# This 3,700-Year-Old Tablet is the Oldest Example of Applied Geometry

It uses "Pythagorean triples" 1,000 years before Pythagoras was even born.

Si.427 tablet from Ancient Babylonian era. UNSW Sydney

Many of the world's most important mathematical theories have been attributed to Pythagoras of Samos. The Pythagorean theoremthe Theory of Proportions, and the sphericity of the Earth are just a few things that came from the brilliant mind of Pythagora. Or at least, we thought they did. As our understanding of history improves, there are a few holes that are surfacing in these attributions.

The most recent one shows the use of "Pythagorean triples" a thousand years before Pythagoras was even born and in ways that are more akin to pure mathematics.

The Pythagorean theorem is neatly represented in the equation a2 + b2 = c2 . Here, a and b are twoparts of a right-angled triangle that are the same size and c is the hypotenuse, the longest side of the triangle (opposite the right-angled triangle). But a recently published study in the Foundations of Science reveals applied geometry records that date back to the Old Babylonian period (1900 - 1600 BCE).

Authored by Dr. Daniel Mansfield, a researcher at the School of Mathematics and Statistics at the University of New South Wales in Australia, the study speaks about two archaeological discoveries, Plimpton 322 and Si.427. These are contemporary tablets from about 3,700 years back that contain inscriptions that are currently the oldest records of applied geometry that we currently have.

The Plimpton 322 was found earlier, and in 2017, Mansfield and his team had hypothesized that this "zoo of right-angle triangles with different shapes" was a unique kind of trigonometric table that had some practical purpose, such as constructing palaces and temples, building canals, or surveying fields. While it is believed that Greeks used trigonometry to study the sky, their predecessors, the Babylonians were using it to solve matters on the ground.

Mansfield had read about another similar tablet but much older than the Plimpton 322, Si.427, in excavation records. He spent many years looking for the Si.427 but only managed to track it down in 2018 in a museum in Turkey. The inscription on the Si.427 is a survey of the land showing its extent and ownership, which is the only known record of that time.

But together with the Plimpton 322, it clears the air on why trigonometry tables were needed. "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.”

Since the Babylonians used a base 60 number system, only a few numbers could be used in the calculations. The table in the Plimpton 322 artifact goes through all the numbers that could be used.

"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," Mansfield further clarified. The discovery and analysis of the tablet have important implications for the history of mathematics.

Mansfield still has a mystery to solve with the Si.427. Behind the inscription, the tablet reveals two numbers in a big font, written as 25:29. He has stated that he is eager to talk to anyone who can help with this.