Part I. from Arithmetic to Pythagoras' Theorem

MATHEMATICS IN OLD GREECE FROM THE VIth TO THE IVth CENTURY Before Common Era (BCE) FROM PYTHAGORAS TO EUCLID

History of mathematics, Bologna, October 2013

Part I

From Arithmetic to Pythagoras’ theorem

I. Introduction.

Before beginning the course, we need to understand its interest, the main problems and difficulties, and the background.

As very young people, something which happened more than 25 centuries ago, for most of you more than hundred times your age, could seem very, very, very

boring.

No blackboard, no pens, no cars and even no mobiles, what is it possible to expect from people living like this? Since mathematics has made huge progress from this time, why learning what was done then? Since we have no more any book from this time, how is it possible to say anything about it?

- What: Contrary to nowadays, at this time there were no clear borders between such different fields as Mathematics, Philosophy, Astronomy, Physics and even Music. As such it is a crucial period to try understanding what mathematics is. - Why: To understand what something is we need to know its origins. It is true for our present world, and it is not different for mathematics. To know what we are doing with mathematics, we need to know something about its origins, how people began to think such abstract objects like numbers, forms and space. - How: The first complete mathematical treaty we have is the Euclid’s Elements, from the 4th century BCE. Before, not only we have no treaty, even incomplete, but we have no mathematical text. Moreover, the most ancient mathematical passages are not found in mathematical books, there are no such books, but in texts from completely different fields, such as philosophy or rhetoric or even literature. We will have to study and interpret them with all the difficulties it implies. - Contrary to mathematics where every (solvable) problem seems to have one solution, we are never sure here of even the existence of a solution. At best we have more or less good hypotheses, so everyone has to make his own inquiry, to check and finally to decide which one is the most credible one.

A claim, found from time to time, such the following: ‘This was proved by XXX in her dissertation YYY. Her results in this respect seem absolutely certain and have been universally accepted’1 is certainly wrong (‘universally accepted’), and shows rather a refusal to discuss the question (since it is solved once and for all), implying (almost) surely the weakness of the argumentation.

1 Excerpt of an article of Kurt von Fritz, The Discovery of Incommensurability by Hippasus of Metapontum, Annals of mathematics, 46, 2, 1945, otherwise an extremely interesting article. 2

II. Pythagoras and the Pythagorean School.

The difficulties to study Pythagoras and his school are due, among other things, to the following hindrances:

- We have no texts of him, not even indirectly, and he is said not to have ever written anything, though some authors as Diogenes Laertius disagree. The information we get are more and more precise as time is passing, a sure indication of a legend in building. - For instance, almost 1000 years after his death, we have a life of Pythagoras with a wealth of details written by Iamblichus. But till the 4th century BCE, we have only few sentences on him. Nevertheless, the absence of information on people especially mathematicians (including Euclid) is the norm in the Antiquity. - Since he was the chief of a school, his followers were keen to give him all the discoveries by members of the school. Later, when commentators wanted to consider some result of unknown author(s), it was convenient to put his name for discoveries made at his epoch. Because of the structure of mathematics, it is much more difficult to give a wrong period for the obtention of a result than to give a wrong name for the one who got it.

So we have to carefully check everything attributed to him. Fortunately, we are more interested in the dating of mathematical results than in their authors. Another problem is his school lasts 2 centuries (roughly from the beginning of the 6th century to the beginning of the 4th century BCE). Even if we agree a result was known by the Pythagoreans, its dating is not obvious. So once again we need to compare the testimonies, to oppose them each other and to what was possibly known at this time in mathematics.

We will proceed as the following: since even people who lived thousand years after Pythagoras were much closer from him and the sources on him than us, we have to take seriously their claims. But we have also to be very circumspect with these texts and to discuss their plausibility.

The very existence of Pythagoras is sometimes put in doubt. Nevertheless many early authors are alluding directly or indirectly to him, among them: Herodotus, Heraclites and Empedocles (cf. Diogenes Laertius, Lives and Opinions…, book VIII).

Plato spoke twice in the Republic about him or his school: ‘The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences - as the Pythagoreans say, and we, Glaucon, agree with them?’ (Republic,, VII, 530d) ‘But, if Homer never did any public service, was he privately a guide or teacher of any? Had he in his lifetime friends who loved to associate with him, and who handed down to posterity a Homeric way of life, such as was established by Pythagoras, who was so greatly beloved for his wisdom, and whose followers are to this day quite celebrated for the order which was named after him?’ (ib., X, 600b).

In another book, Plato refers probably to him or some of his followers, when he spoke about ‘some smart fellow, a Sicilian, I daresay, or Italian’ (Gorgias, 493a).

There are also many texts of Aristotle on the ‘Pythagoreans’, but few on Pythagoras himself.

Since his partisans as well as the commentators were eager to give the paternity of many discoveries to the chief of their school, it is preferable to understand the name ‘Pythagoras’ as indicating rather than the man himself the school of the ‘old Pythagoreans’ i.e. a group of men2 around the time of Pythagoras. Moreover in mathematics it is much more difficult to make wrong claims concerning the dating of results rather than its author. So without strong reasons running against them, we may accept at least the dating for the theorem attributed to Pythagoras or his school i.e. the 6th century BCE.

Inversely some keep extremely sceptical views about Pythagoras (and the Pythagoreans) with respect to mathematics, as seen in the below quotation extracted from the Stranford Encyclopedia of Philosophy:

‘Pythagoras, one of the most famous and controversial ancient Greek philosophers, lived from ca. 570 to ca. 490 BCE. (…) Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries. By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato's and Aristotle's mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view. The Pythagorean question, then, is how to get behind this false glorification of Pythagoras (…) The early evidence shows, however, that, while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested.’

2 And of women, since if Iamblichus (Pythagoras’ life, 267) and (maybe) Diogenes Laertius (Lives and Opinions…, VIII, 41) are right, women were accepted in his school 4

III. About Pythagoras’ life.

Everything about Pythagoras’ life is extremely controversial, more or less legendary. Nevertheless, keeping this in mind, we will give some indications on his life with which most scholars agree.

- He lived in the 6th century (some authors give more precise date like 570-495 BCE). He was born at Samos (a Ionian Greek city)

and died at Metapontum (near the actual Taranto)

in so-called ‘Great Greece’ i.e. the south of actual Italy. - He traveled and stayed many years in Egypt, then in Babylon, according to Iamblichus, as a ‘prisoner of war’, but this is highly disputed. - Around 520 BCE he settled in ‘Great Greece’ firstly in Croton then in Metapontum where he died.

In Croton, he founded a school with some strict rules of secrecy, at once religious, philosophical and political. These rules are often connected to the secrecy of religious ceremonies in Egypt (Herodotus, Histories, II, 81), but they could as easily be Greek, for instance to the so-called ‘Orphic mysteries’.

To summarize, let us give once again a skeptical point of view, as the following quotation from Luc Brisson:

‘From an historic point of view, we know very little on the origins, the formation and the activity of Pythagoras. He would be born at Samos at the beginning of the 6th century and would emigrate to Croton, where he would have taken up the power. A revolt would overturn the Pythagoreans at the end of the century, a little after 510, but the eclipse was short-lived since it seems they control a solid bloc of territory between Metaponte and Locres till 450.’3

3 p. 39, Eikasia. Revista de Filosofía, 12, Extraordinario I, 2007, p. 39-66. 6

IV. About Pythagoras’ (or Pythagorean) doctrine.

According to most authors in the Antiquity, Pythagoras wrote nothing. In any case we have no writings of him, but we have a wealth of texts on him highly contradictory, some of them being impossible to believe. Herodotus (beginning of the 5th century BCE) who wrote one of the first testimonies on Pythagoras is already highly skeptical of some stories told on him (The Histories, IV, 95-97). He is presented as a highly estimated man, mixing Egyptian religious beliefs, (ib., II, 81) with ‘philosophy’ and knowledge (Herodotus says of him he was ‘not the least able philosopher’ (ib., IV, 80), in agreement with Plato who told us he had many almost fanatical followers (Republic, 600a-b). Xenophanes (end of 6th century BCE) reports their doctrine of the immortality of the souls and of transmigration (in Diogenes Laertius, Lives and Opinions …, VIII, 36). According to Heraclitus (also at the end of the 6th century BCE), he was a man of extremely large knowledge: ‘Pythagoras, son of Mnesarchus, practiced inquiry beyond all other men’ (ib., VIII, 6).

Aristotle (4th century BCE) gave many texts about the scientific, especially mathematical, interests of the Pythagorean school (Physics, III, 4, 203a1-16; 4, 204a33-35; IV, 6, 213b22- 28; On the Heavens, I, 1, 268a10-15; II, 2, 284 b6-11; 285a10-15, b23-26; II, 9, 291a7-14; II, 13, 293a19-b6; III, 1, 300a15-17; Metaphysics, I, 5, 985b23-986b8; 987a9-27; 6, 987b10, 22; 8, 989b32-990a32; III, 1, 996a4-8; 4, 1001a9-14; VII, 11, 1036 b 17-20; X, 2, 1053b11- 12; XII, 7, 1072b31-35; XIII. 4, 1078b21-24; 6, 1080b16-20; 8, 1083b9-19; XIV, 2, 1090a20- 24; Nicomachean Ethics, I, 6, 1096b5; II, 5, 1106b29; V, 8, 1132 b 21; ).

The critics of the ‘Pythagoreans’ by Aristotle are often connected to Plato4, because they put numbers everywhere and did not try to save the phenomena: ‘At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth. In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own.’ (On the Heavens, II, 13, 293a19-b6)5.

4 ‘After the systems we have named came the philosophy of Plato, which in most respects followed these thinkers, but had peculiarities that distinguished it from the philosophy of the Italians.’ (Metaphysics, A, 6, 987a29-31). 5 Inversely Simplicius, a Platonician of the 6th century CE, quoting Sosigenes (an astronomer of 1st century BCE), told us Plato set to the astronomers the following purpose: ‘Sw/vzein ta; fainovmena’ (‘to save the phenomena [or the appearances]’) (Commentary on Aristotle’s treaty On the Heavens, 488.21). 7

In later sources (1st century BCE-5th century CE) Pythagoras was sometimes presented as a semi-divine figure by some authors6, but this was met with much skepticism by many others.

6 Cf. for instance the Stanford Encyclopedia of Philosophy as quoted in the previous paragraph. 8

V. Pythagoras and arithmetic.

‘Even earlier the so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.’ (Aristote, Metaphysics, I, 5, 985b23-26).

i) Which number is a number (‘ajriqmov"’)?

The fundamental division of numbers in Greek mathematics was between the ‘odd’ (‘perissov"’) and ‘even’ (‘a[rtio"’) so that the science of odd and even was often used as a synonym of arithmetic. In book VII of Euclid’s Elements, we have the following definitions:

Definition 6. An even number is one (which can be) divided in half. Definition 7. And an odd number is one (which can) not (be) divided in half, or which differs from an even number by a unit.7

These definitions are certainly very ancient, and we find for instance analogous definitions in Plato.

The problem of the ‘one’ or the ‘unity’.

It can be understood in the background of the opposition even/odd. According to Aristotle and other authors, for some Pythagoreans, the unity was neither odd nor even, and for some others it was odd-and-even. Since only numbers can be either odd or even, this reflects a more fundamental question: is one a number?

The one is a generator of numbers. As a consequence, according to Aristotle, it cannot be a number, since it is not right to confuse what measures (the unit) and what is measured (the numbers): ‘Hence too it stands to reason that unity is not a number; for the measure is not measures, but the measure and unity are starting-points.’(Metaphysics, N8, 1, 1088a6-8). It is also a consequence of the definition of number given by Euclid in book VII: ‘A number is a multitude composed of units’ (definition 2). According to some Pythagoreans even 2 could be considered not being a number since it generates the even numbers.

7 Some mathematicians criticize this definition as non-rigorous. Can you guess why? 8 Sometimes the books of Aristotle’s Metaphysics are referred by numbers, sometimes by letter. The correspondence is the following: book 1 is called Alpha (Α); 2, little alpha (α); 3, Beta (Β); 4, Gamma (Γ); 5, Delta (Δ); 6, Epsilon (Ε); 7, Zeta (Ζ); 8, Eta (Η); 9, Theta (Θ); 10, Iota (Ι); 11, Kappa (Κ); 12, Lambda (Λ); 13, Mu (Μ); 14, Nu (Ν). 9

Thus in most books of history of mathematics, it is claimed one was not be a number for the ancient Greeks.

This is an appearance rather than a reality. Depending of the situation, one was considered to be or not to be a number. For instance, according to Plato: - ‘the smallest [number] comes into being first and that is the one’ (Parmenides, 153a), then again till 153e. - One is said odd and two is said even, but odd and even are characters of numbers (according to Plato, arithmetic is the science of the numbers but also of ‘the odd and even’) (Major Hippias, 302a). - In Theaetetus, any integer n is figured as a rectangle whose sides are integers m and p, and of surface equal to n (i.e. n = m  p). So that, for a prime number n, one of the numbers m or p need to be equal to 1, thus 1 is a number.

Even in Euclid we find the same double use for one. For instance, the proposition 15 of the book VII of the Elements claims: ‘If a unit measures some number, and another number measures some other number as many times, then, also, alternately, the unit will measure the third number as many times as the second (number measures) the fourth’, The meaning of the proposition is:

Let 1 be the first number, m the second, p the third, q the fourth: if m/1 = q/p then p/1 = q/m, so that 1 is a number.

Even the definition VII.2 (cf. supra) is not so clear since the term ‘multitude’ is the translation of the Greek word ‘plh'qo"’ which could also mean a quantity containing just one element: for instance Iamblichus speaks of ‘plh'to" e{n’ (‘a multitude of one’ [element], In Nicomachi Arithmeticam Introductionem Liber, p. 11, 8-9).

Of course, it is an example of late Antiquity, but there are many of them before and after Euclid: Herodotus, The histories, 3, 6, 1; Epitomis 978b-c (vs. 977c where an opposite claim is made); Speusippe, fragment 18, in Iamblichus, Theologoumena Arithmeticae 82.10-85.32 Nichomachus of Gerasa (neo-Pythagorean 60-120 CE): the unit is ‘the natural starting of all numbers’ (Introduction to Arithmetic, VIII, 1) and so on.

The discussion about the question of one as a number is answered differently by different authors and even by the same author in different contexts. Thus, the double sense of the one was present in the old Pythagorean School as it was later.

The Aristotelian strict distinction between the generator and what is generated, between the one and the other numbers, was not observed by the Greeks, at least in mathematics.

In conclusion, we can either say the ancient Greeks were not rigorous with respect to ‘one’, or they had two conceptions of it, a larger one and a stricter one, depending of the context9.

9 Such ambiguities are not so uncommon in mathematics even in modern one. For instance we can think about the meaning of the order relation ‘greater’ (or ‘smaller’) which can mean either strictly greater or greater-or- equal, depending of the context 11

ii) An example of the study of numbers by the Pythagoreans.

Nicomachus (a neo-Pythagorean around the first century CE) tells us the old Pythagoreans divided the even numbers in 3 classes:

the even-even i.e. the numbers of the form : 2n the even-odd i.e. of the form : 2u (u is odd) the intermediates i.e. numbers of the form : 2ku (u is odd).

The Pythagoreans also defined what they called ‘friendly numbers’ (Iamblichus, In Nicomachi…, p. 35, 1-7, credits directly Pythagoras for this). Two numbers m and n are ‘friendly’ if the sum of the divisors of m (except m itself) is equal to n and the sum of the divisors of n (except n itself) is equal to m.

It is not even evident at all such a couple exists. But let put m = 220 and n = 284. The divisors of m are: 1, 2, 4, 5, 10, 11, 20, 32, 44, 55 and 110 The divisors of n are: 1, 2, 4, 71, 142. The sum of the divisors of m is: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 32 + 44 + 55 + 110 = 284 The sum of the divisors of m is: 1 + 2 + 4 + 71 + 142 = 220. So 220 and 284 are a pair of ‘friendly numbers’.

It is not easy to find such pairs, and it was the only one such pair known until Fermat found another one (17296 and 18416)10, then Descartes11 (9363584 and 9437056). Later Euler found at once 59 new pairs. Till now, it is not known if there exist an infinite number of such pairs. Beware: in modern arithmetic, ‘friendly number’ has another meaning12, so what was till recently called ‘friendly numbers’ is now often called ‘amicable numbers’.

It is an example showing how deeply the Pythagorean School was working on numbers.

10 17296 = 16×23×47 and 18416 = 16×1151. 11 9363584 = 128×191×383 and 9437056 = 128×73727. This pair and the above one as well are said to have been discovered several centuries before, by Arab mathematicians (for example, Song Y. Yan, Perfect, Amicable, and Sociable Numbers: A Computational Approach, Scientific World) 12 It is the set of numbers whose ratio between the sum of all divisors and this number is the same. For instance, in this sense, 6 and 28 are friendly numbers since the divisors of 6 are: 1, 2, 3 ; the divisors of 28 are 1, 2, 4, 7, 14; so that (1 + 2 + 3)/6 = 1 and (1 + 2 + 4 + 7+ 14)/28 = 1. 12

iii) The representations of numbers.

1. The Pythagorean School represented usually the numbers by pebbles, arranging them into geometric figures (Aristotle, Metaphysics N, 5, 1092b12; Z, 1036b7- 19). It was then possible to arrange numbers as triangles, squares or other figures, and to deduce some properties from them.

For instance, we have the following triangular diagrams:

Figure 1

On the left side, there is the sum of the four first numbers (1, 2, 3, 4,5) represented by blue circles and the triangle is completed by adding the same triangle but without the last line of 5 pebbles (it is the red triangle on the right in the above figure). The figure on the right is a square composed of 5 lines of 5 pebbles, so we get:

(1 + 2 + 3 + 4 + 5) + (1 + 2 + 3 + 4) = 2 (1 + 2 + 3 + 4) - 5 = 52, thus 1 + 2 + 3 + 4 + 5 = (52 + 5)/2 = (5×6)/2 = 15.

This representation can be done for any integer n, so we obtain the formula: The sum of the n first integers is equal to n(n+1)/2 i.e (1 + 2 + … + n) = n(n+1)/2.

2. Representations by square.

From Aristotle (Physics, III, 4, 203a13-15), we can deduce the Pythagoreans used the following figure:

Figure 2

We see immediately on this figure the sum of the 4 first odd integers is a square 4 × 4 so that 1 + 2 + 5 + 7 = 42. The part added every time to the preceding square is called a ‘gnomon’ (‘gnwvmwn’), meaning roughly a ‘set square’. This same representation can be done for any odd integer so that, we obtain the following formula: The sum of the n first odd integers is equal to n2, or : 1 + 3 + 5 + … + (2n-1)+ (2n+1) = (n+1)2. Since 1 + 3 + 5 + … + (2n-1) = n2, another evident consequence read in the above figure is: (n+1)2 = n2 + 2n +1.

Using other geometric figures, it is possible to obtain more formulas for other arithmetic series.

iv) Pythagorean triples.

1. Pythagoras and its triples.

Definition. Pythagorean triple It is a triple (m, n, p) of integers such that the square of the last number is equal to the sum of the squares of the first13 i.e. m2 + n2 = p2.

Using the formula of §iii.2 (1 + 3 + 5 + … + (2n+1) = (n+1)2), it is easy to obtain such triples.

Let j be an odd integer and a (perfect) square, for instance j := 9 = 32. According to the formula, we have: 1 + 3 + 5 + 7 is a (perfect) square as well as 1 + 3 + 5 + 7 + 9 so that, without any other computation, we know: (9, (1 + 3 + 5 + 7), (1 + 3 + 5 + 7 +9)) is the square of a Pythagorean triple.

Verification: 1 + 3 + 5 + 7 = 16 = 42 1 + 3 + 5 + 7 +9 = 25 = 52 and 32 + 42 = 9 + 16 = 25 = 52, and we get the well-known first Pythagorean triple: (3,4,5).

The next one is j := 25 = 52. Then the sum 1+ 3 + …+ 19 + 23 is a (perfect) square as well as 1 + 3 + … + 23 + 25, so that (25, 1+ 3 + …+ 19 + 23, 1 + 3 + … + 23 + 25) is the square of a Pythagorean triple. Verification: 1 + 3 + … + 19 + 23 = 144 = 122 1 + 3 + … + 19 + 23 + 25 = 169 = 132 and 52 + 122 = 25 + 144 = 169 = 132. As you see, it is not necessary to compute the sum to know we have a Pythagorean triple!

This is completely general, and if j is any odd (perfect) square i.e. j = m2, since (from the above formula) the numbers: 1 + … + (j-1) = h and 1 + … + (j-1) + j= k are (perfect) squares so that h = n2 and k = p2, we have: h + j = k i.e. m2 + n2 = p2 and (m, n, p) is a Pythagorean triple14.

13 To get the unity of the triple, we will also ask the following inequalities: m < n < p. 15

Proclus told us Pythagoras got this formula (Commentary of the first book of Euclid’s Elements I, 428-429) which can be written in modern symbolism: for m odd, m2 + ((m2 - 1)/2)2 = ((m2 + 1)/2)2.

A natural question arises: does the formula give all the Pythagorean triples? Since we have: 82 + 152 = 64 + 225 = 289 = 172, the triple (8, 15, 17) is Pythagorean, and since m = 8, thus j = 82 = 64 is not odd15, they are not obtained from the preceding formula. Thus, the answer to the question is negative.

In the same text, Proclus attributes to Plato (another author, Boethius (end of the 5th century) attributes it to the mathematician Archytas) the formula giving Pythagorean triples of the second form, which in modern symbolism would be written as the following: (2m)2 + (m2 – 1)2 = (m2 + 1)16. For instance, for m = 4, we obtain: 82 + 152 = 172.

If (m, n, p) is a Pythagorean triples for any integer t, then (tm, tn, tp) is also a Pythagorean triple. To differentiate non-trivial different Pythagorean triples, such a triple is called primitive if the three numbers m, n, p are relatively prime.

The general formula is given by Euclid (Lemma 1 of the proposition 28, book X)17.

14 According to our definition, this is not completely true, we has also to prove the inequalities of the preceding note. The only non-trivial one is m < n i.e. j < h. According to the formula of §2, it means: j < ((j+1)/2)2 i.e. 4j < j2 + 2j + 1 or 0 < (j-1)2 i.e. 1 < j. 15 Beware, the fact m is the smallest number of the triple is important to be able to conclude immediately. 16 It is the above formula for the odd numbers, but multiplied by 4. 17 In modern notation, it consists for a and b relatively prime and a > b, in the following formula: (a2 – b2, 2ab, a2 + b2). This formula gives all the primitive Pythagorean triples, but it is not made explicit by Euclid. It gives also some non-primitive triples, could you see why and which ones? 16

2. Babylonians and the Pythagorean triples.

As for many results, the knowledge of Pythagorean triples may have been found much before Pythagoras and even the ancient Greeks. According to the interpretation of many scholars working on Babylonian mathematics such triples were known 1000 years before Pythagoras. Their claim is funded on a clay tablet called ‘Plimpton 322’ below:

Plimpton 322

which gives a list of numbers. There is immediately a problem for its interpretation since, without any notation for zero, there is always an ambiguity to understand the meaning of a number in Babylonian representation. In their sexagesimal numeral system (base 60), ‘2:16’ for instance could mean either 2 × 60 + 16 = 136 or 2 × 602 + 16 = 7216 or even the fraction 2 + 16/60 = 34/15 (see also the study of the tablet YBC7286, infra §vii).

1:59 2:49 1 56:07 1:20:25/3:12:1 2 1:16:41 1:50:49 3 3:31:49 5:09:01 4 1:05 1 :37 5 5:19 8 :01 6 38:11 59:01 7 13:19 20:49 8 8:01/9 12:49 9 1:22:41 2:16:01 10 45 1:15 11 27 :59 48:49 12 2:41/7:12:1 4:49 13 29:31 53:49 14 56 ? 1:46/53 15

Tableau 1

According to the usual interpretation the first two rows aim to give respectively the smallest and the greatest numbers of a Pythagorean triples. But there is no Pythagorean triple as such on the tablet.

There are also some mistakes: if it is true the tablet aims to give indeed such a list of parts of Pythagorean triples, some corrections are needed. They are written in red in the above table.

The mistakes could be explained as follows:

- first column 8 instead of 9, a copying mistake - first column 2:41 = 161 and 7:12:1 = 25200 + 720 + 1 = 25921 = 1612, so the square is written instead of the number - 1:46 = 106 is the double of 53 - for the second row of the second column no explanation has been found.

The two first columns contains numbers in sexagesimal numeral, for instance the first one would be (according to the simplest way to translate the numbers): 1 × 60 + 59 = 119; 2 × 60 + 49 = 169; and we get: 1692 – 1192 = 14400 (without a calculator, we can compute as the following: 1692 – 1192 = (119 + 50)2 – 1192 = 502 + 2 × 50 × 119 = 100 (25+119) = 100 × 144) = 1202 such that (119, 120, 169) is a Pythagorean triple. Moreover all ‘triples’, except one (which one?18) are primitive.

Right or not, this interpretation gives a nice transition between arithmetic and geometry, since the Pythagorean triples can be seen as forming right triangles whose sides are integers. Of course it is a consequence of the famous ‘Pythagoras’ theorem’. Before studying this theorem, we will give a brief presentation of how Greek mathematicians considered geometry as opposed to arithmetic. Then we will present a very simple but famous theorem of geometry.

18 The 11th. 18

v) Arithmetic and Geometry.

The fundamental difference between arithmetic and geometry was the character of their objects. Arithmetic was the science of discrete objects (i.e. ‘separated’ each other according to the Greeks), the ‘numbers’; geometry was the science of continuous objects (non-separated and divisible indefinitely), the ‘magnitudes’.

As a matter of fact, there is no definition of ‘magnitudes’ even in Euclid (though there is a definition of numbers, cf. supra, §V.i) even if they can be found almost everywhere in Greek mathematics. For instance, book V of the Elements deal exclusively with magnitudes, even if we find nowhere what they are. It is no so much strange if we think to our notion of ‘set’ never really defined, and used everywhere in mathematics. Moreover, there is a notion of homogeneity for the ‘magnitudes’ which essentially gives some sort of order on such ‘homogeneous’ magnitudes. And though we do not know what they are, such homogeneous ‘magnitudes’ have necessary a fundamental property. Two homogeneous 'magnitudes' are always able to compose a ratio, and all ratios are comparable (i.e. in modern term, there exists relation of total order on the ratios of any magnitudes). The book V of the Element gives then another property (through the definition 5) which was till Dedekind a substitute of the construction of the real numbers19.

Let us just give some examples of magnitudes: Segments of line or of curves, 2 or 3-dimensional figures, angles20 are homogeneous magnitudes. Numbers were opposed to magnitudes, nevertheless any number had a geometrical representation as a segment (or a 2 or 3-dimensional figure) i.e. a magnitude of the same length (surface, volume). Thus, it was possible to work geometrically on numbers.

One characteristic of Greek mathematics was the importance given to geometry, and well before Euclid the mathematicians seem to have considered geometry to be almost all mathematics.

Since magnitudes were not defined, some operations which are evident in modern mathematics, were sometimes difficult in the ancient Greek geometry.

For example, the product of two magnitudes is not generally possible. What is possible is the product of two segments of line, which gives a surface. But it is not possible to geometrically define a product of two surfaces. So, it is not easy to define the product of four magnitudes, since there is no geometrical representation for it.

19 We will come back to it in part V. See also infra note 28. 20 As in the case of the 'one' with respect to numbers, there were some discussions about the angles, if they were or not magnitudes, but mathematicians used and computed them as homogeneous magnitudes. 19

It could explain why the mathematicians as Euclid preferred to use, even for numbers, the division instead of the multiplication, for instance to write ‘a number is measured’ i.e. divided by 3, 4, … numbers instead of he is ‘the product of’ 3, 4, … numbers.

vi) The theorem of the angles of a triangle.

This theorem is both one of the simplest geometrical result and one of the favourite models of mathematical propositions for Aristotle, and afterwards for philosophers including modern ones as Kant and Hegel21.

So it could also be one of the oldest result of Greek Geometry, and in any case it is attributed to Pythagoras by Proclus quoting Eudemus of Rhodes (4th century BCE), an Aristotelian philosopher (Commentary on the First Book of Euclid's Elements, 397.2).

The theorem is the following: The sum of the angles of a triangle is equal to two right-angles (or 180° or p radians).

The proof given in Proclus is a little different of the one given by Euclid (proposition 32 of book I).

So let ABC be a triangle, and then let be XY a straight line passing by A and parallel to BC (according to the so-called Euclid postulate, there exists one and only one such line) as in the figure below:

XYA

C B

Since AX is parallel to CB, we have: angle (XAC) = angle (ACB) and angle (YAB) = angle (CBA).

So we have: angle (ACB) + angle (CBA) + angle (BAC) = angle (XAC) + angle (CAB) + angle (BAY) = angle (XAY) = 180° = 2 right angles.

Moreover the Greek geometers knew this property is connected to properties of the straight line. For instance it would not be true if these straight lines were replaced by arcs of circle (or in a non-Euclidian geometry!). As a matter of fact, its simplicity results from its equivalence to the postulate of the parallels or of the alternates angles, both used in the proof.

21 Could you guess why its demonstration could fascinate (or disgust) a non-mathematician? 21

vii) Pythagoras’ theorem.

This theorem is one of the best known in mathematics. It gives the length of the hypotenuse c of a right-triangle with respect to its sides (a,b): c2 = a2 + b2.

1. Another non-Greek theorem? Despite its name, many scholars think it was known in one way or another, well before Pythagoras by non-Greeks mathematicians. For instance as seen previously in §iv.2, many scholars claim the Plimpton tablet is a proof Babylonian mathematicians knew the properties of the right triangle. Other testimonies are found in Indian or Chinese books, but it is not easy to date them. And, as we saw, concerning Babylonians, the relation of the Plimpton tablet to Pythagorean triples, and a fortiori to Pythagoras’ theorem is at best problematic.

But there is another tablet, the YBC 7289 (cf. below), around the same epoch as the Plimpton 322, which gives an interesting testimony.

tablet YBC 7289

Its usual interpretation is that a square and its diagonals are drawn on it. Then the number on the left (30) is the length of the side of the square. The number on the horizontal (1:24:51:10) is a very good approximation of the square root of 2. The last number (42:25:35) is the length of the diagonal given as the product of the length of the side by the square root of 2. More precisely, since in old Babylonian numbering the power of the base is not noted (a number n less than 60 can be understood as any number of the form n × 60h, where h may be any relative number i.e. positive, null or negative), these numbers are understood as the following:

- 1:24:51:10 = 1 + 24/60 + 51/3600 + 10/216000 = 1,414213

- 30 × (1:24:51:10) = 30 × (1 + 24/60 + 51/3600 + 10/216000) = 30 + 12 + 1530/3600 + 300/216000 = 30 + 12 + 25/60 + 30/3600 + 5/3600 = 42 + 25/60 + 35/3600 = 42:25:35 - 42:25:35 is a pretty large number if we consider it as an integer. But it is interpreted as a fraction less than 1, we get : - 42:25:35 = 42/60 + 25/3600 + 35/216000 = 0,7071

Now √2 = 1,414213, √2/2 = 0,7071 so that we can understand the figure as saying the length of the diagonal of a square of side 30 is equal to the product of 30 by √2, which gives a particular case of Pythagoras’ theorem (plus an excellent approximation of √2, exact at least to the sixth decimal).

Conversely the tablet may have nothing to do with Pythagoras’ theorem. For 30 may be written for 30/60 = 1/2, so that the two horizontal lines would be very good approximations of the square root of 2 and of its inverse 1/√2 = √2/2, since it is extremely usual to find Babylonian tables of fractions with their inverses, used for difficult computations (in sexagesimal).

2. Why Pythagoras’ theorem.

Many Greek sources are claiming Pythagoras as its author. Cicero (1st century BCE), Plutarch (1st century CE) who cites Apollodorus the calculator (2nd century BCE) which speaks only of a ‘famous theorem’ of Pythagoras, Atheneus (2nd century CE), Diogenes Laertius (3rd century CE), Porphyry (3rd century CE).

There is also a lot of scepticism both among these authors and the modern ones. These are rejecting these sources as they are all coming from a dubious one (Apollodorus the calculator) and they doubt the theorem was proved at all by Pythagoras or the Pythagoreans. In the Antiquity, it appears the hesitations concerned more a sacrifice of some oxen Pythagoras was supposed to have given to thank the gods for his discovery, as found in some stories22. Nevertheless, as Heath remarks (A History of Greek Mathematics, p. 145), they do not seem to doubt the discovery of the theorem was made by Pythagoras.

For instance Proclus wrote: ‘if we listen to those who wish to recount the ancient history we may find some who refer this theorem to Pythagoras, and say that he sacrificed an ox in honor of his discovery.’ It seems at the time of Proclus (5th century CE), it was common for the writers/historians concerned by the past, to attribute this theorem to Pythagoras. Once again, the skepticism seems to be about the sacrifice itself and, if such a sacrifice had been made, which result was concerned, more than about the discoveries themselves23.

The hesitations and the differences in the different stories could mean either it was a legend or on the contrary the attribution was widely known and accepted some authors trying to fill the gaps. In any way, no other name Greek or non- Greek is suggested as the author of the theorem.

22 One (disputed) argument is the Pythagoreans were vegetarian. Pro: Porphyry citing the mathematician Eudoxus, (4th century BCE, in Life of Pythagoras, 7); con: Diogenes Laertius citing Aristoxenus (4th century BCE, in Lives and Opinions…, VIII. 20). But even Aristonexus is quoted saying there were some interdictions concerning the meat. One reason could be they believed in the transmigration of souls according to Diogenes Laertius quoting Xenophanes (6th century BCE, in Lives and Opinions…,, VIII.36). 23 There is a plural because there are three possible results: - Pythagoras’ theorem - the so-called ‘application of area’ which consists, two figures being given, to construct a third one similar to the second and of same area as the first (Plutarch, Moralia, Symposiacs, VIII, Question 2, 1094) - in modern terms, the irrationality of 2. The doubts seem to concern not so much the question if Pythagoras or the Pythagoreans discovered all these results, but which one of them deserved such a sacrifice (the number of oxen killed varying from one to hundred). 24

Proclus’ text continues in the following way: ‘for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements [Euclid] (…) because he compelled assent to the still more general theorem by the irrefutable arguments of science’.

This sentence shows the following things: - Since the ‘first ones’ whose Proclus speaks are some Pythagoreans, Proclus is not a Pythagorean or at least his judgment is not completely biased in favor of Pythagoras. - He thinks the attribution to Pythagoras himself is doubtful, so, according to him, it is safer to attribute the discovery to his school. - He opposes the ‘observation’ of the theorem to the ‘proof’ by Euclid. It means the first proof was both less rigorous and less general than Euclid’s one.

3. The demonstrations of Pythagoras’ theorem.

Presently, this theorem has so many proofs it is almost impossible counting them. In one book published 70 years ago, 370 of them are presented (Elisha Loomis, The Pythagorean proposition, 1940), 109 are algebraic and 255 are geometric; nowadays they may be numbered in the thousands.

Which one is Pythagoras’ proof (or the original one)? Concerning Euclid, Proclus had clearly in mind the following proposition of book VI of the Element, which gives a generalization of Pythagoras’ theorem:

Proposition 31: In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

Pythagoras’ theorem is a particular case of it, where figures are squares. In this case, we obtain the result of the proposition 47 of book I:

Figure 3

which says that the surface of the big square (in blue) is equal to the surfaces of the two other squares (the ones in yellow and green).

The proof of the proposition I.47 is the following:

- Step 1:

J A D

Figure 4

The triangles AEC and BCD are equal (CD = CA, BC = CE and angle ACD = angle ACB + 1 right angle = DCB) (cf. figure 4 above).

- Step 2:

h A

B C

E G Figure 5

The surface of the triangle AEC = 1/2 (CE×hA) = 1/2 (CE×CF) = 1/2 surface of the rectangle CFGE (cf. figure 5 above).

- Step 3:

J A D B C

i Figure 6

The surface of the triangle BCD = 1/2 (DC×Bi) = 1/2 surface of the rectangle ACDJ (cf. figure 6 above). 27

From the first equality (the triangles AEC and BCD), we get the equality of the surface of the square ACDJ and the surface of the rectangle CFGE.

The same reasoning with the other part of the figure

L A

C FF BB

E G H

Figure 7

gives the equality of the triangles LBC and ABH, and then the equality of the surface of the square LBAK and the rectangle BHGF (cf. figure 7 above).

Finally the sum of the surfaces of the squares ACDJ and LBAK is equal to the sum of the surfaces of the rectangles CFGE and BHGF. Since CFGE plus BHGF is equal to the square CBHE, the theorem is proved.

This proof is extremely subtle; it uses some results of the equality of the triangles, and can certainly not be described as less rigorous or less ‘irrefutable’ as any other proof. So, according Proclus’ text, it was certainly not the original proof of the theorem. Since the proof of the proposition VI.31, so admired by Proclus, is founded on the theory of proportions of triangles and other figures, it is even less possible to be the original one24.

The simplest proof, certainly less rigorous from a strict mathematical point of view25, is the one generalizing the proof for isosceles right-triangle as it is given in Plato’s Meno, the oldest text about this theorem. The demonstration is the following:

24 This point of view is not shared by everyone. Some scholars think the theorem was the consequence of some not rigorous theory, including one on proportions and similar triangles. 25 And maybe how the Babylonians got their formula, if indeed the tablet YBC 7289 is about the ratio of the diagonal to the side of a square and/or the Plimpton tablet is connected to Pythagoras’ theorem (cf. supra §§iv.2 and vi.1). 28

O D B

C Figure 8

In the big square we have 8 equal right-triangles. The smaller square ABCD is formed of 4 such triangles, so that its surface is half the one of the big square. Since the surface of the big square is equal to AC2 and the surface of the square ABCD is AB2 we get: 4OA2 = AC2 = 2 AB2 and then: AB2 = 2 OA2 = OA2 + OD2.

The generalization of this formula to a general right-triangle consists to make a little rotation in the figure 8 above to get the figure 9 below:

A E G

F C H

Figure 9

In the above figure 9, the big square EGHF contains 8 equal right-triangles, all equal for instance to the triangle DEA. Since EGHF = ABCD + DEA + AGB + BHC + CFD, if we put a (resp. b, d) for the length of DE (resp. EA, DA), we get: Surface of EGHF = (a+b)2 (since DF = EA ≔ a) = surface of ABDC + 4 surfaces DEA = d2 + 4[(1/2) ab] = d2 + 2ab, thus : a2 + 2ab + b2 = d2 + 2ab,

and finally we obtain: a2 + b2 = d2 QED (Quod Erat Demonstrandum26).

26 Which had to be demonstrated. 29

Remark. All the proofs given here are graphic proofs, meaning they are completely done on geometric figures. Nevertheless, they are rigorous to the point it is possible to translate them step by step in our hypothetic-deductive axiomatic language27.

27 Nevertheless, in the last demonstration, could you say why it is not as rigorous as it should be, or the arguments are not as irrefutable as Proclus would like them to be (cf. supra, §2)? 30

4. A consequence of Pythagoras’ theorem: the geometric mean.

Definition. Let a and b be 2 magnitudes28. The geometric mean of a and b is the magnitude h verifying: a/h = h/b or (with modern notations) h = ab.

Remark 1. In Greek mathematics, the equality h = ab has a geometrical meaning, not an algebraic one as in modern mathematics (cf. §v) above). It means it is possible (and even we know how) to construct a square (of side h) such that its surface is equal to the surface of a given rectangle (of sides a and b). As shown below, its existence is proved by giving a construction of such a square using Pythagoras’ theorem.

Remark 2. In old Greek mathematics, the equality a/h = h/b is more general than the one h = ab29. Since for any homogeneous magnitudes a, h, b, the ratio a/h = h/b has a sense, but it is not always possible to define the product ab or the square h2 (cf. §v) above).

It was well-known that till we stay in arithmetic, i.e. with the integers, a geometric mean does not exists necessarily. For instance there is no such mean between 3 and 2. But for some couples of integers, there exists one. For example, between 12 and 3, the integer 6 is their geometric mean: 12/6 = 6/3.

In geometry such means always exist and moreover it is easy to construct them using Pythagoras’ theorem. The Epinomis, written around the 4th century BCE (and sometimes attributed to Plato) says this is an almost ‘divine’ property, such important this result was considered by the mathematicians. It was known at the time of Plato and certainly much earlier, as an immediate consequence of Pythagoras’ theorem.

Construction of a geometric mean.

Step 1. In addition to Pythagoras’ theorem, we need

28 The term ‘magnitude’ (in Greek ‘mevgeqo"’) is not defined in ancient Greek geometry. We will study it more in details in part V. Let us just say it is something ‘continuous’ like (finite) lines, surfaces, volumes, weights or times. 29 Contrary to modern mathematics where they are equivalent, or even the first one being a little less general, since the second has a sense even for h = 0 31

i) Firstly that in a circle, the angle subtended by a diameter is a right-angle. It is the proposition III.31 of the Elements, such an old result it was attributed to Thales30.

ii) The following identity: for any numbers (or magnitudes) a and b, we have: (a + b)2 = a2 + 2ab + b2. This is very simply to prove inside the Greek mathematics i.e. geometrically, and Proclus once again refers to Pythagoras for its proof. And it is moreover needed to prove Pythagoras’ theorem.

As a matter of fact, once again it results just from the view of a geometric figure (It is the figure accompanying the proposition 4 book II of the Elements) :

C a b D b b

a a

F a b E

As we see on the figure, the surface of the square DEFC is equal to the surface of a square of side a, plus a square of side b plus twice the surface of a rectangle of sides a and b, such that: (a+b)2 = a2 + b2 + 2ab.

Step 2. Let ABC be a right-triangle, and AH the height drawn from A on the hypotenuse BC.

C H B

Let a (resp. b, h) be the length of the segment CH (resp. HB, AH). From Pythagoras’ theorem applied to the 3 right-triangles ABC, ABH and AHC, we get: CB2 = AC2 + AB2 (1) AC2 = AH2 + CH2 = h2 + a2 (2)

30 Diogenes Laertius quoting Pamphilia, but he adds other authors, including Appollodorus the arithmetician, consider Pythagoras as the author of this result (Life and Opinions …, I, in Life of Thales, 24). 32

AB2 = AH2 + HB2 = h2 + b2 (3).

Since CB = CH + HB = a + b, from the equalities (2) and (3) we obtain by replacing AC and AB in (1): (a+b)2 = CB2 = (h2 + a2) + (h2 + b2) = 2h2 + a2 + b2 and from the formula of the square of a sum, we get: a2 + b2 + 2 ab = 2h2 + a2 + b2 thus: 2ab = 2h2 and finally: h2 = ab.

Step 3. Construction of the geometric mean of 2 magnitudes.

Let a and b be two magnitudes a and b. Let CH and HB be two segments of length respectively a and b.

h H C B ab

We first construct the circle of centre the middle of BC and passing by the points B and C. We then draw the height H from A on the hypotenuse CB. According to step 2, we have: h2 = ab i.e. h is the geometric mean of a and b (QFD).

Remark. It is essentially the construction of the proposition 14 of book II of the Elements. As a matter of fact, Euclid’s proof is shorter and smarter, and moreover, it does not need the first result concerning the angle of a semi-circle. So why bother with the above proof? Since a first proof is neither smart nor elegant, but uses simpler results, we can argue the first demonstration we gave is more probably also chronologically the first one. Because in this case, Euclid uses the following identity: for a > b : a2 – b2 = (a+b) (a-b) which is harder to prove than the one of the square of the sum (compare the proofs of the propositions II.4 and II.5 of the Elements). But since we have no text, it is impossible to conclude with any kind of certitude.