Librado Romero/The New York Times

3,700 years ago, a Babylonian genius took a reed pen to a clay tablet and created what could be the world's first trigonometric table. This was at least a thousand years before Pythgoras ever figured out that in a triangle with a right angle, the longest side is equal to the sum of the square of the other two sides. Whoever etched the markings on the Babylonian tablet figured out the Pythagorean theorem way before Pythagoras did.

The tablet, known as Plimpton 322, was discovered in the early 1900s in southern Iraq by Edgar Banks, the person on whom Indiana Jones was based. It has four columns and 15 rows of numbers in an ancient cuneiform script. The script uses a sexagesimal system, with a base of 60. These writings have puzzled scientists for over seven decades.

Researchers from the University of New South Wales have now discovered that the writings on the tablet represent the oldest and most accurate trigonometric table in the world.

The purpose of the tablet has long been a mystery to scientists. Ancient scribes had taken on the complicated task of generating, sorting, and inscribing the numbers onto the wet clay. Thus, the contents of the tablet must have been important. Scientists believe that the tablet could have been a tool that surveyors used to calculate the construction of buildings like temples, pyramids, and palaces.

“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,” says Dr. Daniel Mansfield, one of the researchers. He continues: “[I]t is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.”

It's also possible that this ancient trigonometric table can teach us something new and can also be relevant to modern applications. Babylonian mathematics, which has long been out of fashion, has a base 60 approach while modern mathematics has a base 10 approach. Still, the contents of the tablet can potentially be helpful to surveying and even computer graphics. The trigonometry on the tablet is also simpler and more advantageous than modern trigonometry, giving us new opportunities for mathematics education.

Photo by Michael Harris

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

The tablet is actually incomplete. Its edge on its left side is broken, leading researchers to rely on previous studies to figure out that the tablet originally had six columns and 38 rows. The remaining 15 rows, however, describe 15 right-angle triangles with steadily decreasing inclinations.

Hipparchus, a Greek astronomer and mathematician considered to be the father of trigonometry, created a trigonometric table over a millennium after the Babylonian tablet was created. However, the Babylonian base 60 system as well as their use of ratios make up a far superior method. It's unclear why Babylonian trigonometry lost its footing in the world, but we now have the opportunity to revive it.

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