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Babylonians figured out trigonometry 3,700 years ago, and their “simpler” math techniques could help students today

Nancy Bilyeau
Author: UNSW/Andrew Kelly
Author: UNSW/Andrew Kelly

Trigonometry is not exactly a simple subject to learn for many math students. But it’s just possible that the Babylonians, whose culture thrived from 2800 to 1700 BC, can be of service.

A Babylonian clay tablet known as Plimpton 322 was discovered early in the 20th century in present-day Iraq. An Australian mathematician, Dr. David Mansfield of the School of Mathematics and Statistics of New South Wales, who has been analyzing its meaning, announced his conclusions in late August, saying that cracking the meaning of the tablet, as big as a hand palm, could simplify our study of triangles.

Up to now, trigonometry has been based on study of angles and irrational numbers. The analysis of Plimpton 322 reveals that the Babylonians used ratios. “This gives us a different way of looking at trigonometry,” Dr. Mansfield told journalists. “The beautiful thing about it is that it’s much simpler.” His paper, co-authored with UNSW associate professor Norman Wildberger, is published in Historia Mathematica.

Trigonometry is not a fusty, esoteric branch of mathematics. It is essential to architecture, engineering, astronomy, surveying, and even oceanography. Up to now, it’s been taught using Greek principles. But this tablet proves that the Babylonians beat the Greeks in the discovery of trigonometry by about 1,000 years.

The tablet was unearthed in southern Iraq, believed to be near the onetime Sumerian city of Larsa, by Edgar Banks, a collector of antiquities beginning in the 1890s and someone who has been regarded as a possible inspiration for the movie character Indiana Jones. (James Henry Breasted, Frederick Russell Burnham, and Roy Chapman Andrews are among the other Indy models.) Banks was not collecting cuneiform tablets as an archaeologist, but while using his position as the American consul to Baghdad. He sold the tablets to universities, libraries, and museums.

The Greek astronomer Hipparchus has been considered the father of trigonometry. But this tablet was created long before Hipparchus lived.

Photograph: UNSW/Andrew Kelly
Photograph: UNSW/Andrew Kelly

Dr. Mansfield said, “Babylonian mathematics may have been out of fashion for more than 3,000 years but it has possible practical applications in surveying, computer graphics and education.”

Larsa was one of the ancient capital cities of Babylonia. During its most prosperous period it was ruled by a dynasty founded by King Naplanum, who was followed by a line of 13 kings. The arts and sciences were encouraged, with the creation of Sumerian scribe schools. More than 400 clay tablets still exist from the time of the Babylonians. The tablets were written on, inscribed while clay was damp, and then baked.

The tablet got its name because an American publisher and collector named George Arthur Plimpton bought it. (He is the grandfather of writer and editor George Plimpton, founder of the Paris Review and author of Paper Lion.) In 1936, not long before he died, Plimpton donated the Babylonian tablet along with many other manuscripts to Columbia University. It is now contained in the Rare Book and Manuscript Library at Columbia University.

In the 1940s, researchers who were studying the tablet concluded that the cuneiform numbers on it corresponded to the Pythagorean Theorem, which as math-friendly students can tell you says that the square of a right triangle’s hypotenuse is equal to the squared lengths of the other two sides. But that was far as they got until Dr. Mansfield and his team took up the challenge.

The tablet “is a fascinating mathematical work that demonstrates undoubted genius,” says Dr. Mansfield. “The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.”

Related story from us: The British Museum has issued the first online 3D scan version of the ancient Rosetta Stone

Dr. Mansfield added: “A treasure-trove of Babylonian tablets exists, but only a fracture of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.