Once again, archaeology has shown that ancient humans were far more advanced than we like to give them credit for—in this case, showing us that math principles that weren’t developed until the 14^{th} century in Europe were actually in use 1,400 or more years earlier in Babylon.

There exist some 110 cuneiform known tablets created by the Babylonians that explain how they computed things such as planetary positions using math—but for years, researchers have been puzzled by four mostly complete tablets mentioning Jupiter and some unusual geometry.

That is, until a new tablet was recently uncovered, which was revealed to depict new information that clarified what was being explained in the other four.

According to the paper, which is published today in *Science* (DOI 10.1126/science.aad8085), these five tablets all date to between 350 and 50 BCE, and track the motion of Jupiter as it moves across the sky over the course of 60 and 120 days. The tablets used a moderately complex bit of geometry to achieve this—involving creating a trapezoid to determine Jupiter’s total degrees of motion in a time period.

Such an idea didn’t spring up overnight, however.

“They observed and recorded Jupiter’s distance to nearby reference stars along its path,” author Mathieu Ossendrijver, a professor at Humboldt-Universität zu Berlin who focuses especially on Babylonian mathematics and astronomy, told redOrbit by email. “These observational reports are known as astronomical diaries, which were written between 700 and 50 BCE. From all these data they eventually, after 400 BCE, were able to construct mathematical models of Jupiter’s motion.”

**It’s all Babylonian to me**

Some may recall that the ancient Greeks also actively used geometry (looking at you, Euclid)—and may be wondering what makes the Babylonians’ math so unique that it wasn’t used again for 1,400 years. As it turns out, the difference comes in terms of levels of abstraction.

The Greeks used geometry to describe the configurations of figures in physical space, but the Babylonians added in an entirely new dimension to the mix: *time*.

“The Babylonian geometry in these tablets is essentially an application of methods that were developed by the Babylonians near 1800 BCE, but they now apply them in a totally new manner, because the trapezoid figures are not defined in real space (so the speak the space in which we live and in which the planets move) but in a more abstract mathematical space

obtained by drawing velocity against time,” Ossendrijver told redOrbit.

It took the rest of the Western world over a thousand years to catch up. In the 14^{th} century CE, scholars from Oxford theorized and philosopher **Nicole Oresme** of Paris confirmed that one could compute the displacement of an object by creating a trapezoid using velocity. From there, it became canon.

For those curious, an explanation of the math used is below.

**Math alert**

To make a long story short, both the Greeks and the Babylonians used this figure and this formula in various applications of math and science:

But where a Greek trapezoid used measurements of, say, length for *a*, *b*, and *h *to determine the area of a trapezoid, the Babylonians used velocity for *a* and *b*, measured in degree shift per day, and used the total number of days for *h* to determine the total shift of Jupiter.

Or, in other words, “You get the total distance traveled by Jupiter after 60 days by adding up the distances that it travels on each of the 60 days,” wrote Ossendrijver. “Think of these little distances per day as columns with a width of 1 day and a height equal to the distance. Then imagine arranging all these 60 columns vertically next to one another, that gives you the trapezoid figure.”

Which looks like this:

And the Babylonians used this to figure out the total distance traversed by Jupiter: “So the total distance travelled after 60 days is the total length of all these columns, which equals the area of the trapezoid,” wrote Ossendrijver.

In terms of the formula, the area of this trapezoid equals the total displacement, and works like this:

For those curious, the displacement over 60 days was 10 degrees and 45 minutes, and for 120 days it was 16 degrees and 15 minutes. The math for that works out like this for the 60-day period:

The math here different than we normally expect, as we work on a base 100 system, whereas the Babylonians worked in a base **60 system**. Regardless, it turns out to be correct.

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Feature Image: Five spots – one colored white, one blue, and three black – are scattered across the upper half of the planet. Closer inspection by NASA’s Hubble Space Telescope reveals that these spots are actually a rare alignment of three of Jupiter’s largest moons – Io, Ganymede, and Callisto – across the planet’s face. In this image, the telltale signatures of this alignment are the shadows [the three black circles] cast by the moons. Io’s shadow is located just above center and to the left; Ganymede’s on the planet’s left edge; and Callisto’s near the right edge. Only two of the moons, however, are visible in this image. Io is the white circle in the center of the image, and Ganymede is the blue circle at upper right. Callisto is out of the image and to the right. This image was taken March 28, 2004, with Hubble’s Near Infrared Camera and Multi-Object Spectrometer. (Credit: NASA, ESA, and E. Karkoschka (University of Arizona))

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