Pythagorean Theorem Found on Ancient Babylonian Clay Tablets Predating Pythagoras by 1,000 Years

Researchers have identified the Pythagorean Theorem on ancient Babylonian clay tablets, rewriting science history.

The Pythagorean theorem is one of the most famous and fundamental mathematical formulas that every student learns in school. It states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side of the triangle) is equal to the sum of the squares of the other two sides. The formula is usually written as a² + b² = c², where a, b and c are the lengths of the sides.

The theorem is named after Pythagoras, a Greek philosopher and mathematician who lived in the 6th century BC. However, recent discoveries have shown that Pythagoras was not the first to know about this relationship between the sides of a right-angled triangle. In fact, ancient Babylonians had already figured out the Pythagorean theorem more than 1,000 years before Pythagoras was born!

How do we know this? The evidence comes from clay tablets that were found in what is now Iraq, dating back to the Old Babylonian period (1900-1600 BC). These tablets contain mathematical calculations and diagrams that use the principles of the Pythagorean theorem to solve problems related to land surveying, construction and astronomy.

One of these tablets, known as Si.427, is considered to be the oldest known example of applied geometry. Unearthed in 1894 in present-day Iraq, it shows a diagram of four adjacent right-angled triangles inside a square, along with a description of the land boundaries and features in cuneiform script. The tablet uses number sets called Pythagorean triples, which are sets of three positive integers that satisfy the Pythagorean equation. For example, 3, 4 and 5 are a Pythagorean triple, because 3² + 4² = 5².

Daniel Mansfield, a mathematician at the University of New South Wales in Australia, who analyzed Si.427, said in a statement: “With this new tablet, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries.” He published his findings in the journal Foundations of Science in 2021.

Another tablet, known as Plimpton 322, contains a table of 15 rows and four columns of numbers that represent Pythagorean triples. The tablet was discovered by an American archaeologist named Edgar Banks in 1922 and is now kept at Columbia University in New York. Mansfield and his colleague Norman Wildberger published a paper in 2017 arguing that Plimpton might have been used in construction of canals, palaces and temples, or perhaps in land surveying.

The solution to the puzzle was revealed through Si.427. Mansfield came across the clay artifact at the Istanbul Archaeology Museums, where it had been quietly stored for many years, largely unnoticed.

“With this new tablet, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries,” says Mansfield in the statement. “This is from a period where land is starting to become private—people started thinking about land in terms of ‘my land and your land,’ wanting to establish a proper boundary to have positive neighborly relationships.”

Additional supporting evidence comes in the form of tablet IM67118, dating back to 1770 BC. This discovery comprises textual content, a diagram, and tools for calculating the area of a rectangle as well as the length of its diagonal. The accompanying text outlines a solution employing Pythagoras’ theorem.

It’s noteworthy that the Pythagorean theorem had early recognition in ancient India, emerging in the eighth century BC in the Vedic Sanskrit text Sulbasutra, authored by Baudhāyana. The theorem, also identified in China as GouGu, may have roots predating common understanding, potentially transmitted through oral tradition as far back as 2000 BC, according to some scholars.

But then, why is Pythagoras credited with the Pythagorean theorem? Pythagoras, born around 570 BC, lived over 1000 years after the creation of the tablets described above. His teachings are derived from fragments of works attributed to Eudemus of Rhodes, Philolaus of Croton, and archivists from Tarentum in Italy.

Since no original writings of Pythagoras have survived, information was orally transmitted within the Pythagorean school, primarily by its members in the southern territories of present-day Italy. Many discoveries made by the Pythagoreans were attributed to Pythagoras out of reverence for their teacher.

The Greeks are usually credited with inventing trigonometry (the branch of mathematics that deals with angles and distances in triangles), but Mansfield and Wildberger claim that the Babylonians had their own version of trigonometry that was based on ratios rather than angles. They used a base-60 number system (similar to how we use minutes and seconds to measure time) to express these ratios with great accuracy.

Mansfield said: “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.”

These findings reveal that the ancient Babylonians were not only skilled mathematicians, but also practical problem-solvers who applied their knowledge to real-world situations. They also challenge the common assumption that Pythagoras was the first to discover the Pythagorean theorem, and show that mathematical ideas can emerge independently in different cultures and times.

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