18/08/2021

In Folha de S.Paulo, Marcelo Viana talks about Plimpton 322.

A tabuleta Plimpton 322/ Wikipedia

Reproduction of Marcelo Viana's column in Folha de S.Paulo

I once saw a carpenter mark two perpendicular cuts in wood: he took a piece of string 12 dm long, formed a triangle with sides of lengths 3, 4, and 5 dm, and used the fact that the angle between the shorter sides is a right angle. I don't know if he knew why it works, but I bet he was unaware that the technique was already being used 4,000 years ago.

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

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

A reader brought to my attention another recently identified Babylonian mathematical document. The piece, a circular clay tablet called Si.427, dates from 1900 to 1600 BC. It was excavated in Baghdad in 1894 but was thought to be lost until Australian researcher Daniel Mansfield located it in the Istanbul Archaeological Museum.

Si.427 contains one of the earliest examples of the application of trigonometry to one of the problems that most motivated the advancement of mathematics in Egypt and Mesopotamia: the redistribution of land. It is a kind of property record, containing legal and geometric information about a plot of land that was divided so that half of it could be sold.

To make this division accurately, it's important to know how to draw perpendiculars to a given line. That's where the two documents connect: a practical method for obtaining perpendicularity is by constructing triangles whose sides have lengths given by some Pythagorean triple, just like my carpenter did.

To read the full text, visit the newspaper's website.