AMA1D01C Ancient Greece

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1 Hong Kong Polytechnic University 2017

2 References These notes mainly follow material from the following book: Katz, V. A History of Mathematics: an Introduction. Addison-Wesley, and also use material from the following sources: Burton, D. The History of Mathematics: an Introduction. McGraw-Hill, Cajori, F. A History of Mathematics. MacMillan, MacTutor History of Mathematics Archive, University of St Andrews.

3 Introduction Some of the major players: Pythagoras ( BC), Plato ( BC), Aristotle ( BC), Euclid ( BC), Archimedes ( BC), Appolonius ( BC), Ptolemy (AD ), Diophantus (AD )

4 Early Greek Mathematics No complete text dating earlier than 300 BC Fragments exist and later works refer to earlier works Most complete reference can be found in the commentary to Book I of Euclid s Elements written by Proclus in the 5th century AD Thought to be a summary of a history written by Eudemus of Rhodes in around 320 BC Original of Eudemus work was lost

5 Thales Thales Earliest Greek mathematician mentioned was Thales From Miletus in Asia Minor (Asian part of modern day Turkey) Many stories recorded about him: prediction of a solar eclipse in 585 BC, application of the angle-side-angle criterion for triangle congruence, proving that the base angles of an isosceles triangle are equal, proving the diameter of a circle divide the circle into 2 equal parts The proofs themselves are lost, but it looks like there s a strong logical flavour to his mathematics

6 Pythagoras From the island of Samos, off the coast of Asia Minor No surviving works All we know about the Pythagoreans was learned through later writers He was friends with Polycrates, the ruler of Samos Polycrates had an alliance with Egypt, and Pythagoras visited the country in 535 BC with a letter of introduction from Polycrates

7 Pythagoras When Persia invaded Egypt in 525 BC (Pythagoras was possibly on a different visit in the country), Pythagoras was taken as a prisoner of war and sent to Babylon He was influenced by thoughts in both countries Left Samos in 518 BC, in order to escape his diplomatic responsibilities and have more time to think Settled in Crotona, a Greek town in southern Italy Note: The area of Greek influence was much bigger than modern day Greece Gathered a group of disciples later known as the Pythagoreans

8 Greek Influence Figure: Extent of Greek Influence. Source: Greek_Colonization_Archaic_Period.png

9 Pythagoras They believed numbers (positive integers) form the basis of all physical phenomena Motions of the planets can be given in terms of ratios of numbers Same for the musical harmonies Today, Pythagorean tuning refers to a tuning system based on products of the simple ratios 2 3 and 1 2. For comparison, equal temperament is a system based on the number 12 2.

10 Pythagoras Using pictures, they managed to prove = 2 2, = 3 2, = 4 2, and so on Construction of Pythagorean triples: there is evidence to show that they know if n is odd, then (n, n2 1 2, n ) form a Pythagorean triple Also, if m is even (m, m2 2 triple Not a table, but a rule 1, m ) form a Pythagorean

11 Sum of consecutive odds Figure: = 4 2

12 Incommensurability of the Diagonal Two lengths are said to be commensurable if they are both multiples of some shorter length In modern language, it means that the ratio of the two lengths is a rational number It was discovered around 430 BC that the diagonal and side of a square are not commensurable Note if the year was correct, it was after Pythagoras death A big shock - as it went against their belief that integers form the foundation of the universe But it was not enough to give something as simple as the length of the diagonal of a unit square

13 Incommensurability of the Diagonal How was it discovered? Hint was in Aristotle s work He noted that if the side and the diagonal are commensurable, then one may get an odd number which is equal to an even number A 1 = Area of AGFE, A 2 = Area of DBHI A 1 = 2A 2, so A 1 is even, and side AG is even, so A 1 is a multiple of 4 Therefore A 2 is even, which implies side DB is even Looks like the Greeks had the notion of a proof Very different from Egyptian or Babylonian mathematics, which emphasized on calculations

14 Incommensurability of the Diagonal Figure: Incommensurability of the diagonal

15 Incommensurability of the Diagonal A similar proof in more algebraic language: Let ( a b )2 = 2, where a and b are integers with no common factor other than 1 We have a 2 = 2b 2, so a 2 is an even number Since 2 is a prime number, if a 2 is an even number, a is an even number. Let a = 2k, we have 4k 2 = 2b 2, so b 2 = 2k 2. Therefore b 2 is an even number and b is an even number. Contradiction.

16 Plato Major legacy was his philosophy on mathematics Founded the Academy in 385 BC An unverifiable story states that the line AΓEΩMETPHTOΣ MH EIΣ EIΣITΩ ( Let no one ignorant of geometry enter ) was inscribed over the door to the Academy Plato distinguished between ideal, non-physical mathematical objects (e.g., The Circle ) and daily approximations (e.g., any circle we draw on paper) Platonism is the school of philosophy, inspired by Plato, that believes in existence of abstract objects independent of the human mind Allegory of the Cave: We are all chained prisoners in a cave who are not allowed to turn our heads. There are objects outside of the cave of which we can see only the shadow (a circle we draw on paper), but not the object itself ( The Circle ).

17 Plato Those who are to take part in the highest functions of state must be induced to approach it, not in an amateur spirit, but perseveringly, until, by the aid of pure thought, they come to see the real nature of number. They are to practise calculation, not like merchants or shopkeepers for purposes of buying and selling, but with a view of war and to help in the conversion of the soul itself from the world of becoming to truth and reality. (Note the practical purpose of war.) As we were saying, it has a great power of leading the mind upwards and forcing it to reason about pure numbers, refusing to discuss collections of material things which can be seen and touched. (Plato, The Republic. Translated by F. Cornford.)

18 Zeno Zeno of Elea Born in the 5th century BC Most famous for the paradoxes he proposed Version 1: To move from point A to point B, a person must first reach the midpoint, but he must also reach the midpoint between A and that midpoint, and so on. The person must cover an infinity of midpoints, so it is impossible for him to move from point A to point B.

19 Zeno Zeno of Elea Version 2: For a quicker runner to overtake a slower runner, he must first reach the point where the slow runner started, by which time the slower runner will have run a distance, so the quicker runner must reach the point where the slower runner is now, by which time the slower runner will have run another distance, and so on. Therefore the quicker runner will never be able to catch up to the slower runner. His paradoxes were refuted by Aristotle

20 Aristotle Aristotle ( BC) Studied at Plato s Academy from the age of 18 until Plato s death in 347 BC Later invited to the court of Philip II of Macedon to teach his son Alexander (later Alexander the Great) Then returned to Athens to found his own school, the Lyceum (root of the French word lycée, secondary school) List of works include: Physics, Metaphysics, Prior Analytics, Posterior Analytics, Poetics, Rhetoric, Politics, The Athenian Constitution, On Dreams, Virtues and Vices, On Youth and Old Age, On Life and Death, On Breathing

Aristotle Figure: Aristotle tutoring Alexander, by J L G Ferris 1895.

21 Aristotle Figure: Aristotle tutoring Alexander, by J L G Ferris Source: alexander/index.htm

22 Aristotle Logic Aristotle believed that arguments should be built out of syllogisms Syllogism: Discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so Aristotle, Prior Analytics I, 1, 24, 19 Translation: Great Books Edition, Chicago: Encyclopedia Britannica 1952 A syllogism therefore contains certain statements that are taken as true and some other statements which must be true by consequence

23 Aristotle Syllogism example All men are mortal Socrates is a man Therefore, Socrates is mortal

24 Aristotle Logic Allows one to use old knowledge to produce new knowledge However one cannot obtain all knowledge as results of syllogisms We must start somewhere with truths which we accept without argument Postulate: Basic truth peculiar to a particular science Example: Euclid s (One is always able) To draw a straight line from any point to any point. Axiom: Basic truth common to all sciences Example: Euclid s Things which are equal to the same thing are also equal to one another.

25 Aristotle How did Aristotle refute Zeno s paradoxes? While space can be infinitely divided according to Zeno, time can also be divided likewise. He said while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect to divisibility, for in this sense time itself is also infinite. He distinguished between infinite and infinitely divisible. However he did not have the tool of convergent infinite series, where a sum of infinitely many terms may be finite.

26 Chrysippus Later on, Chrysippus ( BC) analyzed more forms of argument Modus ponens ( the method that affirms ) Modus tollens ( the method that denies ) Hypothetical syllogism Disjunctive syllogism

27 Modus ponens Modus ponens If P, then Q. P. Therefore, Q.

28 Modus ponens Modus ponens example: If this drink contains sugar, then this drink is sweet. This drink contains sugar. Therefore, this drink is sweet.

29 Modus tollens Modus tollens If P, then Q. Not Q. Therefore, not P.

30 Modus tollens Modus tollens example: If this drink contains sugar, then this drink is sweet. This drink is not sweet. Therefore, this drink does not contain sugar.

31 Hypothetical syllogism Hypothetical syllogism If P, then Q. If Q, then R. Therefore, if P, then R.

32 Hypothetical syllogism Hypothetical syllogism example: If this drink contains sugar, then this drink is sweet. If this drink is sweet, then Emma will not drink it. Therefore, if this drink contains sugar, then Emma will not drink it.

33 Disjunctive syllogism Disjunctive syllogism P or Q. Not P. Therefore, Q.

34 Disjunctive syllogism Disjunctive syllogism example: Emma s car is red, or Emma s car is blue. Emma s car is not red. Therefore, Emma s car is blue.

35 Museum of Alexandria A research institute Built around 280 BC by Ptolemy I Soter (not to be confused with Ptolemy the astronomer) Buildings were destroyed in 272 AD in a civil war under the Roman emperor Aurelian Fellows of the museum received stipends, free meals, and were exempt from taxes The famous Library of Alexandria is part of it Museum Temple of the Muses Muses nine goddesses inspiring learning and the arts; daughters of Zeus

36 Muses The nine muses: Calliope (epic poetry), Clio (history), Euterpe (lyric poetry), Thalia (comedy), Malpomene (tragedy), Terpsichore (dance), Erato (love poetry), Polyhymnia (sacred poetry; hymns), Urania (astronomy)

37 Muses Figure: Nine Muses, by Samuel Griswold Goodrich. Source: Nine_Muses_-_Samuel_Griswold_Goodrich_(1832).jpg

38 Alexandria Figure: Alexandria on a modern map. Source: Google Map

39 Euclid Not much is known about his life It is believed that he taught and wrote at the Museum of Alexandria Died in Alexandria in 265 BC

40 The Elements Thirteen books Definitions, axioms, theorems, proofs His way of thinking influenced modern mathematics, which follow an axiomatic approach. To the extreme, you forget the meaning of a mathematical object and just consider how they interact with other mathematical objects. For example, you don t care what lines and points mean, all you care about is that two points give a line (the one that joins them) and two (non-parallel) lines give a point (where they intersect)

41 The Elements Some of the definitions from Book I: 1. A point is that which has no part. 2. A line is breadthless length 4. A straight line is a line which lies evenly with the points on itself. 15. A circle is a plane figure contained by one line such that all the straight lines meeting it from one point among those lying within the figure are equal to one another; 16. and the point is called the centre of the circle. 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

42 The Elements Note by modern standard the definitions are not good definitions It was not clear what the terms part, breadthless, evenly mean. In modern mathematics, the word set remains undefined, but we have a list of well-defined operations which we are allowed to perform on sets.

43 The Elements Postulates (truths peculiar to the science of geometry): 1. (One is always able) To draw a straight line from any point to any point. 2. (One is always able) To produce a finite straight line continuously in a straight line. 3. (One is always able) To describe (construct) a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line intersecting two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

44 The Elements Common notions (axioms, truths common to all sciences, logical truths): 1. Things which are equal to the same thing are also equal to one another. 2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part.

45 The Elements Book I, Proposition I: To construct an equilateral triangle on a given finite straight line. (A possibility kind of proposition) This is the very first proposition, so Euclid could only use the definitions, postulates and axioms By Postulate 3, he could construct one circle with centre A and radius AB and another with centre B and radius BA The two circles intersect at a point C By Postulate 1, he could draw the lines AC and BC By Definition 15, AC equals AB and BC equals BA By Common Notion 1, AC, AB and BC are equal Gap: How did Euclid know the two circles intersect? Some postulate of continuity (if a line crosses from one side of a line to the other side, the two lines must intersect) is necessary Such problems will be dealt with in 19th-century mathematics

46 The Elements Figure:

47 The Elements Some of the definitions from Book VII: 1. A unit is that by virtue of which each of the things that exist is called one. 2. A number is a multitude composed of units. 3. A number is a part of a number, the less of the greater, when it measures the greater; 4. but parts when it does not measure it. 11. A prime number is that which is measured by the unit alone. 12. Numbers prime to one another are those which are measured by the unit alone as a common measure. 15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

48 The Elements Book IX, Proposition XX: Prime numbers are more than any assigned multitude of primes. (In more modern mathematical language: Given any finite set of prime numbers, there is at least one prime number which does not belong to the set.) Given any fixed number of prime numbers, you can always find one more, i.e., there are infinitely many prime numbers. Let A, B, C be three prime numbers Consider ABC + 1 If ABC + 1 is prime, we have a new prime If not, then ABC + 1 has some prime factor G. If G is either A, B or C, then G is a factor of 1, a contradiction Therefore G is a prime distinct from A, B or C Note: Euclid gave his proof with three primes, but the same proof may be given for any finite number of primes p 1, p 2,..., p n. Consider p 1 p 2... p n + 1.

49 The Elements Book XIII Devoted to the study of regular polyhedra (also known as Platonic solids ) Each face is a regular polygon An equal number of faces meet at each vertex There are five: tetrahedron (four triangles, three meeting at each vertex), cube (six squares, three meeting at each vertex), octahedron (eight triangles, four meeting at each vertex), dodecahedron (twelve pentagons, three meeting at each vertex), icosahedron (twenty triangles, five meeting at each vertex) Book XIII contained a proof that those are the only ones

50 The Elements Figure: Platonic solids. Source:

51 Archimedes Archimedes ( BC) Born in Syracuse Highly probable that he studied in Alexandria Familiar with all work previously done in mathematics Later returned to Syracuse where he helped King Hieron by applying his knowledge to construct war-engines Finally the Romans took the city and Archimedes was killed by a Roman soldier Last words were said to be Don t disturb my circles, referring to a picture he was contemplating when the Roman soldier approached him The Roman general Marcellus admired him and constructed a tomb in his honour, with a sphere inscribed in a cylinder

Archimedes Figure: Death of Archimedes, by Thomas Degeorge 1815.

52 Archimedes Figure: Death of Archimedes, by Thomas Degeorge Source: Degeorge/degeorge.png

53 Archimedes On the Measurement of the Circle Proposition 1: The area of any circle is equal to the area of a right triangle in which one of the legs is equal to the radius and the other to the circumference. Exhaustion argument: Let K be the area of the given triangle and A be the area of the circle. Suppose A > K. By inscribing in the circle polygons of increasing numbers of sides, eventually gets a polygon with area P with A P < A K. Therefore P > K The perpendicular from the centre of the circle to the midpoint of a side of the polygon is shorter than the radius, and the perimeter of the polygon is less than the circumference. Therefore P < K. CONTRADICTION. Therefore A must be less than equal to K Similarly assuming A < K will lead to another contradiction Therefore A = K

54 Archimedes On the Measurement of the Circle Proposition 3: The ratio of the circumference of any circle to its diameter is less than but greater than 3 71 Proved by finding the ratios of the perimeters of the inscribed and circumscribed 96-sided polygons to the diameter

55 On the Equilibrium of Planes On the Equilibrium of Planes: Mathematical theory of the lever

56 On the Equilibrium of Planes Some Postulates: 1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline toward the weight which is at the greater distance. 2. If, when weights at certain distances are in equilibrium, something is added to one of the weights, they are not in equilibrium but incline toward the weight to which the addition was made 3. Similarly, if anything is taken away from one of the weights, they are not in equilibrium but incline toward the weight from which nothing was taken 6. If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances

57 On the Equilibrium of Planes Some Propositions: 3. Suppose A and B are unequal weights with A > B which balance at point C. Let AC = a, BC = b. Then a < b. Conversely, if the weights balance at a < b, then A > B 6, 7. Two magnitudes, whether commensurable (Prop 6) or incommensurable (Prop 7), balance at distances inversely proportional to the magnitudes.

58 Archimedes Figure: On the Equilibrium of Planes. Proposition 3

59 Appolonius Appolonius Born in Perga, studied at Alexandria under successors to Euclid, and composed the first draft of The Conic Sections there Later moved to Pergamum, which had a new university and library modeled after those in Alexandria

60 Appolonius Conic Sections Eight books First four books have been passed down to us in the original Greek, and the next three books were unknown in Europe until Arabic translations were found. The eighth book is lost. Intersection of a plane and cones gives three types of curves: ellipses, parabolas and hyperbolas

61 Appolonius Figure: Conic sections. Source:

62 Appolonius Conic Sections Appolonius discovered what were equivalent to modern equations of the parabolas, ellipses and hyperbolas Studied asymptotes to the hyperbolas (in Greek, asymptotos means not capable of meeting ) Showed how to construct a hyperbola given a point on the hyperbola and its asymptotes Also studied tangent lines (a line which touches the curve but does not cut the curve)

63 Ptolemy Ptolemy Native of Egypt Major works include Geography and Mathematicki Syntaxis ( Mathematical Collection ) Later Mathematicki Syntaxis became known as Megisti Syntaxis ( The Greatest Collection ), and the Arabs called it al-magisti. Now people refer to it as the Almagest. First recorded observation was made in 125 AD, last one was in 151 AD

64 Ptolemy Almagest Composed of 13 books and is considered the culmination (peak) of Greek astronomy Contains a table of chords from 1 2 degree to 180 degrees in intervals of 1 2 degree Ptolemy did all his computations in a base-60 system Square roots were involved but Ptolemy did not describe how he calculated them A commentary by Theon in the fourth century explained a method Ptolemy could have used Also contains work on plane and spherical trigonometry (with obvious astronomical implications)

65 Diophantus Lived in Alexandria Major work is called Arithmetica, which has 13 books, but only 6 survived in Greek Four others (4 to 7) were recently discovered in an Arabic (translated) version

66 Diophantus Arithmetica Like the Rhind Papyrus, it is a collection of problems Only positive rational answers were allowed For example, 4x + 20 = 4 has no solution We look at two examples (given in modern notation)

67 Diophantus Arithmetica Example 1. Book I, Problem 17: Find four numbers such that when any three of them are added together, their sum is one of four given numbers. Say the given sums are 20, 22, 24, and 27. Solution: Let x be the sum of the four numbers. The four numbers are, respectively, x 20, x 22, x 24 and x 27 We have x = (x 20) + (x 22) + (x 24) + (x 27). Therefore x = 31 and the numbers are 11, 9, 7 and 4.

68 Diophantus Arithmetica Example 2. Book II, Problem 8: Divide a given square number, say 16, into the sum of two squares. Let x 2 be one of the squares 16 x 2 = (2x 4) 2 The 4 is meant to cancel the 16, the choice of 2 was arbitrary The equation becomes 5x 2 = 16x. The positive solution is x = 16 5 Therefore one square is , and the other is =

69 Diophantus In modern mathematics, a Diophantine equation is an equation for which only integer solutions are allowed.

70 Decline of Greek Mathematics The Romans held a utilitarian view towards mathematics Focus was on application of arithmetic and geometry to engineering and architecture The Greeks held the geometer in the highest honour; accordingly nothing made more brilliant progress among them than mathematics. But we have established as the limits of this art its usefulness in measuring and counting. Cicero, Roman politician

AMA1D01C Ancient Greece

AMA1D01C Ancient Greece Hong Kong Polytechnic University 2017 References These notes follow material from the following books: Burton, D. The History of Mathematics: an Introduction. McGraw-Hill, 2011. Cajori, F. A History of