# Babylonians used geometry to track Jupiter’s movements

## Advanced sky-watching calculations came long before Europeans did the same thing

By Bruce Bower

Ancient Babylonians charted Jupiter’s heavenly motion in a surprisingly modern, mathematically abstract way — a feat that until now was thought to have originated among European scholars who lived roughly 1,400 years later.

Analyses of cuneiform writing on four largely intact clay tablets show that innovative geometric calculations enabled ancient astronomers to track the giant planet’s movement across the sky, Mathieu Ossendrijver reports in the Jan. 29 *Science*. These tablets were excavated more than a century ago and are now housed at the British Museum in London.

Researchers did not document exactly where these and thousands of other clay tablets were uncovered in 19th century excavations in Iraq. But most scholars today consider tablets from those digs that contain astronomical tables and calculations to have been found in Babylon, the ancient capital of Babylonia, says Ossendrijver, a historian of ancient science at Humboldt University of Berlin. Based on previous age estimates for other Babylonian tablets dealing with mathematical astronomy, the geometry-bearing tablets were probably written between 2,366 and 2,066 years ago, he says.

“Babylonians applied geometric methods that had been used for 1,000 years to develop a very modern way of studying motion,” Ossendrijver says.

The new findings show that Babylonians made a mental leap from describing planetary motion in concrete, arithmetic terms to representing those movements in an abstract, geometric space, says University of Chicago historian of ancient science John Wee.

More than 400 tablets recovered at Babylonian sites contain calculations about planets and the moon, mostly based on addition and other arithmetic operations, or instructions for carrying out those calculations.

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Four Babylonian tablets at the British Museum preserve portions of a different, poorly understood astronomical calculation system. Researchers call these calculations trapezoid procedures, because of references to four-sided shapes with two parallel sides of differing lengths.

Ossendrijver deciphered those tablets with the aid of a fifth Babylonian tablet held by the British Museum that he identified as containing nearly complete instructions for carrying out trapezoid procedures. While studying photographs of the museum’s tablets in January 2015, Ossendrijver noticed mentions of Jupiter and trapezoid procedures on that crucial piece of impressed clay.

Armed with this Babylonian “Rosetta stone,” Ossendrijver determined that Babylonian astronomers used trapezoid-shaped graphs to determine Jupiter’s movements. They calculated how the planet’s velocity changed from day to day and the distance covered by Jupiter over two consecutive 60-day intervals. Ancient Babylonians also divided trapezoidal graphs into two smaller trapezoids of equal area to determine the time when Jupiter traveled half the distance it would eventually cover over 60 days, Ossendrijver finds.

British and French scholars in the 14th century developed similar methods for tracking planets’ movements in an abstract, geometric space. But Babylonians pioneered those astronomical calculations, Ossendrijver says.

Geometry, defined broadly as any mathematical procedure concerning shapes, sizes, lines and angles, was widely used in the ancient world by the time of Babylonian astronomy, Wee says. Ancient Greek astronomers used geometry, although not the trapezoidal procedures developed by Babylonians to study planetary motion.

Still, Greek astronomers applied geometry in inventive ways (*SN: 8/30/08, p. 10*). Aristarchus, for instance, estimated distances to the sun and moon using geometric calculations unknown to the Babylonians, Wee says.