Almost four millennia ago, two wealthy Mesopotamian landowners quarreled over a plot of land, each claiming they were the rightful owner. The dispute was solved not through sheer force and violence — this was the kingdom that laid out the very first written laws after all — but instead through rather modern-style mediation. A skilled surveyor arrived at a site and with his trusted tools, he divided the disputed lands at the border into equal plots and the two landowners were back to their happy neighborly selves.

Such Babylonian surveyors were in charge of writing up the first cadastral documents in known history, during a time when citizens were entrusted with private property which had to be delineated from common lands. These ancient surveyors, known as scribes, didn’t have total stations and GPS at their disposal, and frankly, they didn’t need them. They were very well capable of accurately measuring and dividing plots of land using a yardstick and their mathematical skill.

A 3,700-year-old clay tablet, known as Si.427, is illustrative in this regard. It shows how Babylonian surveyors must have performed geometric operations, even using Pythagorean triples to accurately make right angles, more than a thousand years before the mighty Greek philosopher was born.

In a new study published today in Foundations of Science, Dr. Daniel Mansfield, a mathematician at the University of New South Wales in Australia, explains the rich significance behind what may very well be the oldest example of applied geometry in the world.

Proto-trigonometry: the geometry for the ground

Although Mansfield is a mathematician, his research into Si.427 looked more like that of an archaeologist. The tablet was discovered in Baghdad at the end of the 19th-century but had since changed hands many times and its location remained an enigma. However, Mansfield had heard about it while studying thousands of Babylonian fragments relating to mathematical applications in the old Mesopotamian kingdom.

In 2017, Mansfield studied another similar tablet from the same period, known as Plimpton 322, revealing that its purpose was that of a trigonometric table of sorts. Babylonians did not actually use trigonometry as we know it, as in the branch of mathematics concerned with specific functions of angles and their application to calculations. In fact, these ancient scribes understood only one angle: the right angle.

While Plimpton 322 isn’t a trigonometric table in the conventional sense, it lists a table of rectangles useful in practical measurements. Specifically, it lists Pythagorean triples, right triangles whose three sides are all integers where the square of the hypotenuse equals the sum of the squares of the other two sides.

For instance, a rectangle with sides 3 and 4, and a diagonal of 5 can be divided into two equal halves at the diagonal, leaving two perfect right-angle triangles.

Plimpton 322 doesn’t list all possible Pythagorean triples but rather compiles a number of triples, both as rectangles and right triangles, that were likely commonly encountered in surveying work. It was very much practical rather than theoretical work.

This limitation was owed to the sexagesimal (base 60) Babylonian number system, which means only some Pythagorean shapes can be used in practice. In this system, numbers are written by adding symbols that represent either 10 or 1, in that order. For instance, the number 5 is written as ‘blank space’ to signify no 10s and by five 1s. The number 16 is written as one 10 followed by six 1s. Then these number signs 1–59 can in turn be strung together to write numerals of any length.

The base 60 number system is actually still in use in some instances of our lives, despite the ubiquity of base 10. For instance, we still count sixty minutes in an hour and sixty seconds in a minute, and measure angles in multiples and fractions of 60. This is the legacy of Greek astronomers who adopted the Babylonian base 60 system because their own system was not as suited for astronomical calculations.

But since it is difficult to write and calculate with prime numbers bigger than 5 in base 60, only some Pythagorean triangles were used. This is why Mansfield calls the Babylonian geometry proto-trigonometry, an intermediate step towards modern trigonometry involving sin, cos, and tan.

However, it was not clear how tables such as those found on Plimpton 322 were actually used in practice. Mansfield had heard about another tablet that contained triangles and rectangles, but despite his best efforts to track it down, speaking with many officials at Turkish government ministries and museums (the last known leads for the tablet), he couldn’t find it. However, one day in mid-2018, the mathematician received a photo of Si.427 in his inbox.

“I ran out of my office and found two colleagues in the middle of a meeting. I burst into their meeting and I rambled exciting things about “Pythagoras” and “Babylon”, and my colleagues were kind enough to smile while I got all my excitement out,” he recounted.

Together, Plimpton 322 and Si.427 paint a picture of how mathematics was used in ancient Babylon. Rather than using trigonometric concepts to study the night’s sky, as the ancient Greeks had in the second century BC, the alternative proto-trigonometry employed by Babylonians seems to mostly solve problems related to the ground.

“We knew that the Babylonians were mathematically advanced. They knew all about the geometry of right triangles, but we didn’t know why. What were they doing with right triangles? What were they using them for? This question of “why” motivated me to look at Babylonian artefacts from museums, libraries and private collections around the world. What I discovered is that the Babylonians were applying their understanding of right triangles to accurately measure and subdivide land,” Mansfield told ZME Science.

“The way we understand trigonometry harks back to ancient Greek astronomers. I like to think of the Babylonian understanding of right triangles as an unexpected prequel, which really is an independent story because the Babylonians weren’t using it to measure the stars, they were using it to measure the ground. Perhaps some aspects of this knowledge were transferred to other civilizations, but I’ve not seen any evidence of this,” he added.

Although the discovery of Plimpton 322 prompted some to speculate that its purpose was linked to the construction of palaces and temples, canals, and other practical works, it was only with the discovery of Si.427 that all the jigsaw pictures came together. During the period that these tablets were etched, Babylon was undergoing social change where much land moved became private. Designating proper boundaries without affecting neighborly relationships was essential, which is where the surveyors and their right triangles came in.

Next, Mansfield plans on studying what other applications besides surveying the Babylonians had for their proto-trigonometric tablets. He’s also interested in whether there are any real-world applications for these simple but fast techniques in our modern era. “For example, this approach might be of benefit in computer graphics or any application where speed is more important than precision,” he said.

And as a comical illustration of the essential role surveyors had in the Old Babylon period, here’s a hilarious poem in which an older student berates a younger one for his incompetence in surveying a field. “It’s essentially a 4000-year-old diss track,” Mansfield told me.

Go to divide a plot, and you are not able to divide the plot;
go to apportion a field, and you cannot even hold the tape and rod properly.
The field pegs you are unable to place; you cannot figure out its shape,
so that when wronged men have a quarrel you are not able to bring peace,
but you allow brother to attack brother.
Among the scribes, you (alone) are unfit for the clay.