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Ancient math book? The Babylonians wrote sophisticated mathematical procedures in cuneiform script on clay tablets.

Ancient math book? The Babylonians wrote sophisticated mathematical procedures in cuneiform script on clay tablets.

Mathieu Ossendrijver, Science/AAAS

Math whizzes of ancient Babylon figured out forerunner of calculus

Tracking and recording the motion of the sun, the moon, and the planets as they paraded across the desert sky, ancient Babylonian astronomers used simple arithmetic to predict the positions of celestial bodies. Now, new evidence reveals that these astronomers, working several centuries B.C.E., also employed sophisticated geometric methods that foreshadow the development of calculus. Historians had thought such techniques did not emerge until more than 1400 years later, in 14th century Europe.

The study “is an extremely important contribution to the history of Babylonian astronomy, and more generally to the history of science,” says astronomy historian John Steele of Brown University, who was not part of the work.


Astroarchaeologist Mathieu Ossendrijver of Humboldt University in Berlin bases his findings on a reexamination of clay tablets, one of them unknown until recently, dating from 350 B.C.E. to 50 B.C.E. One week each year for the past 14 years, Ossendrijver has made a pilgrimage to the British Museum’s vast collection of tablets inscribed in the Babylonian cuneiform script. He was trying to solve a puzzle posed by two tablets dealing with astronomical calculations: They also contained instructions for constructing a trapezoidal figure that seemed unrelated to anything astronomical.

Between 2002 and 2008, Ossendrijver, an astrophysicist turned historian, studied two other tablets that also prescribed the drawing of a trapezoid, and in these he thought he could make out a reference to Jupiter. The giant planet was a favorite among the Babylonians, who equated the orb with their main god, Marduk, patron deity of the city of Babylon. But the Jupiter link was tentative.

Then, late in 2014, retired Assyriologist Hermann Hunger of the University of Vienna visited Ossendrijver, bringing photos taken decades ago of an uncatalogued Babylonian tablet from the British Museum that described some kind of astronomical computation. Alone in his office a few months later, Ossendrijver perused the photos. The images were blurry and the inscriptions slanted, making them hard to read, but he realized the numbers were identical to those in the trapezoid inscriptions he had been scrutinizing. By comparing the photos with fragments of other Babylonian texts, he discovered that the computations described the motion of Jupiter.

Marduk, the patron god of Babylon, was equated with Jupiter, so ancient astronomers charted the planet’s path with care.

Marduk, the patron god of Babylon, was equated with Jupiter, so ancient astronomers charted the planet’s path with care.


Examining all of the tablets at the British Museum, Ossendrijver figured out that the trapezoid calculations were a tool for calculating Jupiter’s displacement each day along the ecliptic, the path that the sun appears to trace through the stars. The computations recorded on the tablets covered a period of 60 days, beginning on a day when the giant planet first appeared in the night sky just before dawn.

During that interval, Jupiter’s motion across the sky appears to slow. (Such erratic apparent motion stems from the complex combination of Earth’s own orbit around the sun with that of Jupiter.) A graph of Jupiter’s apparent velocity against time slopes downward, so that the area under the curve forms a trapezoid. The area of the trapezoid in turn gives the distance that Jupiter has moved along the ecliptic during the 
60 days. Calculating the area under a curve to determine a numerical value is a basic operation, known as the integral between two points, in calculus. Discovering that the Babylonians understood this “was the real ‘aha!’ moment,” Ossendrijver says. 

Although elated, Ossendrijver wasn’t ready to publish, because a second part of the trapezoid prescription remained unclear. By delving into older, purely mathematical Babylonian texts written between 1800 B.C.E. and 1600 B.C.E., which also described computations with a trapezoid, he realized that the astronomers who made the tablets had gone a step further. To compute the time at which Jupiter would have moved halfway along its ecliptic path, the astronomers divided the 60-day trapezoid into two smaller ones of equal area. The vertical line dividing the two trapezoids marked the halfway time; because of the different shapes of the trapezoids, it indicated not 30 days but slightly fewer.

The Babylonians had developed “abstract mathematical, geometrical ideas about the connection between motion, position and time that are so common to any modern physicist or mathematician,” Ossendrijver says.

Indeed, compared with the complex geometry embraced by the ancient Greeks a few centuries later, with its cycles and epicycles, the inscriptions reflect “a more abstract and profound conception of a geometrical object in which one dimension represents time,” says historian Alexander Jones of New York University in New York City. “Such concepts have not been found earlier than in 14th century European texts on moving bodies,” he adds. “Their presence … testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy.”

After cuneiform died out around 100 C.E., Babylonian astronomy was thought to have been virtually forgotten, he notes. It was left to French and English philosophers and mathematicians in the late Middle Ages to reinvent what the Babylonians had developed.

The new discovery may hint that Babylonian geometry did not die out completely after all. Either way, Jones says, learning how the Babylonians astronomers acquired their geometric acumen “would tell us something about why human beings do science in the first place, and from time to time do it very well indeed.”