Swirling gas clouds on Jupiter
Ancient tablets describe math that was thought to have been invented over 1,000 years later, rewriting the history books.
The Babylonians’ techniques outshine those used by contemporary Greek and Egyptian astronomers—and shockingly mirror the mean speed theorem, a mathematical description of motion developed by a 14th-century English group known as the Oxford Calculators.
And the findings probably represent the tip of a mathematical iceberg. “There are thousands of tablets in various museums that were never translated,” says Ossendrijver, “and often we can translate a tablet but don’t understand what is going on until much later.”Researchers have long known that the Babylonians, who lived in what is now Iraq, had considerable mathematical skill: They successfully approximated the square root of 2 and understood the Pythagorean theorem nearly 4,000 years ago—more than a millennium before Pythagoras was born.
They were also talented astronomers, maintaining nightly catalogs detailed enough to record the passage of Halley’s comet. Babylonians regularly used arithmetic to boost their astronomical predictions.
But no one had ever found a Babylonian astronomical calculation that leveraged their impressive knowledge of pure geometry, until Mathieu Ossendrijver of Germany’s Humboldt University of Berlin spent 13 years deciphering what he described as a “small bunch of four weird trapezoid computations” between 2,000 and 2,400 years old.
Ossendrijver was the first to notice that the tablets—stored in the British Museum since the 1880s—had something to do with the planet Jupiter. However, they didn’t make much sense without knowing how the Babylonians encoded aspects of Jupiter’s motion, such as its appearances on the horizon.
Such a find also speaks to the human spirit of discovery—both of the ancient astronomers who gazed at the heavens, and the modern researchers who seek to reconstruct their understanding of the cosmos.
Like in Egypt, priests encouraged much of the development of Babylonian science. Babylonians used a numeral system with 60 as its base, which allowed them to divide circles into 360 degrees. The use of 60 as a base of a mathematical system is not a minor issue: 60 is a number that has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), which simplifies the representation of fractions: 1/2 (30/60), 1/3 (20/60), 1/4 (15/60), 1/5 (12/60), 1/6 (10/60), and so forth. As early as 1800 BCE, Babylonian mathematicians understood the properties of elementary sequences, such as arithmetic and geometrical progressions, and a number of geometrical relationships. They estimated the value of pi as 3 1/8, which is about a 0.6 percent error. It is highly likely that they also were familiar with what we today call the Pythagorean Theorem which states that the square of the longest side of a right triangle equals the sum of the squares of the other two sides. However, we have no evidence that the Babylonians proved it formally, since their mathematics rested on empirical knowledge rather than formal proof.
It was in astronomy where Babylonians showed a remarkable talent, and magic, mysticism, astrology, and divination were its main drivers. They believed that the movement of the heavenly bodies forecasted some terrestrial event. Since the reign of Nabonassar (747 BCE), the Babylonians kept complete lists of eclipses and by 700 BCE, it was already known that solar eclipses could only be possible during new moons and lunar eclipses only during full moons. It is possible that by this time Babylonians also knew the rule that lunar eclipses take place every six months, or occasionally every five months. By the time Nebuchadnezzar ruled Babylon, the priests had also calculated the courses of the planets and plotted the orbits of the sun and the moon. – Cristian Violatti
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.”