# Si. 427: A 3,700-Year-Old Tablet Depicts Earliest Example of Applied Geometry

Si. 427 – a 3700-Year-Old tablet that belongs to the Babylonian period consists of the earliest example of applied geometry that a mathematician could discover. The ancient clay tablet belongs to an era that is millennium prior to the birth of Pythagoras.

The tablet is known as Si.427 possesses features that could alter the history of mankind and are currently positioned in a museum in Istanbul.

“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BCE,”

Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Australia.

“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.”

Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Australia

That plan uses sets of numbers known as Pythagorean triples to derive accurate right angles, or sets of numbers that fit trigonometric models for calculating the sides of a right-angled triangle. This makes the timing of the artifact particularly interesting, with important implications for the history of mathematics.

Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Australia

The recent discovery has been published in various newspapers describing the tablet. The tablet has been analyzed and findings have been developed concerning the Si.427, known as Plimpton 322. Back in 2017, it has been revealed by Mansfield and colleagues that Plimpton 322 was an early trigonometric table, showing a whole list of Pythagorean triples.

The researchers could not understand the purpose of the list back then. However, now they have proposed a theory that it might date to slightly later than Si.427, and contain only Pythagorean triples that would be relevant for making rectangular measurements of the ground. If we put it into simpler words, it is believed by researchers that it’s a planning manual.

This theory is in contrast with the trigonometry laid out by Pythagoras. It was invented by looking at the stars back in the 2nd Century BCE. The number of Pythagorean triples that can be utilized for making land measurements by Babylonian surveyors is quite a few.

It is known to all that a Pythagorean triple fits the equation a2 + b2 = c2, where the sides defining a triangle that is adjacent to the right angle are a and b, and the hypotenuse (the longest side) is c. The easiest example to understand the concept is 32 + 42 = 52.

Using this theorem, it is easy to determine that these sets of numbers can be used to draw triangles and rectangles with perfect right angles. But the sexagesimal, or base 60, Babylonian number system made it difficult to work with prime numbers larger than 5.

“This raises a very particular issue – their unique base 60 number system means that only some Pythagorean shapes can be used.”

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

Now, with Si.427, we finally know what they wanted to use these Pythagorean triples for – laying down land boundaries.

“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 several other tablets from the same period that implies the relevancy of the same. One of the discovered tablets regards a dispute over date palms on the border between two properties, in which the local administrator had agreed to dispatch a surveyor to settle the matter. It is quite simple to understand that why the ability to measure land and plots accurately was of utmost importance to the ancient civilization.

The tablet depicts a reliable understanding of geometry. The depictions are not as advanced as modern mathematics or the mathematics discovered by ancient Greeks, but it indicates that our understanding of mathematics may have been more incremental than current historical knowledge tells us.

“Nobody expected that the Babylonians were using Pythagorean triples in this way. “It is more akin to pure mathematics, inspired by the practical problems of the time.”

Daniel Mansfield

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