The Music of Pythagoras (13 page)

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 center. 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 turned 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 traveled 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 labeled 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.

The text tablet labeled Plimpton 322

In 1945, a text that archaeologists have labeled 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.
*
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.

In the 1950s, the Iraqi Department of Antiquities excavated a site known as Tell Harmal near the location of ancient Babylon
^†
—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
.).
*
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 center 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, disorganized collection of tablet fragments. A piece labeled 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.)

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
*
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
.