Mathematician reveals world’s oldest example of applied geometry

The significance of the tablet was unknown until Dr. Mansfield’s detective work was revealed. Credit: UNSW Sydney

A UNSW mathematician has revealed the origins of geometry applied to a 3,700-year-old clay tablet that has been hiding in plain sight in an Istanbul museum for more than a century.

The tablet, known as Si.427, was discovered at the end of the 19e century in what is now central Iraq, but its significance was unknown until the detective work of the UNSW scientist was revealed today.

More interestingly, Si. 427 is considered the oldest known example of applied geometry – and in the study published today in Foundations of science, research also reveals a fascinating human history of land surveying.

“Si.427 dates from the Old Babylonian (OB) period —1900 to 1600 BCE,” says senior researcher Dr. Daniel Mansfield of the UNSW School of Mathematics and Statistics.

“This is the only known example of a cadastral document from the OB period, which is a plan used by surveyors to define land boundaries. In this case, it gives us legal and geometric details on a field which is divided after the sale of a part. “

This is an important object because the surveyor uses what are now called “Pythagorean triples” to make precise right angles.

“The discovery and analysis of the tablet has important implications for the history of mathematics,” says Dr Mansfield. “For example, it is over a thousand years before the birth of Pythagoras.”

In the wake of another global discovery

In 2017, Dr Mansfield conjectured that another fascinating artifact from the same period, known as Plimpton 322, was a unique type of trigonometric table.

Credit: University of New South Wales

“It is generally believed that trigonometry – the branch of mathematics that concerns 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 the problems of measuring the ground, not the sky.”

The tablet revealed today is believed to have existed even before Plimpton 322 – in fact, surveying issues likely inspired Plimpton 322.

“There is a whole zoo of right triangles of 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 useful shapes,” says Dr Mansfield.

Tablet Purpose Revealed: Land Survey

In 2017, the team speculated on the purpose of the Plimpton 322, hypothesizing that it was likely to have a practical purpose, possibly used to build palaces and temples, build canals or fields for study.

“With this new tablet we can actually see for the first time Why they were interested in geometry: establishing precise boundaries of land, ”says Dr Mansfield.

“This is from a time when the earth is starting to become private – people started to think of the earth in terms of” my land and your land “, wanting to set an appropriate boundary to have positive neighborly relations. And that’s what this tablet immediately says. It’s an area being divided and new frontiers being created. “

There are even clues hidden on other tablets from this period about the stories behind these borders.

Australian mathematician reveals world's oldest example of applied geometry

Si. 427 is a hand-held tablet from 1900-1600 BC. AD, created by a Babylonian surveyor. It is made of clay and the surveyor wrote on it with a stylus. Credit: Must credit UNSW Sydney

“Another tablet refers to a dispute between Sin-bel-apli – a prominent individual mentioned on many tablets, including Si.427 – and a wealthy landowner,” says Dr Mansfield.

“The dispute is over valuable date palms on the border between their two properties. The local administrator agrees to send a surveyor to resolve the dispute. It is easy to see how important precision was in resolving disputes. between such powerful individuals. “

Dr Mansfield says the way these boundaries are made reveals real geometric understanding.

“No one expected the Babylonians to use Pythagorean triplets in this way,” says Dr. Mansfield. “It is more akin to pure mathematics, inspired by the practical problems of the time.”

Create right angles, easier said than done

A simple way to make a precise right angle is to make a rectangle with sides 3 and 4 and a diagonal 5. These special numbers form the 3-4-5 “Pythagorean triple” and a rectangle with these measures has right angles. mathematically perfect. This is important to ancient surveyors and still used today.

“The ancient surveyors who made the Si.427 did something even better: they used a variety of different Pythagorean triples, both as rectangles and right triangles, to construct precise right angles,” explains Dr Mansfield.

However, it is difficult to work with prime numbers greater than 5 in the Babylonian base 60 number system.

“This raises a very particular problem: their unique base 60 numbering system means that only certain Pythagorean forms can be used,” says Dr Mansfield.

Australian mathematician reveals world's oldest example of applied geometry

Si.427, pictured here held by Dr Daniel Mansfield at the Istanbul Archaeological Museum, is believed to be the earliest known example of applied geometry. Credit: UNSW Sydney

“It seems that the author of Plimpton 322 went through all of these Pythagorean forms to find the ones that are useful.

“This deep and highly numerical understanding of the practical use of rectangles deserves the name ‘proto-trigonometry’, but it is completely different from our modern trigonometry involving sin, cos, and tan.”

Hunt Si.427

Dr Mansfield first discovered Si.427 while reading excavation records. The tablet was unearthed during the Sippar expedition of 1894, in what is now Baghdad province in Iraq.

“It was a real challenge to find the tablet from these documents and find it physically. The report said the tablet had gone to the Imperial Museum in Constantinople, a place that clearly no longer exists.

“Using this information, I embarked on a quest to find him, speaking to many people in Turkish government ministries and museums, until one day in mid-2018 , a photo of Si.427 finally landed in my inbox.

“That’s when I learned that it was actually on display at the museum. Even after locating the object, it still took months to fully understand how important it is, and it So it’s really satisfying to finally be able to share this story. “

Next, Dr. Mansfield hopes to find out what other applications the Babylonians had for their proto-trigonometry.

The only mystery remains that Dr. Mansfield hasn’t unlocked: On the back of the tablet, at the very bottom, it lists the sexagesimal number “25:29” in large print — think 25 minutes and 29 seconds.

“I can’t figure out what these numbers mean – it’s an absolute conundrum. I look forward to discussing any leads with any historians or mathematicians who might have a hunch as to what these numbers are trying to tell us! ”

Mathematical mystery of ancient Babylonian clay tablet solved

More information:
Daniel F. Mansfield, Plimpton 322: A Study of Rectangles, Foundations of science (2021). DOI: 10.1007 / s10699-021-09806-0

Provided by the University of New South Wales

Quote: Mathematician Reveals World’s Oldest Example of Applied Geometry (2021, Aug 4) Retrieved Jan 8, 2022 from

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