New light on the roots of mathematics: Even the ancient Babylonians used geometric principles for land surveying, reports a mathematician. This emerges from his research on a 3700-year-old clay tablet that has slumbered unnoticed in a museum in Istanbul. His results shed light on the importance of a previously famous clay tablet with mathematical tables.
How, where, and when did people start doing advanced math? Analyzes of clay tablets show that the ancient Babylonians made amazingly sophisticated calculations. The approximately 3700 year old cuneiform tablet with the designation “Plimpton 322”, which was discovered around 1900 in southern Iraq, was already regarded as outstanding evidence of this. A few years ago, the mathematician Daniel Mansfield from the University of New South Wales in Sydney devoted an analysis to this find. He underpinned the explanation that the characters represent a trigonometric table. “It is widely believed that trigonometry – the branch of mathematics that studies triangles – was developed by the ancient Greeks,” says Mansfield. “But apparently the Babylonians had already developed a special form of ‘proto-trigonometry'”, says the mathematician.
Why did the Babylonians study trigonometry?
As early as 2017, he speculated about the possible purpose of the Plimpton 322 board and hypothesized that it probably had a practical background: perhaps it was used for calculations in connection with the construction of buildings and canals or for surveying fields. The latter now confirms the current investigation of the clay tablet with the designation Si.427, which was discovered in 1894 in what is now the province of Baghdad in Iraq. As Mansfield reports, he first had to track down this artifact in an almost detective manner. He became aware of this plaque through a description in old excavation reports. According to the information, she got to Istanbul at that time. “So I went looking and spoke to a lot of people in Turkish ministries and museums,” reports Mansfield. The effort was ultimately crowned with success: the scientist was able to locate the tablet in the Istanbul Archaeological Museum and then study it.
As he reports, Si.427 comes from the Old Babylonian period from 1900 to 1600 BC. And thus from the same era as Plimpton 322. From his analysis of the signs it appears that Si.427 is a kind of cadastral document – a plan that was used by surveyors to establish land borders. “It’s about legal and geometric details related to a field that was split after a sale,” said Mansfield. The special thing about it is that the surveyor used the system known today as the Pythagorean triplet to determine exact right angles. Mansfield emphasizes that this tablet was created over a thousand years before the Greek scholar Pythagoras was born.
Pythagorean triples are made up of three natural numbers, which can occur as lengths of the sides of a right triangle, for example by the triplet 3, 4, 5. With the side lengths of such triangles it is also possible to construct right angles. As Mansfield explains, however, the Babylonians had peculiarities. Because they did not use a decimal system, but a 60-based mathematics that was similar to our time division. This, in turn, according to him, sheds light on the Plimpton 322 board: “It seems like the maker went through Pythagorean forms to find the useful ones,” says Mansfield. He therefore suspects that land surveying challenges formed the basis for the tables on Plimpton 322.
Land surveying right angles
“Si.427 makes it clear for the first time why the ancient Babylonians were interested in geometry – namely to define precise land borders,” says Mansfield. “This plaque comes from a time when land was gradually becoming private property – people began to think about land in the sense of ‘my land and your land’ and wanted to define a precise boundary to facilitate good neighborly relations. This is exactly what this board shows: A field is divided and new boundaries are drawn, ”says the scientist.
According to him, other tablets also provide indications of these procedures in the delimitation of land holdings in the ancient Babylonian culture: “For example, in a case of a dispute over valuable date palms on the border between two properties. The local administrator agrees to send a surveyor to settle the conflict. It shows the importance of accuracy in resolving disputes between people who were powerful at the time, ”says Mansfield.
He now wants to continue researching the Babylonian “proto-trigonometry”: He hopes to be able to uncover further cases of applied geometry in ancient Mesopotamia. According to him, there is also a riddle at Si.427 that he would like to solve: There are mysterious numbers on the back of the board. “I can’t figure out what they mean yet. I’m very interested in discussing with historians or mathematicians who might have an idea what these numbers are telling us! ”Said Mansfield.
Source: University of New South Wales