ISTANBUL – An ancient Babylonian tablet with the uninspiring name “Si.427” may be re-writing mathematical history. The 3,700-year-old hand-sized clay tablet is inscribed with cuneiform markings that demonstrate the Babylonians were applying the Pythagorean theorem 1,000 years before the Greek philosopher Pythagoras, who has been credited with the discovery that bears his name. The research was published this month in Foundations of Science, in a follow-up to research published last year in the Journal of Cuneiform Studies.
The clay tablet has been in an Istanbul museum since it was discovered in 1894 during French archeological excavations of the ancient Babylonian city of Sippar. The remains of the ancient city are in modern-day Iraq at the divergence of the Euphrates river. Researchers have only recently begun to uncover the secrets of the tablet.
The tablet appears to have been used by surveyors to draw accurate boundaries, and is now the earliest known example of applied geometry. Those boundaries required the use of the Pythagorean Theorem.
A separate table from a clay tablet of the same period, called Plimpton 322, contains what appears to be a series of solutions related to the Pythagorean Theorem. Dr. Daniel Mansfield, a mathematician at the University of New South Wales in Australia and the discoverer of the tablet’s meaning, concludes that the table is “a mathematical study of the individual sides of Pythagorean triples.”
Pythagorean triples are three positive integers that meet the Pythagorean theorem condition of a2 + b2 = c2. The set of numbers (3, 4, 5) is an example of such a triple, as 32 + 42 = 52 or 9+16 = 25.
As for the Pythagorean Theorem itself, it is a fundamental consequence of Euclidean geometry. It states that the area of the square whose side is the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the other two sides of the triangle. There are diverse proofs of it using both geometric and algebraic methods. The theorem has evoked mathematical, scientific, and even mystical discourse, like sacred geometry, over the centuries.
The table conveyed on Plimpton 322 appears to show solutions from a right angle – solutions that would be later unified in the field of trigonometry.
Plimpton 322 [Courtesy of the Rare Books and Manuscripts Library, Columbia University; photograph by Andrew Kelly]
Suggesting that trigonometry was used, however, is misleading and anachronistic.
“It is generally accepted that trigonometry — the branch of maths that is concerned with the study of triangles — was developed by the ancient Greeks studying the night sky” in the second century B.C., Mansfield said in a statement about the research. “But the Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky.”
Previous research has shown that “Mesopotamian mathematics is fundamentally about lists.” Writing this month in Foundations of Science, Mansfield notes that the ancient scribes used a base 60, or sexagesimal, number system, and built the tablet possibly to assist in surveying. The tablet appears to contain the mathematical solutions to right triangles only, possibly a limitation of their base 60 system.
“Scribes measured and understood just one angle: the right angle,” writes Mansfield. “The Akkadian word mutarrittum, meaning ‘direction of the plumb line’ was used as a metaphor for the perpendicular side of a shape.” Mansfield adds that researchers have identified that “an interest in perpendicularity is apparent from early sketches in education and surveying.” The triples were used by surveyors to construct perpendicular sides.
The resulting solutions would provide for accuracy in land measurements. But measuring and establishing perpendicular lines would help also with other tasks and the Babylonian surveyors and scribes were identifying a means of establishing those lines. Plimpton 322 could have been used to build canals, temples, or simply designate land boundaries because this period of time also sees privatization of land spaces.
“This is from a period where land is starting to become private,” says Mansfield. “People started thinking about land in terms of ‘my land and your land’, wanting to establish a proper boundary to have positive neighbourly relationships. And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.”
Si.427 is a hand tablet from 1900-1600 BC, created by an Old Babylonian surveyor. It’s made out of clay and the surveyor wrote on it with a stylus [University of New South Wales – Sydney]
“Plimpton 322 is an investigation into rectangles with regular sides,” Mansfield concludes. “It could have been motivated by a particular practical need, or by a purely theoretical interest in geometry. Although it is more likely that the answer lies somewhere between these two extremes.”
Si.427 appears to predate Plimpton 322 while also going further. “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 says.
The tablet mentions a prominent individual name, Sin-bel-apli, and a wealthy female landowner. “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,” Mansfield says.
The resolution of accurate boundaries for the resolution of the dispute required the geometric understanding of Pythagorean triples. Regardless, historians and scientists are slowly converging on the reality that Pythagoras did not discover the theorem and that Babylonians and likely Indians were aware of the theorem and its solutions centuries earlier.
Si.427 still has one more mystery. It lists the sexagesimal number ‘25:29’ in a large script on the bottom of the back of the tablet. “I can’t figure out what these numbers mean – it’s an absolute enigma,” says Mansfield. “I’m keen to discuss any leads with historians or mathematicians who might have a hunch as to what these numbers trying to tell us!”
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