Pythagoreans in possession of the triple 3–4–5, wherever they learned it, and recognising its usefulness, would not have let matters rest there. And if they set their minds to looking for a meaningful connection among the three numbers, it would arguably not have taken long to find the theorem. They could not have done it with pebbles, which they used as counters rather than as units to measure distances, but the same visualisation that made pebbles interesting would soon have arrived at something like the diagram above, where squaring the numbers in the triple reveals the hidden relationship.
If the Pythagoreans found this relationship, having already discovered the harmonic ratios, they must have felt as though lightning had struck twice, for here was another stunning example of the hidden numerical rationality of the universe. Believing so strongly in a unity of all things, they would have been quick to jump to the correct conclusion that this same pattern of hidden connections had to apply to all right triangles – perhaps even to the incorrect conclusion that it had to be true of all triangles.
The second part of the tradition that has Pythagoras and his early followers discovering the theorem was that afterwards a sword of Damocles hung over their heads. The universe had a cruel surprise in store for them, ‘incommensurability’. Most right triangles have no whole-number triplet like 3–4–5. For example, according to the theorem, a right triangle with sides measuring 3 inches and 3 inches, each of which squared is 9, must have a third side – a hypotenuse – the length of which squared is 18.
However, it is no simple matter to find the square root of 18 and the length of that hypotenuse, for the square root of 18 does not exist among whole numbers or fractions. An isosceles triangle like this one was a nightmare for a community of scholars who believed in a rational universe based beautifully and neatly on numbers. They could see that it did indeed exist and was a right triangle. It was not something hypothetical hanging fuzzily out in conceptual space. It was the triangle they got when they drew a diagonal from corner to corner of a square. But no subdivision of the length of the sides (neither inch, nor centimetre, nor mile, nor any fraction thereof) divided evenly into the length of the diagonal. More generally, though it might seem that for any two lengths you might try to measure there would be some unit that would divide into both of them and come out even with no remainder – maybe not the inch or centimetre or any length that has a name, but some unit, however small – the fact is that this is not the way reality works in this universe. Nor does the problem of incommensurability exist only with isosceles triangles. It was also true of the ‘gate’ measured on the Babylonian tablet. The Babylonians knew about the triples and also apparently accepted that the units measuring the diagonal most of the time did not come out even.
The early Pythagoreans may well have discovered the problem, but it is far less likely that they found the solution – irrational numbers – or that they would have liked it if they had.[9] Irrational numbers are not neat or beautiful like whole numbers. An irrational number has an infinitely long string of digits to the right of the decimal point with no regularly repeating digit or group of digits.
The suggestion that the only information Pythagoras learned elsewhere was a vestige of the theorem – the triple 3–4–5 – has in its favour that it solves a problem with the sequence of the Pythagoreans’ discoveries. In order to have had a devastating crisis of faith resulting from the discovery of incommensurability, they had to have had the faith first, not the crisis, and indeed the tradition is strong that this was the order in which the discoveries were made. However, it is difficult to imagine anyone discovering the theorem from scratch (without the triple) while not simultaneously discovering the problem. Another possible sequence would have the Pythagoreans discovering incommensurability first, as they struggled with right triangles, and only later realising that a few right triangles were, in fact, not incommensurable; but that is not the traditional sequence. A similar issue undermines the possibility that Pythagoras learned the whole theorem elsewhere: He would also have learned about incommensurability, so that it could have come as no surprise later.
Eleanor Robson is convinced, from evidence in the mathematics itself, that Old Babylonian mathematics was not the arithmetical precursor to early Greek mathematics; but a triple used in construction and surveying – the origin of which no one remembered – hardly represented the bulk of Old Babylonian mathematics.14 It could have waited on the shelf while the Pythagoreans investigated harp string lengths and discovered musical ratios. The study of those was a new kind of thinking about numbers, what Aristoxenus meant when he wrote: ‘The numbers were withdrawn from the use of merchants and honoured for themselves’. A love of this sort of thinking could have led the Pythagoreans, after their musical discovery, to consider the 3–4–5 triple more carefully. Granted, this particular triangle was, by Pythagorean standards, not very interesting. Five does not show up in the basic ratios of music; 3, 4, and 5 do not add up to 10 or make the tetractus. No one is going to swear by this triangle! But the hidden connection . . . that was another reason to fall to one’s knees, and perhaps to have a huge crisis of faith when you began looking at other right triangles.
Some Pythagoreans reputedly found other beauty in the triplet: They designated 5 as ‘marriage’. The 5-unit side of this right triangle connected the 3-unit side and the 4-unit side. Thus, 5, or ‘marriage’, connected 3 (which is odd) and 4 (even). ‘Odd’ was male and ‘limiting’; ‘even’ was female and ‘limitless’. So this triangle was a manifestation of the harmony reconciling limiting and limitless. In the modern world, we associate such weak links with a different sort of mind from that which would come up with the Pythagorean theorem. In the ancient world, whose people were taking the first tentative steps towards understanding nature and the cosmos and the human condition, that distinction is invalid.
If Pythagoras discovered or knew the rule, did he prove it? Most historians of mathematics believe that the concept of ‘proof’ as later understood was unknown before the Alexandrian Euclid introduced it in his Elements, around 300 B.C. The decision that something would ‘be true for every right triangle’, for someone living as early as Pythagoras, would probably have been made on grounds other than a Euclidian proof. It would either have been an unsupported assumption or a guess, or a decision made in a scientific rather than a mathematical way – by testing it, as many times and with as many different examples as possible. The idea that mathematical statements should apply generally, though taken for granted today and implied in the Babylonian work, was not usually part of the ancient mind-set before Pythagoras. It is considered to be one of the great contributions of early Greek mathematics and probably a contribution of Pythagoras and his followers. With them, it might have been only an assumption based on their belief in the unity of all being, not something they could demonstrate at all or even thought it necessary to demonstrate.
There are, nevertheless, simple proofs of the theorem that some would like to attribute to the Pythagoreans, and one argument that they used such proofs is that these same thought sequences are good ways to discover the theorem, even if you had no concept of ‘proofs’ after the fact. There is a geometry lesson in Plato’s Meno that some think is traceable to Pythagoras. The clue is that Plato used it to demonstrate the ‘recollection’ of what one learned before birth, an idea related to the Pythagorean doctrine of memory of past lives. The triangle in the Meno proof is the troublesome isosceles triangle, but the proof sidesteps the problem of incommensurability by using no numbers. It is admittedly difficult to imagine Pythagoreans being satisfied with any ‘truth’ that used no numbers. It would have seemed the universe was thumbing its nose at them, with this triangle that provided such a clear and unequivocal demonstration of their rule, and that contained incommensurability. The discussion of Plato’s proof fits better in the context of a later chapter. Bronowski, in the book from his television series, showed another well known numberless proof that h
e believed Pythagoras may have used. This clever proof is in the Appendix.
The right triangle was not the only pitfall in Pythagorean thinking where incommensurability lurked, but it was the most obvious. The scholarly argument about whether the Pythagoreans discovered it, and whether that caused a crisis of faith in the rationality of the universe, rambles on until it resembles that string of digits after the decimal place in an irrational number. However, in truth, an intelligent person thinking along Pythagorean lines and dealing with right triangles could hardly have missed discovering incommensurability. But only someone who reverenced numbers and the rationality of the universe would have been deeply troubled. Some have thought that Pythagoras and his followers reacted by retreating to a geometry without numbers – that what had early been an ‘arithmetised geometry’ was reformulated in a nonarithmetical way, and this carried over into Euclid. In spite of the passage in Plato’s Meno, and the suggestion that it reflected Pythagoras’ proof of his theorem, nothing could seem more blatantly un-Pythagorean than a retreat from numbers!15
Porphyry would have been pleased to learn that the earlier Mesopotamians knew about right triangles, the triples, and the theorem. His choice of possibilities would almost certainly have been that the theorem was known earlier but that Pythagoras’ was an independent discovery, for he believed that several ancient peoples – he named the Indians, Egyptians, and Hebrews – possessed primeval, universal wisdom (prisca sapientia was the later term), and Pythagoras was the first to possess it in the Greek world.16 The theorem is so intrinsic to nature, so beautifully simple, that it would be odd if no earlier triangle user in prehistory or antiquity got curious and figured it out.
What about a more startling suggestion: that Pythagoras had nothing whatsoever to do with the discovery? Could it be that it was later credited to him only because such legends tend to become associated with famous people? Over two thousand five hundred years, numerous achievements that were not remotely Pythagorean have been carelessly credited to Pythagoras. ‘Pythagorean’ or ‘of Pythagoras’ have become descriptive words connoting something clever that shows mathematical insight, with an overlay of wisdom, fairness, or morality. A ‘Pythagorean cup’, sold on Samos, punishes the immoderate drinker who fills it above a marked line, by allowing the entire cup of wine to drain out the bottom. Modern citizens of Samos are surprised – or at least pretend to be – that anyone would doubt this was an invention of Pythagoras. A ‘Pythagorean’ formula predicts which baseball teams in America are likely to win. No one is insulted by doubts about that one.
A worse possibility for Pythagoras’ image is that he took the theorem from the Babylonians and claimed it as his own. According to Heraclitus, he ‘practised inquiry more than any other man, and selecting from these writings he made a wisdom of his own – much learning, mere fraudulence.’[10] It would certainly not have surprised Heraclitus if Pythagoras had stolen the Pythagorean theorem lock, stock, and barrel from the Babylonians. However, the fragments in which Heraclitus dismissed him as an impostor also placed Pythagoras high in the echelon of thinkers. Two of Heraclitus’ other targets, Xenophanes and Hecataeus, were renowned polymaths. ‘Inquiry’ meant not study in general but Milesian science. Most scholars think that Heraclitus had no basis for his attacks. He had an aversion to polymaths, and he was simply an ornery and contentious man being ornery and contentious. On another occasion he commented that ‘Homer should be turned out and whipped!’
If the Pythagoreans did come up with the theorem independently, the question remains whether credit should go to Pythagoras and his contemporaries or to later generations of Pythagoreans. Intemperate Heraclitus would not have been pleased to know that evidence coming from his own work places the appearance of the Pythagorean mathematical achievements in Pythagoras’ lifetime: Heraclitus followed up on Pythagorean ideas about the soul and immortality and continued to develop the idea of harmony. For him, the lyre and the bow – Apollo’s musical instrument and weapon – symbolised the order of nature. The bow was ‘strife’, the lyre harmonia. The significance of the bow (‘strife’) was original with Heraclitus, but the role of the lyre and harmonia were developments from Pythagorean thought, which suggests that the idea of connections between numerical proportions, musical consonances, and the Pythagorean numerical arrangement of the cosmos dated from the time of Pythagoras himself. Heraclitus was only one generation younger than Pythagoras.
In the first century B.C., the theorem seems to have been widely attributed to Pythagoras. A case in point: The great Roman architect Marcus Vitruvius Pollio, better known as Vitruvius, knew it well, attributed it without question to Pythagoras, and, in Book 9 of his ten-volume De architectura, mentioned the sacrifice to celebrate it. Apparently Vitruvius could write about Pythagoras as the discoverer of the theorem and assume that no one would gainsay him. He knew other methods of forming a right triangle, but found Pythagoras’ much the easiest:
Pythagoras demonstrated the method of forming a right triangle without the aid of the instruments of artificers: and that which they scarcely, even with great trouble, exactly obtain, may be performed by his rules with great facility.
Let three rods be procured, one three feet, one four feet, and the other five feet long; and let them be so joined as to touch each other at their extremities; they will then form a triangle, one of whose angles will be a right angle. For if, on the length of each of the rods, squares be described, that whose length is three feet will have an area of nine feet; that of four, of sixteen feet; and that of five, of twenty-five feet: so that the number of feet contained in the two areas of the square of three and four feet added together, are equal to those contained in the square, whose side is five feet.17
Where, then, does this discussion end? In spite of the certainty that Vitruvius and his contemporaries shared, the most sceptical modern scholars think Pythagoras had nothing to do with the theorem at all. Others do not close the door to the possibilities that Pythagoras and/or his early followers may have made the discovery independently, unaware of previous knowledge of the theorem, or that they learned it elsewhere but were the first to introduce it to the Greeks.
My own conclusion is that there is no good reason to decide that Pythagoras and the Pythagoreans had nothing to do with the theorem, and several meaningful hints that they did, including the fact that Plato chose to assume that right triangles were the basic building blocks of the universe when he wrote his Timaeus, the dialogue most influenced by Pythagorean thinking.[11] If earlier knowledge of the theorem had indeed been lost, then someone had rediscovered it at about the time of Pythagoras. Of all those who were aware of right angles and triangles and used them in practical and artistic ways, the Pythagoreans were unique in their approach to the world, apparently having the motivation and leisure to give top priority to ideas and study. Their intellectual elitism kept them focused beyond the nitty-gritty of ‘what works’ on the artisan level, and their musical discovery led them to think beyond number problem solving for its own sake – causing them to turn their eyes beneath the surface and view nature in an iconic way. For the Pythagoreans (as for no others among their contemporaries), the theorem would have represented an example of the wondrous underlying number structure of the universe, reinforcing their view of nature and numbers and the unity of all being, as well as the conviction that their inquiry was worthwhile, and that their secretive elitism was something to be treasured and maintained. Has any other ruling class – and the Pythagoreans seem also to have been busy ruling – had that same set of priorities? Regarding the possibility that they began with the triple, I like the fact that their having it would not imply a continuum with the Old Babylonian mathematical tradition – a continuum that scholars like Robson have convincingly argued did not exist. And in this scenario, the Babylonian evidence, instead of pulling Pythagoras off the pedestal, actually suggests a way that he and his followers could have rediscovered the theorem in the time and place tha
t tradition has always said they did without being disillusioned too soon by the discovery of incommensurability.
Bronowski credited Pythagoras with discovering the link between the geometry of the right triangle and the truth of primordial human experience. He echoed Plato’s reverence for right triangles as the basics of creation when he wrote, ‘What Pythagoras established is a fundamental characterisation of the space in which we move. It was the first time that was translated into numbers. And the exact fit of the numbers describes the exact laws that bind the universe.’ Bronowski for that reason thought it not extravagant to call the ‘theorem of Pythagoras’ ‘the most important single theorem in the whole of mathematics.’18
But what of the ox? Did Pythagoras sacrifice it, or perhaps forty of them (some stories say), in thanksgiving for the discovery of the theorem? That Apollodorus referred to this ‘famous’ story does not necessarily mean he believed it. Many dismiss the tale as impossible on the grounds that Pythagoras, who ate no meat, would not have sacrificed an ox. However, there is plenty of evidence that he had no objection to the slaughter of animals for ritual purposes. If vegetarianism is a clue, it may point in a different direction: Later Pythagoreans were more ready than early ones to believe that Pythagoras was a strict vegetarian. Burkert thought the existence of the sacrifice story ‘ought rather to be considered an indication of antiquity’, weighing in on behalf of the argument that Pythagoras or his earliest followers made the discovery that spawned the tale. A later generation would not have made up this story about their hero.
[1]‘Theorem’ has implications, in modern terminology, that do not apply to the earliest knowledge of this rule. With that in mind, this book will nevertheless continue to use ‘theorem’ to avoid seeming to mean something different from what everyone calls the Pythagorean theorem.
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