[1]Think of having the lowest string tuned to C on the piano, the fourth string tuned to F above the C, the next to G a whole step above that, and then the top string tuned at C an octave above the lower C.
[2]Musical instruments and human voices, because of intricate differences in the way their structures resonate and amplify sound, emphasise or ‘bring out’ certain overtones more than others, and that is what causes the great variety of sounds they make. That is how a trumpet ends up sounding like a trumpet while a clarinet sounds like a clarinet.
[3]On the piano, equivalent notes might be, for example, middle C (ground note); c (octave above that ground note); g (fifth above that octave); c (fourth above that g). For a demonstration using the piano: Press down gently on the c above middle C without allowing it to sound (removing the damper from the strings). Strike middle C (the ground note) and you will clearly hear the octave. Press carefully on the g above that octave. Strike middle C and you will hear that fifth above the octave. A piano is not tuned to the Pythagorean system, but it is close enough for you to hear these overtones.
[4]A gnomon is an instrument for measuring right angles, like the device used by carpenters called a ‘carpenter’s square’.
[5]Not all pyramids have only four sides. The Great Pyramid that Pythagoras may have seen in Egypt is not a pyramid of this sort. It has five sides: a square base and four triangular sides.
[6]In some later ancient mathematics, whose roots can be traced to the ‘Pythagorean’ tradition and which by some scholars’ interpretation existed separately and in parallel with the Euclidean tradition, the number 2 also had no status as a ‘number’. It was not considered even or odd or prime. Like ‘1’, it was not a number at all, but the ‘first principle of number’.
[7]Heracleides Ponticus is not to be confused with the earlier Heraclitus who so severely criticised Pythagoras. Heracleides Ponticus lived in the fourth century B.C. and was a pupil of Plato.
[8]Part of their ‘present condition’ was an economy that was more primitive than Croton’s. They used no coinage, and would not until more than a century later. See W. K. C. Guthrie (2003), p. 178 n.
CHAPTER 6
‘The famous figure of Pythagoras’
Sixth Century B.C.
In the first or early second century A.D., Plutarch, the author of the famous Parallel Lives, and his team of researchers tried to find the earliest reference connecting Pythagoras with the ‘Pythagorean theorem’.[1] They came upon a story in the writing of a man named Apollodorus, who probably lived in the century of Plato and Aristotle, that told of Pythagoras sacrificing an ox to celebrate the discovery of ‘the famous figure of Pythagoras’.[2] Plutarch concluded that this ‘famous figure’ must have been the Pythagorean triangle. Unfortunately Apollodorus was no more specific than those words ‘the famous figure of Pythagoras’ – which probably indicates that it was so famous he had no need to be.
A modern author could also write ‘the famous figure of Pythagoras’ and be as certain as Apollodorus apparently was that no reader would think of anything but the ‘Pythagorean triangle’. Even nonmathematicians can often recall the ‘Pythagorean theorem’ from memory: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. For millennia, anyone who had reason to know anything about this theorem thought Pythagoras had discovered it.
For many who learned the formula in school and always thought of it only in terms of squaring numbers, rather than involving actual square shapes, it came as an almost chilling revelation when Jacob Bronowski in his television series The Ascent of Man attached a square to each side of a right triangle and showed what the equation really means. The space enclosed in the square ‘on the hypotenuse’ is exactly the same amount of space as is enclosed in the other two squares combined. The whole matter suddenly took on a decidedly Pythagorean aura. Clearly this was something that might indeed have been discovered and is true in a way that does not require a trained mathematician or even a mathematical mind to recognise. In fact, using numbers is only one of several ways of discovering it and proving it is true.
Bronowski pointed out that right angles are part of the most primitive, primordial experience of the world:
There are two experiences on which our visual world is based: that gravity is vertical, and that the horizon stands at right angles to it. And it is that conjunction, those cross-wires in the visual field, which fixes the nature of the right angle.1
Bronowski did not mean that experiencing the world in this way necessarily leads immediately, or ever, to the discovery of the Pythagorean theorem. Indeed, all over the ancient world, long before Pythagoras, right angles were used in building and surveying, and right triangles appeared decoratively.[3] Without drawing tools, a draftsman can produce right triangles, and a skilled draftsman can produce right triangles that no human eye can see are not absolutely precise – this without knowledge of the Pythagorean theorem. Just as tuning a harp is an ‘ear thing’ – and was, long before anyone understood the ratios of musical harmony – the use of right triangles in design was an ‘eye thing’. Such judgements of harmony, figures, and lines are intuitive for human beings, and the mathematical relationships that lie hidden in nature and the structure of the universe often manifest themselves in the everyday world in useful ways long before anyone thinks of looking for explanations or deep relationships.
Yet at certain times and places in history and prehistory – for reasons about which it is only possible to speculate – circumstances have been right to call forth a longing to look beyond the surface. Among the Pythagoreans there was a strong and unusual motivation. Investigation like this was the road by which one could escape the tedious round of reincarnations and rejoin the divine level of existence. One cannot summarily dismiss the tradition that they discovered the theorem, though, contrary to popular belief for centuries, they were definitely not the first to do so.
No one knows how or when the ‘Pythagorean theorem’ was first discovered, but it happened long before Pythagoras. Archaeologists have found the theorem on tablets in Mesopotamia dating from the first half of the second millennium B.C., a thousand years before his lifetime. It was already so well known then that it was being taught in scribal schools. In other regions, evidence of early knowledge of the theorem is less conclusive but still interesting. Egyptian builders knew how to create square corners with an astounding and mysterious degree of precision, perhaps by using a technique that earned them the nickname ‘rope pullers’ among their Greek contemporaries. There is a hint about what that meant, perhaps, from circa 1400 B.C. in a wall painting in a tomb at Thebes, where Porphyry’s story had Pythagoras spending most of his time while in Egypt. The painting shows men measuring a field with what looks like a rope with knots or marks at regular intervals.2 Possibly they were using the rope to create right angles, taking a length of rope 12 yards long, making it into a loop, and marking it off with three notches or knots so as to divide it into lengths 3, 4, and 5 yards long. Three, four, and five are a ‘triple’ of whole-number unit measurements that create a right triangle, and holding the loop at the three marks and pulling it tight would have given them one. The knots or marks in the Thebes wall decoration are not clearly spaced at those intervals, but that could be because the artist was no surveyor.
1400 B.C. wall painting at Thebes depicting men measuring a field
The Egyptians left no instructions about ‘rope puller’ techniques, and knowledge of the 3–4–5 triplet is no clear indication that they knew the theorem that made deeper sense of it. They had another method of getting right angles that involved no ropes at all. The groma was a wooden cross suspended from above so that it pivoted at the centre. A plumb bob was hung from the end of each of the four arms; a surveyor or builder sighted along each pair of plumb bob cords in turn, then t
urned the entire device ninety degrees and repeated the sighting, and finally adjusted one of the cords to make up half of the difference. The result was a precise right angle.
In India, right triangles appeared in the designs on Hindu sacrificial altars dating from as early as 1000 B.C.3 A collection of Hindu manuals called the Sulba-Sûtras (‘Rules of the Cord’), dating from between 500 and 200 B.C., told how to construct these altars and how to enlarge them while retaining the same proportions. In times of trouble, enlarging the altar was a way of seeking surer protection from the god or gods, and getting the right response depended on keeping the exact proportions. Builders attached cords to pegs set in the ground, as bricklayers do today, hence ‘rules of the cord’. The Pythagorean theorem does not appear in the manuals, but the writers seem to have been aware of it. Knowledge originating in Greece in the sixth century B.C. could possibly have reached India, for instance with Alexander the Great’s armies in about 327 B.C. It is not too far-fetched to speculate that it did. The Cynic philosopher Onesicritus travelled with Alexander, and in his records he mentioned being questioned by an Indian wise man about Greek learning and doctrine. One of the matters they discussed was the Pythagorean avoidance of eating meat.4 By the time Onesicritus had that discussion, the Pythagorean theorem was well known in the Greek world and almost certainly known to him. However, there is more to the Indian case. Though the written manuals date from after Pythagoras’ lifetime, records exist of similar altars, and of their proportional enlargement, from several centuries earlier. No instruction manuals survive from that time and it is plausible that the writers of the later manuals were applying new understanding to an ancient art. One only need witness the astounding facility with which the humblest, most isolated, illiterate Indian woman today is able to create highly elaborate symmetrical geometric designs with painted powders on her doorstep, referring to a small pattern held down by a stone nearby, to question whether an understanding of mathematical geometry was necessary to create an intricate design and enlarge it while retaining the original proportions.
In Mesopotamia, however, the evidence is irrefutable that the theorem was known and understood in the early second millennium B.C.5 We have not the vaguest hint about who discovered it or how, or how useful it was. School lessons on tablets measured gates and grain piles, and one grain pile was so amazingly large that the lesson problem was clearly set out only as an exercise, not with a real pile in mind – though probably with the goal of equipping pupils to put the same number skills to work in real-life, practical situations.6
The twentieth-century discoveries about the theorem’s Mesopotamian origins began in 1916 when Ernst Weidner studied a Mesopotamian school tablet labelled VAT 6598, dating from the Old Babylonian period in the early second millennium B.C. The two final problems that he could read on the tablet, part of which was missing, required calculating the diagonal of a rectangle and showed methods for doing that. These did not include the Pythagorean theorem, but Weidner assessed the accuracy of the methods and compared them with the theorem, alerting archaeologists and mathematicians to the possibility that it was known more than a thousand years before Pythagoras.
In 1945, a text that archaeologists have labelled Plimpton 322 came to light, listing fifteen pairs of what would later be known as Pythagorean triples – three whole numbers that, when used as the measurements of the sides of a triangle, produce a right triangle.7 The smallest Pythagorean triples are 3–4–5 and 5–12–13.[4] The list took the ancient scribes into large numbers. While the Plimpton 322 list was not airtight evidence that its makers knew the Pythagorean theorem, it was further evidence of the possibility.
The text tablet labelled Plimpton 322
In the 1950s, the Iraqi Department of Antiquities excavated a site known as Tell Harmal near the location of ancient Babylon[5] – a town called Shaduppum that had been an administrative complex under kings ruling just before the great lawgiver Hammurabi, during the First Babylonian Dynasty (1894–1595 B.C.).[6] Modern Baghdad has sprawled out so far that the area where Tell Harmal is situated is now one of its suburbs, but during the First Babylonian Dynasty, Shaduppum was a heavily fortified independent community. The Iraqi archaeologists uncovered massive walls buttressed with towers, a temple with life-sized terra-cotta lions at its entrance, captured in mid-roar, and, across the street from the temple, buildings that had been the primary administrative centre and had included a school for scribes. The cuneiform documents buried among its rubble were not only administrative texts, letters, and a law code, but also long lists of geographical, zoological, and botanical terms, and mathematical material. Many of these tablets were, like Weidner’s VAT 6598, school texts, used and copied by pupils with differing degrees of skill and sloppiness at the scribal school. They amounted to a cross section of Babylonian knowledge at its height, four thousand years ago. One tablet revealed that the scribes of that era understood right triangles, square roots, and cube roots, and were using them in a manner that implied familiarity with the Pythagorean theorem.8
In the 1980s, Christopher Walker of the British Museum made an extraordinary find, not at an archaeological dig but in the museum’s vast, disorganised collection of tablet fragments. A piece labelled BM96957 turned out to be a ‘direct join’ to the tablet Weidner had written about in 1916. The two pieces together present three problems and three methods of solving them. The third method, found only on Walker’s BM96957, is the Pythagorean theorem. (See the box for a near translation of a part of the text.)
The Babylonians used the sexigesimal place value system, not the decimal – that is, their number system was based on sixes, not tens. (The modern system of counting hours, minutes, and seconds is derived from it.) In the drawing and text below, the portions in brackets are a conjectural reconstruction by Eleanor Robson, based on the contents of the rest of the tablet. The italicised numbers in brackets give the equivalents in the decimal system. The drawing is not to scale, nor was it on the tablet. The length of the diagonal is an irrational number. It is 41 plus an infinite string of numbers after the decimal point. The author of the tablet satisfied himself with an imprecise measurement of the diagonal. The measurement is of a rectangular gate, lying on its side, so that ‘height’ refers to the longest side.9
[What is the height? You:] square [41. . . . . . , the diagonal]. 28 20 (1700) is the squared number. Square [10, the breadth]. You will see 1 40 (100)
[Take] 1 40 from 28 20 (1700 minus 100) [26 40 (1600) is the remainder.]
What is the square root? The square root is 40.
This solution definitely used the theorem we now call Pythagorean. In modern terminology: The breadth of the gate is 10, which squared is 100. The height of the gate is 40, which squared is 1600. The length of the diagonal of the gate is a number close to 41. The square of that number is 1700. 1600+100=1700
Was the mathematical knowledge that scribal students were mastering in the first half of the second millennium B.C. still available in Babylon in the sixth century B.C., in the neo-Babylonian era, when Iamblichus thought Pythagoras visited Babylon? We tend to assume that knowledge once discovered stays discovered, but much can happen to knowledge in a thousand years, particularly in as politically unstable a region as this. For example, sophisticated building techniques used routinely by the Romans were unknown to even the most brilliant architects and builders in the Middle Ages and early Renaissance, and were being discovered as though for the first time as late as the fifteenth century.10 Knowledge of a thirty-geared, hand-operated mechanical computer known as the Antikythera Mechanism used by the Hellenistic Greeks in 150–100 B.C., and the technological understanding necessary to manufacture and use it, were likewise lost, and a thousand years passed before anyone even thought of the possibility of such an invention again.11 [7] The tablets at Shaduppum disappeared in the rubble before 1600 B.C. and were, in the time of Pythagoras, lying right where archaeologists would find them in the twentieth
century A.D.
There are very few school mathematics tablets dating from 1600–1350 B.C., and another evidential gap 1100–800 B.C. The historian and Assyriologist Eleanor Robson, who has given these issues more thought than perhaps any other modern scholar, listed several possible explanations, but concluded that ‘the collapse of the Old Babylonian state in 1600 B.C.E. entailed a massive rupture of all sorts of scribal culture. Much of Sumerian literature was lost from the stream of tradition, it seems, and most of Old Babylonian mathematics too.’12 [8]
Although Robson believes that the later Babylonians were probably ignorant of the achievements of Old Babylonian mathematics, it is likely that useful fallout from that lost knowledge, such as a triple that was handy for finding right angles, would have remained in use in Mesopotamia and elsewhere for centuries, without those who utilised it remembering the hidden relationship among the numbers.13 And even if Pythagoras never visited Babylon, Greece was no wasteland when it came to building and surveying: Eupalos’ astounding water tunnel on Samos was built in Pythagoras’ century, as were many magnificent Greek temples. Though Pythagoras and his followers were not the first to know the theorem, their discovery might have been an independent discovery, or linked only by some surviving vestige of the more ancient, lost knowledge.
Think of land: Pythagoras had, after all, grown up in a Geomoroi family on Samos, and the Geomoroi got their name from the way they laid out their land. Take 9 plots of land, add 16 more, and you have 25 plots, as you can see if you draw them.
Pythagorus Page 10