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Sumerian Mathematics
Thanks to a clue from a 50-year old photograph, a historian has decoded a mysterious trapezoid described on ancient Babylonian astronomical tablets.

 

That previously unexplained description is a scheme to predict Jupiter’s place in the zodiac – and it shows that ancient Mesopotamian astronomers beat Europeans by at least 1500 years in grasping the ideas that led to integral calculus. The codebreaking tablet from the photograph had laid unnoticed in the cuneiform collection at London’s British Museum for decades.

 

Museum visitors looking for cuneiform astronomy texts typically use a list compiled by a former curator. The list includes four tablets that refer to a trapezoid shape while discussing Jupiter, which had always been confusing. Although geometry was often used in Babylonian mathematics, historians thought their astronomy relied only on arithmetic – so the trapezoid didn’t make sense. The breakthrough came in 2015, when a colleague visiting from Vienna in Austria handed a stack of mid-20th-century photographs to Mathieu Ossendrijver of Humboldt University in Berlin.

 

Cuneiform Tablet

Those pictures showed cuneiform tablets from around 100 or 200 BC that had been excavated from Babylon and Uruk in the 19th century and transported to the British Museum. Ossendrijver knew the collection well. But one tablet, inscribed with a calculation involving Jupiter, was unfamiliar. It wasn’t included on the list of astronomy texts, but he realised right away that it offered a chance to crack the meaning of the trapezoid.

 

“When I found this tablet last year, I immediately thought of these other tablets that I knew about, a few of which I translated myself,” Ossendrijver says. “But I never understood them.” Planetary tracking. The new calculation and the mysterious trapezoid drawing appear to be recipes for the same process. They show how to predict Jupiter’s place in the zodiac – constellations that the Babylonians used as a coordinate system for calculating celestial positions. Such a prediction would have been important to the ancient Babylonians, who looked to astrology to figure out everything from weather forecasts to the price of goods.

 

But more interesting to historians is how the Jupiter-tracking algorithm works. Ossendrijver thinks the slanted top of the trapezoid is like a graph of Jupiter’s speed across the sky in the days after it first appears above the horizon. By calculating the area inside the trapezoid, Babylonian astronomers could find where the planet would be in the sky – exploiting the same link between velocity and displacement taught in introductory calculus classes. That makes it the only known geometrical method in Babylonian astronomy. It’s also different from Greek astronomy, in which shapes were used to represent dimensions of real space and time, but nothing as conceptual as velocities.

 

“It’s a nice example of this ability the Babylonians had to think in a very abstract way about real problems,” says John Steele of Brown University in Rhode Island, who was not involved in the research.

The method was prescient, too. Scholars at Oxford’s Merton College and in Paris during the 14th century are typically credited with the same insight about velocity and displacement. They even connected it to the trapezoid shape. These ideas were the antecedents of the calculus developed by Newton and Leibniz – but the Babylonians had them far earlier.

 

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