Used geometry that hints at calculus 1,500 years before Europeans.

Even when a culture leaves behind extensive written records, it can be hard to understand their knowledge of technology and the natural world. Written records are often partial, and writers may have been unaware of some technology or simply considered it unremarkable. That’s why the ancient world can still offer up surprises like the Antikythera Mechanism, an ancient mechanical computer that highlighted the Greeks’ knowledge of math, astronomy, and the mechanical tech needed to tie them together.

It took several years after the discovery for the true nature of the Antikythera Mechanism to be understood. And now something similar has happened for the Babylonians. Clay tablets, sitting in the British Museum for decades, show that this culture was able to use sophisticated geometry to track the orbit of Jupiter, relying on methods that in some ways pre-figure the development of calculus centuries later.

We already knew that the Babylonians tracked the orbits of a variety of bodies. There are roughly 450 written tablets that describe the methods and calculations that we’re aware of, and they date from 400 to 50 BCE. Most of the ones that describe how to calculate orbital motion, in the words of Humboldt University’s Mathieu Ossendrijver, “can be represented as flow charts.” Depending on the situation, they describe a series of additions, subtractions, and multiplications that could tell you where a given body would be.

(Complicating matters, Babylonian astronomy worked in base-60, which leads to a very foreign-looking notation.)

The Babylonians did have a grasp of geometric concepts—Ossendrijver calls them “very common in the Babylonian mathematical corpus”—but none of them appeared in their known astronomical calculations.

In the British Museum, however, he located a tablet that hadn’t been formally described, and it contained parts of the procedure for tracking Jupiter. Combined with other tablets, it starts with Jupiter’s first morning rising, tracks it through its apparent retrograde motion, and finishes with its last visible setting at dusk. Again, it’s procedural. Different sections are used to predict the planet’s appearance at different segments of its orbit.

Ossendrijver took the procedure for calculating the first 120 days and showed that calculating its daily displacement over time produces a trapezoid. In this case, the shape was largely a rectangle but with its top side angled downward over time in two distinct segments. A series of other tablets treated the calculations explicitly as producing a trapezoid.

Things get interesting in the next procedure, which is used to calculate when Jupiter reaches the midpoint in the first half of this stage of its motion. This procedure involved taking the left half of the trapezoid and dividing it into two pieces of equal area. The location of the dividing line (labelled vc above) then produces the answer. As Ossendrijver describes it, “They computed the time when Jupiter covers half this distance by partitioning the trapezoid into two smaller ones of ideally equal area.”

Figuring this out obviously required some sophisticated geometry. European scholars wouldn’t develop similar methods until the 14th century, when they became used at Oxford. The Greeks did use geometry for some astronomical work, but this involved calculations of actual space. The Babylonians here are working in an abstracted time-velocity space.

It’s also striking that this general approach is similar to some aspects of calculus. There, the area under a curve is calculated by mathematically creating an infinite number of small geometric figures and summing their areas. There’s no indication that the Babylonians were anywhere close to taking this intellectual leap given that they only divided this shape up a few times. But it does show that they recognised the value of the general approach.

By John Timmer via Ars Technica