An Australian mathematician has revealed the origins of applied geometry on a 3700-year-old clay tablet that has been hiding in plain sight in a museum in Istanbul for over a century.
The tablet – known as Si.427 – was discovered in the late 19th century in what is now central Iraq, but its significance was unknown until a scientist’s detective work was revealed today.
Most excitingly, Si.427 is thought to be the oldest known example of applied geometry – and in the new study released in Foundations of Science, the research also reveals a compelling human story of land surveying.
“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BCE,” says lead researcher Dr Daniel Mansfield, from the University of New South Wales. This is a significant object because the surveyor uses what are now known as “Pythagorean triples” to make accurate right angles.
“The discovery and analysis of the tablet have important implications for the history of mathematics,” Dr Mansfield says. “For instance, this is over a thousand years before Pythagoras was born.”
In 2017, Dr Mansfield conjectured that another fascinating artifact from the same period, known as Plimpton 322, was a unique kind of trigonometric table.
“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 BCE,” says Dr Mansfield.
“But the Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky.”
The tablet revealed today is thought to have existed even before Plimpton 322 – in fact, surveying problems likely inspired Plimpton 322.
“There is a whole zoo of right triangles with different shapes. But only a very small handful can be used by Babylonian surveyors. Plimpton 322 is a systematic study of this zoo to discover the useful shapes,” says Dr Mansfield.
Back in 2017, the team speculated about the purpose of the Plimpton 322, hypothesizing that it was likely to have had some practical purpose, possibly used to construct palaces and temples, build canals or survey fields.
“With this new tablet, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries,” Dr Mansfield says.
“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. And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.”
There are even clues hidden on other tablets from that time period about the stories behind these boundaries.
There’s just one mystery left that Dr Mansfield hasn’t unlocked: on the back of the tablet, at the very bottom, it lists the sexagesimal number ‘25:29’ in big font – think of it as 25 minutes and 29 seconds.
“I can’t figure out what these numbers mean – it’s an absolute enigma. 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!”
AR #63
Opening the Door to Sacred Geometry
by Len Kasten
