# Babylonians Were Using Geometry Centuries Earlier Than Thought

Ancient astronomers were tracking planets using math believed to have first appeared in 14th-century Europe

As one of the brightest objects in the night sky, the planet Jupiter has been a source of fascination since the dawn of astronomy. Now a cuneiform tablet dating to between 350 and 50 B.C. shows that Babylonians not only tracked Jupiter, they were taking the first steps from geometry toward calculus to figure out the distance it moved across the sky.

The discovery suggests that ancient astronomers were using mathematical concepts thought to have arisen centuries later in Europe. It's also the first time anyone has found direct evidence that Babylonians used this kind of abstract mathematics for astronomy.

The tablet "testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy during the second half of the first millennium B.C.," says Alexander Jones, a professor of the history of the exact sciences in antiquity at New York University.

Mathieu Ossendrijver of Humboldt University in Berlin found the tablet while combing through the collections at the British Museum. The written record gives instructions for estimating the area under a curve by finding the area of trapezoids drawn underneath. Using those calculations, the tablet shows how to find the distance Jupiter has traveled in a given interval of time. Until now, this kind of use of trapezoids wasn't known to exist before the 14th century.

"What they are doing is applying it to astronomy in a totally new way," Ossendrijver says. "The trapezoid figure is not in real space and doesn't describe a field or a garden, it describes an object in mathematical space—velocity against time."

Scholars already knew that Babylonians could find the area of a trapezoid, and that they were quite familiar with the motions of planets and the moon. Previous records show that they used basic arithmetic—addition, subtraction, multiplication and division—to track these celestial bodies.

By 400 B.C. Babylonian astronomers had worked out a coordinate system using the ecliptic, the region of the sky the sun and planets move through, Ossendrijver says. They even invented the use of degrees as 360 fractions of a circle based on their sexagesimal, or base 60, counting system. What wasn't clear was whether the Babylonians had a concept of objects in abstract mathematical space.

The trapezoid method involves learning the rate at which Jupiter moves and then plotting the planet's speed against a set number of days on an x-y graph. The result should be a curve on the graph. Figuring out the area of trapezoids under this curve gives a reasonable approximation of how many degrees the planet has moved in a given period.

Babylonian tablets that hinted at this ability had been found before. Otto Neugebauer, a well-known scholar in the field, identified two such tablets decades ago. A notation on Neugebauer's translation says "Jupiter…?", Ossendrijver adds, but it remained inconclusive because the context didn't seem to fit, and the tablets were damaged and incomplete.

In his own forays into the British Museum, Ossendrijver wasn't looking for evidence of new mathematics, just for interesting astronomical tablets. Some, he says, have been sitting in the museum for decades waiting for translation.

Ultimately he found three more with evidence for trapezoid-based calculations, including the one that clearly shows the mathematical feat of applying the technique to observations of Jupiter. All five tablets are described in Ossendrijver's study, appearing this week in *Science*.

While scientists can get approximate ages for the tables, they may never know exactly when they were written. The most recent find is a schematic of how to find Jupiter's position rather than a record of where the planet was on a given day, so the tablets can't be precisely dated.

Nor can Ossendrijver identify the author. Tablets often have a signature or a date, or both. But the piece that should have that information seems to be missing—it might have simply broken off the bottom, he says. Still, the tablets were originally found near a temple to Marduk, a Babylonian deity represented by Jupiter. That makes Ossendrijver pretty sure that the author was one of the astronomer-priests of the temple.

It's also unclear how widespread this method was, or if anyone tried using it on other planets. Jupiter moves very slowly against the background stars, and planets such as Mars or Venus should have been much easier to track this way. Further searches of tablets housed in museums may reveal more clues.

"We cannot exclude the possibility that one day we will find another tablet, maybe applied to Mars," Ossendrijver says.

But it's also possible the author had trouble passing on his revolutionary technique. The math might have been too abstract, while existing methods for observing the heavens worked well enough at the time. "Perhaps his colleagues didn't understand it." Ossendrijver says.