By Kenneth Chang

Clay tablets, including one at the left, revealed that Babylonian astronomers employed a sort of precalculus to describe Jupiter’s motion across the night sky relative to distant background stars. They did this 15 centuries earlier than Europeans were first credited with making such measurements.

For people living in the ancient city of Babylon, Marduk was their patron god, and thus it is not a surprise that Babylonian astronomers took an interest in tracking the comings and goings of the planet Jupiter, which they regarded as a celestial manifestation of Marduk.

What is perhaps more surprising is the sophistication with which they tracked the planet, judging from inscriptions on a small clay tablet dating to between 350 B.C. and 50 B.C. The tablet, a couple of inches wide and a couple of inches tall, reveals that the Babylonian astronomers employed a sort of precalculus in describing Jupiter’s motion across the night sky relative to the distant background stars. Until now, credit for this kind of mathematical technique had gone to Europeans who lived some 15 centuries later.

Additional tablets, including this one, show that the Babylonians realized that the area under the curve of a graph of velocity against time represented distance traveled.

“It’s a figure that describes a graph of velocity against time,” he said. “That is a highly modern concept.”

Mathematical calculations on four other tablets show that the Babylonians realized that the area under the curve on such a graph represented the distance traveled.

“I think it’s quite a remarkable discovery,” said Alexander Jones, a professor at the Institute for the Study of the Ancient World at New York University, who was not involved with the research. “It’s really quite clear from the text.”

Ancient Babylon, situated in what is now Iraq, south of Baghdad, was a thriving metropolis, a center of trade and science. Early Babylonian mathematicians who lived between 1800 B.C. and 1600 B.C. had figured out, for example, how to calculate the area of a trapezoid, and even how to divide a trapezoid into two smaller trapezoids of equal area.

For the most part, Babylonians used their mathematical skills for mundane calculations, like figuring out the size of a plot of land. But on some tablets from the later Babylonian period, there appear to be some trapezoid calculations related to astronomical observations.

In the 1950s, an Austrian-American mathematician and science historian, Otto E. Neugebauer, described two of them. Dr. Ossendrijver, in his recent research, turned up two more.

But it was not clear what the Babylonian astronomers were calculating.

A year ago, a visitor showed Dr. Ossendrijver a stack of photographs of Babylonian tablets that are now held by the British Museum in London. He saw a tablet he had not seen before. This tablet, with impressions of cuneiform script pressed into clay, did not mention trapezoids, but it recorded the motion of Jupiter, and the numbers matched those on the tablets with the trapezoid calculations.

“I was certain now it was Jupiter,” Dr. Ossendrijver said.

When Jupiter first appears in the night sky, it moves at a certain velocity relative to the background stars. Because Jupiter and Earth both constantly move in their orbits, to observers on Earth, Jupiter appears to slow down, and 120 days after it becomes visible, it comes to a standstill and reverses course.

In September, Dr. Ossendrijver went to the British Museum, where the tablets were taken in the late 19th century after being excavated. A close-up look of the new tablet confirmed it: The Babylonians were calculating the distance Jupiter traveled in the sky from its appearance to its position 60 days later. Using the technique of splitting a trapezoid into two smaller ones of equal area, they then figured out how long it took Jupiter to travel half that distance.

Dr. Ossendrijver said he did not know the astronomical or astrological motivation for these calculations.

It was an abstract concept not known elsewhere at the time. “Ancient Greek astronomers and mathematicians didn’t make plots of something against time,” Dr. Ossendrijver said. He said that until now, such calculations were not known until the 14th century by scholars in England and France. These mathematicians of the Middle Ages perhaps had seen some as yet unknown texts dating to Babylonian times, or they developed the same techniques independently.

“It anticipates integral calculus,” Dr. Ossendrijver said. “This is utterly familiar to any modern physicist or mathematician.”

Thanks to Kebmodee for bringing this to the It’s Interesting community.