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.”