New light on the roots of mathematics: Even the ancient Babylonians used geometric principles to survey the lands, says a mathematician. This stems from his search for a 3,700-year-old clay tablet asleep unnoticed in a museum in Istanbul. His results shed light on the importance of a previously popular clay tablet with mathematical tables.
How, where and when did people start doing advanced math? Analyzes of the clay tablets show that the ancient Babylonians performed astonishingly complex calculations. A 3,700-year-old cuneiform tablet called Plimpton 322, discovered around 1900 in southern Iraq, was already considered outstanding evidence of this. A few years ago, mathematician Daniel Mansfield of the University of New South Wales in Sydney devoted an analysis of this discovery. He supported the interpretation that the letters 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 it is clear that the Babylonians had already developed a special form of ‘primitive trigonometry,'” says the mathematician.
Why did the Babylonians care about trigonometry?
As early as 2017, he speculated about the possible purpose of the Plimpton 322 and hypothesized that it may have had a practical background: it may have been used for calculations related to building construction and canals or for surveying fields. The latter now confirms the current investigation of a clay tablet with the designation Si.427, which was discovered in 1894 in what is now Baghdad Governorate in Iraq. As Mansfield reported, he first had to track down this artifact in an almost detective fashion. He learned about this painting from a description in ancient excavation reports. According to the information, she arrived in Istanbul at that time. “So I went to research and talk to a lot of people in Turkish ministries and museums,” Mansfield says. This effort was ultimately crowned with success: the scientist was able to locate and then study the tablet in the Istanbul Archaeological Museum.
As mentioned, Si 427 comes from the Old Babylonian period from 1900 to 1600 BC. Hence from the same era as Plimpton 322. From his analysis of the signs, Si.427 appears to be a type of cadastral document – a plan that surveyors used to establish a land boundary. “It comes down to the legal and engineering details of a field that was split after the sale,” Mansfield said. The remarkable thing about this is that the surveyor used the system known today as the Pythagorean triple to accurately determine right angles. Mansfield asserts that this tablet was created more than a thousand years before the birth of the Greek scientist Pythagoras.
The Pythagorean triple consists of three natural numbers, which can occur as the side lengths of a right-angled triangle, for example through the triangle 3, 4, 5. Using the side lengths of such triangles, it is also possible to construct right angles. As Mansfield explains, the Babylonians had idiosyncrasies. Because they didn’t use a decimal system, but they used 60 based math which was similar to our time division. This in turn, he says, highlights the Plimpton 322: “It seems that the manufacturer went through the Pythagorean models to find the useful shapes,” says Mansfield. It is therefore suspected that land surveying challenges formed the basis for the tables at Plimpton 322.
Scan the ground at the right angles
“Si.427 demonstrates for the first time why the ancient Babylonians were interested in geometry—that is, defining precise land boundaries,” says Mansfield. “This painting comes from a time when land gradually became private property – people began to think of land in the sense of ‘my land and your land’ and they wanted to define precise boundaries to facilitate good neighborly relations. That is exactly what this tablet shows: the field is divided and new boundaries drawn,” says the scientist.
According to him, other tablets also give indications of these procedures in demarcating land holdings in ancient Babylonian culture: “For example, in the event of a dispute over valuable palm trees on the boundary between two properties. The local official agrees to send a surveyor to settle the dispute. Mansfield explains. The importance of accuracy in resolving conflicts between people who were powerful at the time.
Now he wants to continue researching Babylonian “primitive trigonometry”: he hopes to be able to reveal more cases of applied geometry in ancient Mesopotamia. According to him, there is still a mystery in Si.427 that he would like to solve: there are mysterious numbers on the back of the plate. “I can’t figure out what they mean yet. I am very interested in discussing with historians or mathematicians who might have an idea of what these numbers tell us!” Said Mansfield.
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