“Si.427 dates from the Old Babylonian period (1900-1600 BCE),” said Dr. Daniel Mansfield, a mathematician in the School of Mathematics and Statistics at the University of New South Wales.
“It’s the only known example of a cadastral document from this period, which is a plan used by surveyors define land boundaries.”
“In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”
“This is a significant object because the surveyor uses what are now known as Pythagorean triples to make accurate right angles.”
In 2017, Dr. Mansfield conjectured that Plimpton 322, another fascinating tablet from the same period, was a unique kind of trigonometric table.
Si.427 is thought to have existed even before Plimpton 322 — in fact, surveying problems likely inspired Plimpton 322.
“There is a whole zoo of right triangles with different shapes. But only a very small handful can be used by Babylonian surveyors. Plimpton 322 is a systematic study of this zoo to discover the useful shapes,” Dr. Mansfield said.
Back in 2017, he hypothesized that Plimpton 322 was likely to have had some practical purpose, possibly used to construct palaces and temples, build canals or survey fields.
“With Si.427, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries,” he said.
“This is from a period where land is starting to become private — people started thinking about land in terms of ‘my land and your land,’ wanting to establish a proper boundary to have positive neighborly relationships. And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.”
There are even clues hidden on other tablets from that time period about the stories behind these boundaries.
“Another tablet refers to a dispute between Sin-bel-apli — a prominent individual mentioned on many tablets including Si.427 — and a wealthy female landowner,” Dr. Mansfield said.
“The dispute is over valuable date palms on the border between their two properties. The local administrator agrees to send out a surveyor to resolve the dispute. It is easy to see how accuracy was important in resolving disputes between such powerful individuals.”
The way these boundaries are made reveals real geometric understanding.
“Nobody expected that the Babylonians were using Pythagorean triples in this way. It is more akin to pure mathematics, inspired by the practical problems of the time,” Dr. Mansfield said.
According to the scientist, one simple way to make an accurate right angle is to make a rectangle with sides 3 and 4, and diagonal 5.
These special numbers form the 3-4-5 Pythagorean triple and a rectangle with these measurements has mathematically perfect right angles. This is important to ancient surveyors and still used today.
“The ancient surveyors who made Si.427 did something even better: they used a variety of different Pythagorean triples, both as rectangles and right triangles, to construct accurate right angles,” Dr. Mansfield said.
However, it is difficult to work with prime numbers bigger than 5 in the base 60 Babylonian number system.
“This raises a very particular issue — their unique base 60 number system means that only some Pythagorean shapes can be used,” Dr. Mansfield said.
“It seems that the author of Plimpton 322 went through all these Pythagorean shapes to find these useful ones.”
“This deep and highly numerical understanding of the practical use of rectangles earns the name ‘proto-trigonometry’ but it is completely different to our modern trigonometry involving sin, cos, and tan.”
Dr. Mansfield’s paper appears in the journal Foundations of Science.
D.F. Mansfield. Plimpton 322: A Study of Rectangles. Found Sci, published online August 3, 2021; doi: 10.1007/s10699-021-09806-0