Ancient tablet in Istanbul predates Pythagorean geometry by 1000 years

The 3,700-year-old tablet sitting in an Istanbul museum revealed that Babylonians predated Pythagoras by over 1,000 years.

For more than a century, a Babylonian clay tablet has been sitting in the Istanbul Archaeological Museum in Turkey. It is now identified as the earliest known example of applied geometry, which means the mathematical branch of trigonometry was known 100 before Pythagoras claimed to invent it.

A Sydney mathematician, Dr Daniel Mansfield at the University of South Wales was behind the discovery. In a research published in Foundations of Science on Wednesday, the mathematician said the 3700-year-old artifact was used to measure land boundaries. The tablet details legal and geometric details about a field that has been split after some of it was sold off.

“Sometimes the most amazing discoveries are hiding in plain sight. It’s a discovery that will completely change the way we view the history of mathematics,” the researcher said.

The tablet is also now the only known example of a cadastral document from the OB period, 1900 to 1600 BCE.

The tablet, known as Si.427 was kept in Istanbul since it was found in 1894 during an expedition in Sippar, an ancient Babylonian city that was located in today’s Baghdad province in Iraq. The records showed that it then went to the Imperial Museum of Constantinople, which doesn’t exist now.

After months spent to get a copy of the tablet, Mansfield then finally understood the significance of the artefact as it revealed how they calculated the land: by using what is called a Pythagorean triple.

Pythagoras, a Greek philosopher was thought to be the founder of the theorem that for a right angle triangle (with one of the angles being 90o), the square of the hypotenuse is equal to the sum of the squares of the other two sides.

“This proves that Babylonian surveyors had a rock-solid theoretical understanding of the geometry, rectangles and right triangles and used it to solve practical problems,” Mansfield said.

“This is over a thousand years before Pythagoras was born,” he said. Three Pythagorean triples can be seen on Si.427: 3, 4, 5; 8, 15, 17; and 5, 12, 13.

Discovery leading to another discovery

In 2017, Mansfield and his colleague from UNSW, Norman Wildberg identified another Babylonian tablet that contained the world’s oldest and most accurate trigonometric table. The researchers thought the tablet, known as Plimpton 322, had a practical purpose but they didn’t know how it was used.

Mansfield’s recent discovery also helped to understand that Si.427, which is thought to have existed before, had actually inspired the Plimpton 322. It is now understood that Plimpton, too, was used to calculate precise land boundaries.

The Babylonian mathematical approach was what Mansfield calls an “alternative ‘proto-trigonometry’.” They were interested in measuring the ground, unlike Greeks who studied the night sky.

It means, Babylonians had or started to develop the understanding of private property and boundaries at the time.

“Now that we know what problem the Babylonians were solving, that recolours all the mathematical tablets from this period,” Mansfield said. “You see mathematics being developed to address the needs of the time.”

But there is still something that remains secret to the researcher: The sexagesimal number 25:29 that was engraved on the back of the tablet.

“Is it part of a calculation that they performed? Is it an area that I haven’t come across yet? Is it a measurement of something?” he told the Guardian. “It’s really annoying to me because there’s so much about the tablet that I understand. I’ve given up trying to figure out what that one is.”

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