A clay tablet that dates back 3,700 years to ancient Babylon is engraved with a “remarkably accurate field plan” that is the oldest known example of applied geometry in the world, reports a new study.
The tablet, called Si.427, is the only known example of a cadastral document, which is a land survey map used to define property boundaries, from the Old Babylonian period. The artifact displays insights about right angles and triangles that are normally associated with Pythagoras (of Pythagorean theorem fame, if you remember high school math), a Greek philosopher who lived more than a millennium after the tablet was created.
In this way, Si.427 reveals an early form of proto-trigonometry, which is like a “prequel” of geometric techniques that emerged later, according to Daniel Mansfield, a mathematics researcher at the University of New South Wales and the author of a study about the tablet published on Wednesday in Foundations of Science.
“The Greeks invented their trigonometry because they were studying astronomy, but the Babylonians had their own separate variant of trigonometry which they developed to solve problems about land and boundaries,” Mansfield said in an email.
Si.427 first surfaced in modern archaeological records during the Sippar expedition, a French excavation that took place near Baghdad, Iraq, in 1894. The artifact became part of the collection at the Imperial Museum of Constantinople, an institution that no longer exists, leading Mansfield on a wild hunt for the intriguing tablet. With the help of experts in Turkey, Mansfield was able to finally track down the tablet to the Istanbul Archaeological Museum, which he visited during a sabbatical in 2019.
After so much time searching for the tablet, Mansfield said it was “terrifying” to ultimately hold in his hands. “It's such a culturally significant object and I was afraid that something terrible might happen while I was holding it,” he recalled. “I'm happy this is safely back in the museum where it belongs.”
Mansfield was initially inspired to search for the tablet based on the revelations of another artifact that he had previously studied, known as Plimpton 322, which is about the same age as Si.427 and also displays evidence of sophisticated Babylonian geometry. Plimpton 322 is engraved with several rows of Pythagorean triples, which are sets of integers that are used to draw right angles, according to Mansfield’s 2017 study in Historia Mathematica.
As the oldest known trigonometric table, Plimpton 322 illustrates the mathematical sophistication of Babylonian civilization. However, Si.427 now adds a crucial piece of this puzzle by demonstrating that these calculations were likely used for land surveying and property disputes at this time. The engraving on the tablet visualizes a plot of land that was split into parcels that were separately sold off.
“Like earlier field plans, Si.427 retains the subdivision into rectangles, right trapezoids, and right triangles,” Mansfield wrote in the study. “But unlike earlier field plans, it concerns the sale of private land and the measurements have been made with unusually high precision.”
“The rectangles themselves are most remarkable because they actually have opposite sides of equal length, which is unique and suggests that [Old Babylon] surveyors had devised a way to create perpendicular lines more accurately than before.”
To that point, cadastral maps such as Si.427 may have predated and inspired the creation of trigonometric tables like Plimpton 322, which are written in the base 60 numerical system used by ancient Babylonians. These early documents provided a quick and handy way to solve complex problems, which is a basic strategy that persists in many modern contexts.
“This is a simple, alternative approach to right triangles and rectangles,” Mansfield said. “It's certainly not as sophisticated as modern trigonometry, but sometimes you want a simple answer instead of a sophisticated one. And I'm not just talking about how students want their mathematics exams to be.”
“Simple answers are great because they are fast,” he added. “There are many practical applications, such as computer graphics, where speed is more important than accuracy and I'd like to investigate how this simple approach might be advantageous.”
Mansfield is optimistic that other tablets from this period may contain similar clues about the mathematical techniques used during this early era, which will shed light on the origins of geometry. He is also hoping to unravel some of the remaining mysteries on Si.427, such as an enlarged and currently unexplained inscription of the sexagesimal number ‘25:29’.
“I've spent many evenings trying to figure out what those numbers mean,” Mansfield concluded, “and I still have no idea.”