrus (RMP) About 1650 BCE from writings made 200 years earlier (18 ft x 13 in) 84 (87) mathematical problems Recto Table Moscow Mathematical Papyrus – 1850 BCE (date somewhat uncertain), (18 ft x 1.5 to 3 in) 25 problems Egyptian Mathematical Leather Roll Date ?, (10 in by 17 in) Bought by Rhind A collection of 26 sums done in unit fractions It took 60 years and much work to understand its contents Big disappointment Berlin Papyrus 1800 BCE probably written 150 years before RMP RMP and EMLR were discovered, probably in the ruins of the Ramesseum in Thebes, in the middle of the 19th century and bought in Luxor by Alexander Henry Rhind in Rhind died in 1863 and his executor sold the papyrus, in two pieces, to the British Museum in Some fragments of the RMP turned up in New York in 1862 and are now in the Brooklyn Museum. The first translation was into German in The first English translation, with commentary, was made in 1923 by Thomas Peet of the University of Liverpool. The MMP was purchased around 1893 probably in Thebes by Vladimir Golenishchev and acquired 20 years later by the Moscow Museum of Fine Arts. The first notice of its content appeared in a brief discussion by conservator of the Egyptian section of the museum in He wrote chiefly about problem 14 , the determination of the volume of a frustum of a square pyramid. The MMP was translated into German in 1930 and into English in 1999. Thus by early in the 20th century the basic outline of Egyptian mathematics was understood.

11 Egyptian Mathematics The Egyptian papyri show practical techniques for solving everyday problems The rules in the papyri are seldom motivated and the papyri may in fact only be a study guide for students. However, they demonstrated a solid understanding of the operations of addition, subtraction, multiplication and division and enough geometry for their needs. Did their knowledge remain static for the next 1000 yrs?

12 Early Greek History (Morris Kline)
The Greek civilization dates back to ~ 2800 BCE Settled in Turkey, Greece, southern Italy, Sicily, Crete, Rhodes, Delos, and North Africa ~ 800 BCE replaced various hieroglyphic systems of writing with Phoenician alphabet With the adoption of an alphabet the Greeks became more literate and more capable of recording their history and ideas. Greeks visited and traded with the Egyptians and Babylonians Went to Egypt to travel and study Visited Babylonia and learned mathematics and science there Miletus – birthplace of Greek philosophy, math, and science Ionia fell to Persia about 540 BCE Ionian revolt against Persia 494 BCE crushed Ionia declines in importance Ionia becomes Greek again in 479 BCE, but by then cultural activity shifted to mainland Greece with Athens as its center

13 Greek/Hellenistic Timeline
Classical Period ( BC) - Classical period of ancient Greek history, is fixed between about 500 B. C., when the Greeks began to come into conflict with the kingdom of Persia to the east, and the death of the Macedonian king and conqueror Alexander the Great in 323 B.C. In this period Athens reached its greatest political and cultural heights: the full development of the democratic system of government under the Athenian statesman Pericles; the building of the Parthenon on the Acropolis; the creation of the tragedies of Sophocles, Aeschylus and Euripides; and the founding of the philosophical schools of Socrates and Plato.   Hellenistic Period ( BC) - period between the conquest of the Persian Empire by Alexander the Great and the establishment of Roman supremacy, in which Greek culture and learning were pre-eminent in the Mediterranean and Asia Minor. It is called Hellenistic (Greek, Hellas, "Greece") to distinguish it from the Hellenic culture of classical Greece. Dates vary. M Kline uses 600 BCE – 300 BCE and 300 BCE – 600 CE (end of ancient Greek civilization)

14 Locate Attica, Athens, and Miletus on map
Thales Locate Attica, Athens, and Miletus on map Show Mount Mycale just north of Miletus. Location of battle in 479 BCE where Greeks defeated the Persians. Show Cnidus North and west of Rhodes – birthplace of Eudoxus Eudoxus

15 Missing Syracuse

16 Early Greek Number System
Acrophonic Attic ~ 600 BC to ~ 300 BC Similar to Roman Numerals Acrophonic: symbols for numerals come from the first letter of the number’s name Attic: number name from region of Attica which includes Athens 1 = | not from name of 1 The symbol for 5 that looks like a gamma is the early form of pi Denotes drachma

17 Later Greek Number System
Ionic (Alphabetical) M used for numbers 10,000 and above. M = 50,000 ‘εχοη = 5678

18 Fractions Acrophonic With the exception of monetary amounts, there were no acrophonic numerals for fractions Ionic Used apostrophe to signify fraction ’ = 1/ ’ = 1/32 Addition, subtraction, multiplication and division Acrophonic commerce Coins ├ drachmae │ obol or 1/6 drachmae C half circle = ½ obol Ɔ other half circle = ¼ obol

19 Early Greek Mathematics
Roughly 600 BC to 300 AD

20 How do we know about Greek mathematics?
Initially knowledge was passed from teacher to student orally Probably around 450 BCE chalk boards and wax tablets were introduced for non permanent work. Papyrus rolls were used for permanent records but new copies were required frequently. In about 300 BCE Euclid’s Elements was completed and it was so comprehensive and of such quality that all older mathematical texts became obsolete. In 2nd century CE books of papyrus appeared and became the main form in the 4th century. Also vellum (animal skin) was introduced.

21 How do we know about Greek mathematics?
If the person copying the “Elements” was a mathematician, material not in the original text may have been added The oldest surviving complete copy of the Elements is from 888 CE probably based on a version with commentary and additions produced by Theon of Alexandria in the 4th century CE In addition to the 888 AD document there are numerous fragments some dating to as early as 225 BCE Some surviving texts exist that were written after 888 CE that were based on versions of the elements earlier than 888 CE First print edition in Venice in 1482 For the complete story see the article “How do we know about Greek mathematics” on the Mac Tutor History of Mathematics

22 Major Schools of the Classical Period
Miletus School – founded by Thales (c 640 – c 546 BCE) Anaximander, Anaximenes, and Pythagoras students of Thales Anaxagoras belonged to this school then moved to Athens Pythagoras formed his own large school in southern Italy Xenophanes of Colophon in Ionia migrated to Sicily in southern Italy and founded a center (view contested) Parmenides and Zeno belonged School moved to southern Elea in Italy and became the Eleatic school Sophists active from latter half 5th century in Athens Academy of Plato in Athens – Aristotle student School of Eudoxus School of Aristotle Anaxagoras was the first to bring philosophy to Athens

23 Greek Mathematics – Thales of Miletus
Thales of Miletus 624 BC – 547 BC (Ionia) None of his writings survive (if there were any) Name appears in the writings of others years later The first of the seven sages of antiquity A pupil of the Egyptians Credited by Proclus with five theorems of elementary geometry A circle is bisected by any diameter The angles between two intersecting straight lines are equal Two triangles are congruent if they have two angles and a corresponding side equal The base angles of an isosceles triangle are equal An angle in a semicircle is a right angle

24 More on Thales The development of geometry is preserved in a work of Proclus (412 – 485 CE), A Commentary on the First Book of Euclid’s Elements. Proclus provided a remarkable amount of intriguing information, the vital points of which are the following: Geometry originated in Egypt where it developed out of necessity; it was adopted by Thales who had visited Egypt, and was introduced into Greece by him The Commentary of Proclus indicates that he had access to the work of Euclid and also to The History of Geometry which was written by Eudemus of Rhodes, a pupil of Aristotle, but which is no longer extant. His wording makes it clear that he was familiar with the views of those writers who had earlier written about the origin of geometry. He affirmed the earlier views that the rudiments of geometry developed in Egypt because of the need to re-define the boundaries. Thales is believed to have been a shrewd businessman. During a good season for olive growing, he cornered all the olive presses in Miletus and Chios and rented them out at a high fee. Thales is said to have predicted the eclipse of the sun in 585 BCE, but this is disputed on the grounds that astronomical knowledge was not adequate at the time. He is reputed to have calculated the heights of pyramids by comparing their shadows with the shadows of a stick of known height. By some such use of similar triangles, he is supposed to have calculated the distance of a ship from shore. The discovery of the attractive power of magnets and static electricity is also attributed to Thales.

25 Greek Mathematics – Pythagoras of Samos
Pythagoras (585 BCE – 497 BCE) Founded a religious, scientific, and philosophical brotherhood in Croton in southern Italy It was a formal school with limited membership – teachings were kept secret The Pythagoreans got on the wrong side of local politics and had to flee. There are no written works by the Pythagoreans We know about them through the writings of others including Plato and Herodotus When one speaks of the works of Pythagoras one really speaks of the work done by the group up to about 400 BCE Plato 428 – 348 BCE Herodotus 484 – 425 BCE Greek historian One of the great Greek contribution to mathematics was the recognition that mathematical entities, numbers, and geometrical figures are abstractions, ideas entertained by the mind and sharply distinguished from physical objects or pictures. The Pythagoreans get credit for this.

26 All is Number One of the scientific achievements of Pythagoras was the discovery of the mathematical order in the musical scale and the harmonies so produced. It is believed that he experimented with string instruments and discovered that two tones sound well together when the ratios of their frequencies can be expressed by the use of small numbers and the smaller the numbers the better is the harmony. 20 22/12 24/12 25/12 27/12 29/12 211/12 212/12 Pythagorean scale developed for string instruments. Equal temperament scale developed for keyboard instruments. Different notes in powers of 21/12 . For example: 21/12 is C#/Db One of the first advocates of the equal temperament scale was Vincenzo Galilei, Galileo Galilei’s father. My understanding is that the Equal Temperament Scale dominates today, but there are other scales and some people do not like the Equal Temperament Scale Pythagorean scale may be best for string instruments and equal temperament for piano and such. Table from “Science & Music” by Sir James Jeans

27 All is Number Boethius (~ 500 CE) tells us that Pythagoras investigated the relation between the length of a vibrating string and the musical tone it produced. If a string was shortened to ¾ of its original length, then what is called the fourth of the original tone was heard; if shortened to 2/3, the fifth was heard; and if shortened to 1/2 , the octave. The string lengths were proportional to 12, 9, 8, and 6. The Pythagorean school was able to find many interesting relations between these number including the fact that a cube has 6 faces, 8 vertices, and 12 edges. They were able to convince themselves that since these combinations of string lengths produced sounds that harmonize, the numbers themselves essentially caused it. The Pythagoreans extrapolated that the natural numbers are fundamental to natural science. This was bad science but really good for number theory For a string: velocity (v) = Sqrt (T/μ), v = λf, f = (1/λ) Sqrt(T/μ) For a length of string L the fundamental is λ/2 fundamental = (1/2L)Sqrt(T/μ)

28 Figurate Numbers Triangular Numbers 1, 3, 6, 10, 15 …
Square Numbers , 4, 9, 16, 25 … The sum of two consecutive triangular numbers is a square number The sum of any number of consecutive odd integers, starting with one, is a perfect square

29 Pythagorean “Arithmetica”
Pythagorean triplets – natural numbers that satisfy Pythagoreans knew that when m is odd, then a = m, b = (m2 – 1)/2, and c = (m2 + 1)/2 are such a triplet. For example for m = 3, the triplet is 3, 4, 5. However the formula does not produce all triplets. Euclid solves the general problem is his Elements. a = 2mn, b = m2 – n2, c = m2 + n2 Choose m = 75 and n = 32, a = 4800, b = 4601, c = 6649 One of the sets in Plimpton 322 If one sets n = 1 and divides a, b, and c by 2 one obtains Pythagoras’ expression.

30 More “Arithmetica” Perfect numbers – the number is the sum of its divisors. 6 = 28 = 496 = 8128 The search for perfect numbers continues to today. No odd perfect numbers have been found. Friendly numbers – pairs of numbers such that each is the sum of the divisors of the other – The first friendly pair is 220 & 284 The secondly pair is 17,296 and 18,416 and was independently discovered by al-Banna (Arab 1256 – 1321) and by Fermat in 1636 In the 1970s the phrase “friendly numbers” started to be used differently Friendly pair: factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 sum = 284 In number theory today a friendly pair are two numbers with the same “abundancy” The ratio of the sum of the divisors and the number itself. The abundancy of 6 is ( )/6 =2; the abundancy of 28 is ( )/28 = 2

31 Still More “Arithmetica”
The Pythagoreans studied prime numbers, progressions, and ratios and proportions. They understood that certain sums could be easily calculated. … + n = (n/2)(n + 1) Numbers to Pythagoreans were whole numbers only. The ratio of two whole numbers was not a fraction and therefore another kind of number. Actual fractions were employed in commerce. Arithmetica included the understanding of even and odd numbers The sum of two even numbers is even The product of two odd numbers is odd When an odd number divides an even number, it also divides its half There is a story about Carl Friedrich Gauss (1777 – 1855). Supposedly, when he was a little boy, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience]... To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away. The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result.

32 Pythagorean School Geometry
In the 5th century BCE geometry, the theorem/proof logic took root and a system of theorems was developed where theorems were proved based on theorems previously proved. Studied triangles, parallel lines, polygons, circles, spheres and regular polyhedra Worked on a class of problems known as “application of area” Example – Construct a polygon equal in area to a given one and similar in shape to another given one. Squaring the circle Convert a rectangle of sides L and H into a square of equal area. ___________ _ | | H |__________ |_____ B L D H A From the center of L+H draw a circle of radius (L+H)/2. Extend side H up until it crosses the circle. Denote the length of the extended line from L+H to the circle by S. S = sqrt(LxH) and therefore has the same area as the rectangle. Let the H side of the diameter be point A, the L side point B, and the point where L meets H D. Let the point where H extended meets the circle be point C and let S be the length of CD. Then BD/CD = CD/DA and S2 = LH. A triangle to square - triangle to right triangle to rectangle to half rectangle to square. Squares can be added by Pythagorean sum.

33 Pythagorean School Geometry
Did the Pythagoreans prove their geometric results? During most of the life of the school the members probably affirmed results on the basis of special cases. By 400 BCE they may have given some real proofs Did they prove the Pythagorean Theorem? Proclus credits the proof to Euclid so the Pythagoreans probably did not have a real proof. Proclus 412–485 C.E.

34 Pythagoreans' Symbol Golden Ratio
Will discuss constructing the pentagon in Euclid. If ratios are true, then (b+m)/b = b/m and m2 + bm = b2 or b2 – bm – m2 = b/m = (1 +/- sqrt(5))/2 For the Pythagoreans b/m = (1 + sqrt(5))/2 so theu could construct it but had a problem making it a number because sqrt 5 is irrational. Golden Ratio

35 Golden ratio and Golden Rectangle

36 Pythagorean Theorem For a right triangle with arms a and b
and hypotenuse c, then the area of the square A constructed on a plus the area of the square B constructed on b equals the area of the square C constructed on c. C A a c b B

37 Proof of Pythagorean Theorem
Who knows what the Pythagoreans did. Euclid had a different proof because this method did not fit in his sequence of theorems. The areas of the two large squares are equal. The area of the square on the left is 4 triangles + square A + square B. The area of the square on the right is 4 triangles + square C. Since all eight triangles are identical, The area of A + the area of B = the area of C

38 Rational Numbers The concept of integers comes from counting objects
Life requires us to measure quantities such as length, speed, weight, time , … To satisfy these measurements fractions (ratios) are required. No matter how accurately it is necessary to measure something, that measurement can always be expressed as a fraction of the unit of measure. Early mathematicians thought of numbers as points on a line and believed that all points could be expressed as fractions.

39 Irrational Numbers From the Pythagorean theorem it follows that for a square with its side equal to 1 the hypotenuse equals The Pythagoreans tried to find two natural numbers whose ratio was , but failed. The discovery of “irrational” ratios is attributed to Hippasus of Metapontum (5th cent. B.C.) After the was revealed to be irrational, the Pythagoreans, according to a legend, killed Hippasus, not willing to believe this fundamental number could fail to be a ratio of integers.

40 Proof that is irrational
a/b = where both a and b are integers and the ratio a/b has been reduced to lowest terms. This means that a and b cannot both be even. a2 = 2 b2 a must be even since if it were odd then a2 would be odd. If a is even it can be written as 2c implying 4c2 = 2 b2 or 2c2 = b2 By the same argument as above b must be even. The assumption that is rational leads to a contradiction. Was the first irrational number? Since there are no surviving documents from this era to verify which was the first irrational number found we can not be sure exactly which one it was but there are some doubts that it was sqrt 2. The proof that sqrt 2 is irrational has a relatively high level of abstraction, even though it is easy for us to understand. Specifically, looking at a general rational number as a quotient of two integers that are represented as variables and the proof uses algebraic techniques that are also doubted to have existed at the time. To get a feel for how mathematics was done back then just pick up a translation to Euclid's Elements. The statements are entirely prose, there is no algebraic symbolism in the original text. For us, showing that sqrt 2 is irrational is an easy proof since we grew up with this level of abstraction but for someone living in 500 BC the proof would be all Greek to them. Back then the terms rational and irrational really did not exist. They thought that every two magnitudes (or lengths) were commensurable. This meant that for every two given lengths there is a third length that divides the first two evenly. For example, take these two lengths. __________________________________ _____________________ We can find a third length _______ that divides both of the first two evenly. This is equivalent to saying that the ratio of the first two lengths is a rational number, in our current mathematical lingo. So the Pythagoreans, and all those before them thought that all numbers were rational. Another way to look at this concept of commensurability is to examine the relationship between the ratio of the original lengths a and b and the ratio of the lengths a and b - a, where a is the shortest length. Take our above example, if we continually subtract the smaller length from the larger length and look at the ratios of the resulting lengths, eventually, the lengths are the same and hence the ratio is 1. In fact, when the lengths are the same it is the length that divides the first lengths evenly.

41 Platonic Solids http://www.3quarks.com/en/PlatonicSolids/index.html
Pythagoras Tetrahedron – 4 equilateral triangles Cube (hexahedron) – 6 squares Dodecahedron – 12 equilateral pentagons Theaetetus (417 BC – 369 BC) Octahedron – 8 equilateral triangles Icosahedron – 20 equilateral triangles In book XIII of Euclid’s elements it is shown that there are only five convex regular polyhedra So why are these called the Platonic solids?

42 Platonic solids

43 The Eleatic School Zeno, born c. 490 BCE, was a member of the Eleatic school, and most of what we know personally about him is from Plato's dialogue Parmenides. In defense of Parmenides’ views on relation of the discrete to the continuous, Zeno proposed a number of paradoxes of which four deal with motion. The first view was that space and time are infinitely divisible, in which case motion is continuous and smooth; and the second that space and time are made up of indivisible small intervals, in which case motion is a succession of small jerks Parmenides is one of the dialogues of Plato. The Parmenides purports to be an account of a meeting between the two great philosophers of the Eleatic school, Parmenides and Zeno of Elea, and a young Socrates. The occasion of the meeting was the reading by Zeno of his treatise defending Parmenidean monism against those partisans of plurality who asserted that Parmenides' supposition that there is a one gives rise to intolerable absurdities and contradictions.

44 More Zeno The second paradox is called the Achilles and the Tortoise and addresses the view that space and time are infinitely divisible. Achilles and the tortoise decide to have a race. Achilles is known to be the faster runner of the two, and therefore decides to give the tortoise a head start. Once the race begins, it is true to say that Achilles will take some time to reach the starting point of the tortoise. During this finite time, it is also true to say that the tortoise will have moved forward by some finite distance, and will therefore still maintain a lead in the race. Achilles will once again take some time to reach the tortoise's new position, during which the tortoise will move forward some distance yet again, thus maintaining his lead. This continues on forever. Therefore Achilles never overtakes the tortoise. Zeno makes the fatal assumption that a sum of an infinite amount of terms implies an infinite sum. It was probably not until Georg Cantor (1845 –1918) that this problem was formally addressed mathematically. There are still issues with the nature of physical reality at the quantum level.

45 The Sophist School After the defeat of the Persians by the Greeks at Mycale in 479 BCE, Athens became the major city in ancient Greece. The first Athenian school was the Sophist school and one of their chief pursuits was the use of mathematics to understand the functioning of the universe. Many of the mathematical results obtained were by-products of their efforts to solve three famous construction problems.

46 The Three Famous Geometry Problems of Antiquity
These three problems were to be solved using a straight edge (not a ruler) and compass Squaring the circle (Anaxagoras, 500 BC – 428 BC worked on problem while in prison) Doubling the cube (worked on by Hippocrates of Chios, 470 BC – 410 BC, Sophist). Hippocrates is credited with the idea of arranging theorems so that the later ones can be proven on the basis of the earlier ones. Trisecting an angle (worked on by Hippias of Elis, born c 460 BCE, a leading Sophist). Hippias invented the quadratrix These problems grew out of: Application of area problems Doubling the square Bisecting the angle Quadratrix ~ 420 BCE

47 Curve BFJ is the magical curve. Construct the original angle DAE
Curve BFJ is the magical curve. Construct the original angle DAE. Determine point F where AE crosses BFJ. Draw FH parallel to the sides of the square. Find point K such that KH/FH = 1/3. Draw a line through K parallel to AD. Angle DAJ = 1/3 DAE.

48 Lune of Hippocrates D A B F C E
The area of the semicircle ACE equals the area of the quarter circle ADC. Hence the area of the lune ACEF equals the area of triangle ADC The fact that a shape that appears to be related to a circle can be shown to be equal in area to a triangle led folks to believe that the circle could be squared. Hippocrates of Chios was the most famous mathematician of his century. A contemporary, Hippocrates of Cos (460 BC – 375 BC), is the father of medicine AD = r, AC = r sqrt2, area ACE = pi [(r sqrt2)^2]/ (2*4) = pi (r^2)/4. Area ACD = pi (r^2)/4

49 Famous Problems Squaring the circle
Squaring the circle involves constructing the square root of π which was shown to be impossible in 1882 when Ferdinand von Lindemann proved that π is transcendental. Doubling the cube (Delian problem) & trisecting an angle Both these problems require finding the cube root of a quantity. In 1837 Pierre Wantzel showed the problems are unsolvable by compass and straightedge construction. There are certain angles that can be trisected, e.g., 90 degrees but in general it cannot be done. Name from story of the citizens of Delos consulting the oracle at Delphi about a plague sent by Apollo. Oracle responded that they must double the size of the altar to Apollo which was a regular cube

50 Beginning of the “Elements”
Hippocrates of Chios wrote the first Elements around 430 BC. After Hippocrates at least four other Elements were written each improving on the previous versions Euclid wrote his Elements in about 300 BC

51 The Platonic School Theodorus of Cyrene (born c. 470 BCE) and Archytas of Tarentum (428 – 347 BCE) were both Pythagoreans and both taught Plato. Theodorus is noted for having proved that the square roots of the non-square numbers up to 17 are irrational. Archytas introduced the idea that a curve is generated by a moving point and a surface by a moving curve. He also provided a solution of the duplication of a cube problem. Plato founded his Academy in Athens about 387 BCE.

52 Plato and Mathematics Plato was not a mathematician but his enthusiasm for the subject and his belief in its importance for philosophy and the understanding of the universe encouraged mathematicians to pursue it. Almost all the important mathematical work of the fourth century was done by friends and pupils of Plato Plato himself seems to have been more concerned to improve and perfect what was known. It appears that starting with Plato’s school concepts of mathematics became abstract. Numbers and geometrical concepts were distinct from physical things. Concepts of mathematics have a reality of their own and are discovered, not invented or fashioned. Quotation from The Republic (book VI section p ) And do you know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on – the forms which they draw or make, and which have shadows and reflections in water of there own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind?

53 Contribution of the Platonic School
Deduction What do we do? – probable knowledge Induction Observation Experimentation Most significant discovery probably the conic sections Attributed to Menaechmus a pupil of Eudoxus Looked at the intersection of a plane with various shaped cones Literature suggests it came out of work on the construction of sundials Theaetetus generalized the theory of irrationals The square (cube) root of a whole number is rational if and only if the number is a perfect square (cube) of whole numbers. Deduction: You are aware that students of geometry, arithmetic and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them either to themselves or others; but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion.

54 Eudoxus of Cnidus (408 – 355 BC)
Eudoxus did everything Philosopher Geometer Astronomer Created the first astronomical theory of the heavenly motions Geographer Physician Legislator He is consider to be one of the greatest of the ancient mathematicians, second only to Archimedes

55 Eudoxus of Cnidus Born in Cnidus in Asia Minor
Studied geometry under Archytas in Italy and medicine with Philistion in Sicily At age 23 he went to Athens for two months to attend lectures Traveled to Egypt for 16 months where he learned astronomy from the priests of Heliopolis and made measurements at their observatory (~ 381 – 380 BCE) Founded school at Cyzicus in northern Asia Minor About 368 BCE he and his followers joined Plato Some years later he returned to Cnidus where he died in 355 BCE In Athens he attended lectures on philosophy and oratory, and in particular the lectures of Plato. Cyzicus ˈsɪzɪkəs Rhymes with CNIDUS ˈnī-dəs\ Midas, nidus

56 Astronomy The spherical earth is at rest at the center of the universe. Around this center, 27 concentric spheres rotate. The exterior one caries the fixed stars, The others account for the sun, moon, and five planets. Mercury, Venus, Mars, Jupiter, and Saturn Each planet requires four spheres, the sun and moon, three each.

57 Eudoxus of Cnidus – Theory of Proportion
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 =. Conceptually, this is an infinite process but it was needed to deal with incommensurables. It was now possible to compare the “magnitudes” of rational and irrational ratios. Unfortunately, the concept of magnitude was only applied to geometric concepts such as line segments, angles, areas, etc and not to numbers. Numbers were integers. Eudoxus solved the problem of comparing incommensurables in geometry but forced a separation between number and geometry. It took two thousand years to recover Note that, to the Greeks, ratios were not fractions as we think of them. Ratios compared the properties of geometric shapes. Eudoxus essentially avoided treating rational numbers as numbers. He avoided giving numerical values to the lengths of line segments, sizes of angles, etc.

58 Eudoxus of Cnidus – Method of Exhaustion
Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. Translation: Given a > e Choose a number r1 such that a > ar1 > a/2 Let a1 = a – ar1 = a(1 – r1); choose r2 such that a1 > a1r2 > a1/2 Let a2 = a1 - a1r2 = a1(1 - r2) = a(1 – r1)(1 – r2) an = a(1 – r1)(1 – r2)…(1 – rn) < e

59 Eudoxus of Cnidus – Method of Exhaustion
XII.1 Similar polygons inscribed in circles are to one another as the squares on their diameters XII.2 Circles are to one another as the squares of their diameters XII.6 Pyramids of the same height with polygonal bases are to one another as their bases XII.7 Any pyramid is the third part of the prism with the same base and equal height XII.10 Any cone is the third part of the cylinder with the same base and equal height XII.18 Spheres are to one another in triplicate ratio of their respective diameters.

60 Eudoxus of Cnidus Some consider him to be the greatest of the ancient mathematicians, second only to Archimedes Resolved the difficulty in comparing rational and irrational numbers Put the method of exhaustion on a rigorous basis Similar to the limit concept of calculus Proved the volume of a pyramid (cone) equals 1/3 vol. prism (cylinder) with same base and equal height Area of circles go as the square of their diameters and the volume of spheres go as the cubes of their diameters The first to present a general geometric model of celestial motion.

61 Aristotle ( BCE) A student of Plato and a tutor of Alexander the Great Primarily a philosopher and biologist but a competent mathematician current on the activities of the mathematicians The works of Aristotle are important particularly to Euclid’s Elements because they come just before the Elements and show the innovations in the Elements that are Euclid’s. In particular, Euclid’s formulation of the postulates and many propositions are his own. An example is the proposition that an angle inscribed in a semicircle is a right angle. Aristotle’s proof is much different than Euclid’s

62 Aristotle ( BCE) A major achievement of Aristotle was the founding of the science of Logic. In producing correct laws of mathematical reasoning the Greeks had laid the groundwork for Logic, but it took Aristotle to codify and systematize these laws into a separate discipline.

63 References I have borrowed liberally from:
1) Mathematical Thought from Ancient to Modern Times, Morris Kline A History of Mathematics, Carl B. Boyer The Mac Tutor History of Mathematics archive, University of St Andrews Scotland A History of Greek Mathematics, Sir Thomas Heath Wikipedia and other web sites Famous Problems of Elementary Geometry, Felix Klein Science & Music by Sir James Jeans (Dover) The Works of Archimedes, edited by Sir Thomas L. Heath God Created the Integers – The Mathematical Breakthroughs that Changed History, Edited by Stephen Hawking