Ancient tablet in Istanbul predates Pythagorean geometry by 1000 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 Pythagorean triples were practised 1,000 years before its known inventor, Pythagoras.

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 artefact was used to measure land boundaries. The tablet details legal and geometric details about a field that has been split after some of it were 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, known as Si.427, is also now the only known example of a cadastral document from the OB period, 1900 to 1600 BCE.

"Books on the history of mathematics usually start with ancient Greece. Si.427 is a culturally significant object because it shows there was another chapter before this," Mansfield told TRT World via email. 

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

"We have known for a long time that the Babylonians were mathematically advanced, particularly with regards to right triangles. But until now we didn't know why," Mansfield said. 

"What were they doing with this highly advanced understanding of mathematics? The tablet is important because it tells us why: they used these shapes to make accurate land measurements." 

Three Pythagorean triples can be seen on Si.427: 3, 4, 5; 8, 15, 17; and 5, 12, 13 on the tablet.

What led Mansfield to the finding? He went around the world looking for examples of how the Babylonians might have used what the mathematician calls 'proto-trigonometry' in surveying.  While Babylonians were interested in measuring the ground, Greeks studied the night sky. 

"I'd read about Si.427 from the original 1894 archaeological dig records in Sippar (near modern-day Baghdad), and its accuracy immediately struck me as special," Mansfield said.

The expedition records showed that the tablet first went to the Imperial Museum of Constantinople, which doesn’t exist now.

"This eventually led me to the Istanbul Archaeological Museums where the object has been kept safe all these years," he said.

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.

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.”