# 3,700-Year-Old Babylonian Tablet Decoded

Plympton 322 is a famous Babylonian clay tablet showing numbers that can form the sides of right-angled triangles. UNSW/Andrew Kelly

Almost a century ago, Edgar Banks – the inspiration for Indiana Jones – dug up a clay tablet in southern Iraq, but it took until now for its meaning to be understood. With this explanation has come insight into Babylonian mathematics, which operated on a different, and in some ways preferable, system than our own.

In 1945, it was realized that the tablet, known as Plimpton 322 after it was sold to collector George Plimpton for \$10, had mathematical significance, but the details remained a mystery. New research argues it represents part of a trigonometric table, and one more accurate than those that came afterwards.

Plimpton 322's burial location in what was once the city of Larsa indicates it's 3,700 years old, dating from the time of Hammurabi, who established one the earliest surviving legal codes. “Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realized it contains a special pattern of numbers called Pythagorean triples,” said Dr Daniel Mansfield of the University of New South Wales in a statement. Pythagorean triples are any whole numbers a, b, and c that can form a right-angle triangle through the formula a2 + b2 = c2, with 3, 4, and 5 being the most familiar example.

“The huge mystery, until now, was its purpose – why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet,” Mansfield continued.

Mansfield became interested in the problem and collaborated with his colleague Dr Norman Wildberger to try to unravel it. Wildberger is the inventor of a new way of doing trigonometry, based on the ratio of sides rather than angles. In 2005, he published a book, Divine Proportions: Rational Trigonometry to Universal Geometry, demonstrating that any problem that can be solved using traditional trigonometric methods canalso  be solved using his technique, and often more easily for those who have taken the time to learn it.

The idea of Plimpton 322 as a trigonometric table had been raised before, and eventually rejected, but this was done in the absence of an understanding of Wildberger's methods.

Mansfield and Wildberger concluded that the ancient Babylonians had beaten Wildberger to his ideas by almost four millenia, albeit only for right-angled triangles. They report in Historica Mathematica that instead of using sinΘ, cosΘ, and tanΘ as we do – something we inherited from the ancient Greeks – Plimpton 322 could be used by anyone needing to know the length of one side of a right-angled triangle by finding the closest match to the two known sides.

“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles," Mansfield said. "It is a fascinating mathematical work that demonstrates undoubted genius." The tablet would have been useful to architects or surveyors.

At some point since its making, a section of Plimpton 322 broke off. What remains are the side lengths for 15 right-angle triangles, ordered by inclination. Mansfield and Widlberger believe there were once 38 rows and 6 columns, making a truly impressive store of possible triangles.

The use of ratios in combination with the Babylonian base sixty number system, from which we get the length of our hours and minutes, made for an arguably superior method for calculating trigonometry to the table of chords created by the Greek mathematician Hipparchus more than 1,000 years later.

Mansfield told IFLScience that we have no idea why Babylonian trigonometry was lost. While it is possible that ancient mathematicians decided Hipparchus’ work was superior, it is also possible that Larsa and other centers of this knowledge lost a war, taking valuable knowledge with it. Mansfield noted that there is a gap in our records of the Babylonian civilization lasting several centuries.

When artifacts appear again, what we find comes mixed with influences from other cultures. Still, many Babylonian tablets have yet to be examined in detail, even aside from those that have yet to be dug up, so there may be plenty more we can learn about  Babylonian mathematics now that we have a hint.

For all the merits of Wildberger’s system, it has struggled to gain a foothold among mathematicians and teachers well versed in classical trigonometry. However, Mansfield speculates that Plimpton 322 might change this. The use of ratios rather than angles could become a matter of great interest to historians of mathematics, who may learn more about how it was done. Eventually, it may be taught in schools to show there is more than one way to think about trigonometry.