# World’s oldest example of applied geometry revealed by Australian mathematician.

Pythagoras of Samos is credited with several of the world’s most important mathematical theories. The Pythagorean theorem, the Theory of Proportions, and the sphericity of the Earth are only a few examples of Pythagora’s bright mind. At least, that’s what we believed. As our grasp of history grows, a few gaps in these attributions become apparent.

The most recent one reveals the employment of “Pythagorean triples” a thousand years before Pythagoras was ever born, and in more pure mathematics-like ways.

An Australian mathematician has discovered the origins of applied geometry on a 3700-year-old clay tablet that has been concealed in plain sight in an Istanbul museum for over a century. The tablet, Si.427, was discovered in late 1800s in what is now central Iraq. According to the UNSW scientist’s study, its significance was unknown until now.

“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BCE,” says lead researcher Dr. Daniel Mansfield from UNSW Science’s School of Mathematics and Statistics. “It’s 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 tells us legal and geometric details about a field that’s split after some of it was sold off.”

Si.427 is a hand tablet produced between 1900 and 1600 BC by an Old Babylonian surveyor. It’s made of clay, and the surveyor wrote on it with a stylus. On the front is a diagram of a field. The field is being subdivided, and some of it is being sold. The lines serve to demarcate the individual fields’ borders. The borders are quite accurate, maybe more so than you’d anticipate at the moment.

By using Pythagorean triples, the surveyor was able to reach such accuracy. It enabled him to draw perfectly perpendicular border lines. In its most basic form, a Pythagorean triple has sides of 3, 4, and 5 – creating a perfect right angle.

The text on the tablet’s back is written in cuneiform, one of the earliest writing systems. The only thing we haven’t figured out is why there are so many numbers at the bottom of the tablet’s back.

In 2017, Dr. Mansfield proposed that Plimpton 322, a fascinating artefact from the same time period, was a new sort 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.”

Mansfield had heard about another comparable tablet in excavation records, but it was considerably older than the Plimpton 322, Si.427. He searched for the Si.427 for many years before finding it at a museum in Turkey in 2018. The inscription on the Si.427 is a survey of the land, indicating its area and ownership, and is the sole known document from that historical period.

However, when combined with the Plimpton 322, it clarifies why trigonometry tables were required. “This is from a period where land is starting to become private – people started thinking about the land in terms of ‘my land and your land’, wanting to establish a proper boundary to have positive neighborly relationships,” Mansfield said in a press release. “And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.”

Because the Babylonians employed a base 60 number system, only a few numbers could be utilized in the computations; the table in the Plimpton 322 artifact runs through all the possible numbers.

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

The tablet’s discovery and study have far-reaching consequences for the history of mathematics.

With the Si.427, Mansfield still has a riddle to unravel. The tablet shows two numerals in large type, inscribed as 25:29, underneath the inscription. He has indicated that he is happy to speak with anyone who can assist him with this.

Plimpton 322: A Study of Rectangles, Daniel F. Mansfield

Published: August 2021
DOI: https://doi.org/10.1007/s10699-021-09806-0