# The roots of trigonometry, freshly debated

## 3,700-yr-old Babylonian tablet contains world’s first trigonometric table, claim Australian mathematicians; not all are convinced.

Plimpton 322. Source: Rare Book & Manuscript Library, Columbia University

Around 3,700 years ago, someone in Babylon wrote a table of numbers on a clay tablet. Since the 1930s, when the tablet entered Columbia University’s collection, researchers have puzzled over the numbers: what purpose did they serve? Now, mathematicians at the University of New South Wales believe they have the answer.

It is the world’s first trigonometric table, claim Daniel Mansfield and Norman Wildberger, who have published their observations in Historia Mathematica. In doing so, they have challenged the existing view that attributes the origin of trigonometry to Greek astronomer-mathematician Hipparchus (2nd century BC).; the Babylonian tablet was written more than a millennium before his time. The tablet is named Plimpton 322 for American publisher G A Plimpton, who bought it from a dealer and later bequeathed it to Columbia University. It has four columns and 15 rows intact, with the numbers written in base-60. “The current view is that there are two missing columns broken off from the original tablet, and that are an additional 23 rows that were intended to fill the reverse and bottom,” Wildberger told The Indian Express.

The far-right column simply contains the row number. The next two columns have been found to contain the length of the hypotenuse and one more side of a right triangle, a different triangle for each row. In other words, these two numbers are part of a “Pythagorean triple” — compiled more than 1,000 years before Pythagoras. The three columns on the left — the contents of the two missing columns have been conjectured — contain various ratios involving the sides of the same triangle whose sides are given in the columns to the right (see box; what is written in decimals here is expressed in terms of base-60 numerals on the tablet).

When these relations have been known or conjectured for so long, what does the new research add? To Mansfield and Wildberger, this is trigonometry, based not on angles but on ratios.

“Our contribution is to explain why the numbers were written and arranged in this way,” Mansfield told The Indian Express. “The idea that this is a trigonometric table based on ratios has never been explored (in contrast, the idea that it is a trigonometric table based on angles has been explored and rightly dismissed many times).”

Mansfield and Wildberger call it “exact sexagesimal trigonometry”. In base-60 and without decimals, they note, the Babylonians could not exactly express the ratio between the base b and the diagonal (hypotenuse) d. In modern trigonometry, that ratio is the cosine of the base angle, but the ancient Babylonians did not deal in angles. So, they split that ratio into parts they could express.

The research has been met with some scepticism. Historians quoted by The New York Times and Science magazine have called some of the interpretations speculative. Eleanor Robson, a researcher on Mesopotamian mathematics who had studied Plimpton 322 and concluded that it was a teaching aid, tweeted: “…A Babylonian maths text I wrote about 20 y ago, arguing it’s not as interesting as mathematicians think. But still they come…” In another tweet, she advised journalists not to contact her on Plimpton 322.

Evelyn Lamb, a mathematician with the University of Utah, blogged in Scientific American suggesting that the new research was an attempt to sell Wildberger’s “pet theory” called rational trigonometry. Contacted by The Indian Express, she suggested that questions be addressed instead to “someone who is an expert on this tablet”.

Mansfield and Wildberger, incidentally, bring in an Indian context: a table of sines of various angles composed by astronomer-mathematician Madhava (1340-1425). Using Madhava’s table and the Babylonian table separately to solve a few mathematical problems, they conclude that the former is more accurate.

“Madhava’s table is a remarkable achievement, and a significant milestone in the development of the angle-based trigonometry with applications to astronomy,” Mansfield said, in reply to a question. “But if you use it only to study right triangles, then Plimpton 322 can outperform this much later table.”

kabir.firaque@expressindia.com