# In Babylon, a thousand years before Pythagoras – 08/17/2021 – Marcelo Viana

I once saw a carpenter make two vertical cuts in the wood: he took a string of 12 dm in length, formed a triangle with sides of 3, 4 and 5 dm, and took advantage of the fact that the angle between the short sides was straight is. I don’t know if you knew why it worked, but I’ll bet you didn’t know the technique was used 4,000 years ago.

Plimpton 322, a clay tablet found during excavations in Mesopotamia and dating from 1800 BC. BC is one of the most famous ancient mathematical documents. It has a table with 15 rows and 4 columns inscribed with numbers (in Babylonian sexagesimal notation) that form Pythagorean triples, i.e. integer triples a, b and c (for example a = 3, b = 4 and c = 5) with a2 + b2 = c2.

Most experts believe this is a list of examples to use in the classroom. But the inscription also points to a method for calculating the triples – more than a thousand years before Pythagoras! – showing a knowledge of geometry that was believed to have only been achieved in Greece.

A reader drew my attention to another recently identified Babylonian mathematical document. The piece, a round clay plate called Si.427, dates from 1900 to 1600 BC. It was excavated in Baghdad in 1894 but reported missing until Australian explorer Daniel Mansfield found it in the Istanbul Archaeological Museum.

Si.427 contains one of the oldest examples of the application of trigonometry to one of the problems that most motivated the advancement of mathematics in Egypt and Mesopotamia: land redistribution. It is a type of land register that contains legal and geometrical information about a property that has been divided so that half of it could be sold.

To make this division precisely, it is important to know how to draw perpendiculars to a particular line. This is where the two documents join: A practical way to get squareness is to construct triangles with sides given by a Pythagorean triple, just like my carpenter did.

Thus, the practical problems raised by Si.427 can be solved with the “theory” contained in Plimpton 322. Mansfield suggests that these, instead of a list of didactic materials, actually act as some sort of “glue” for a real estate agent.

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