I once saw a carpenter make two perpendicular cuts in the wood: he took a string of 12 dm length, formed a triangle with sides of lengths 3, 4 and 5 dm, and used the fact that the angle between the short sides is straight. I don’t know if you knew why it works, but I bet you didn’t know that the technique was already used 4,000 years ago.
Plimpton 322, a clay tablet found in excavations in Mesopotamia and dating back 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 Babylonian sexagesimal notation) that form Pythagorean triples, that is, integer triples a, b and c (for example, a=3, b=4 and c=5) such that a2+b2=c2.
Most experts believe this is a list of examples for use in the classroom. But the inscription also points to a method of calculating the triples –more than a thousand years before Pythagoras!– which shows a knowledge of geometry that was thought to have only been achieved in Greece.
A reader drew my attention to another recently identified Babylonian mathematical document. The piece, a circular clay plate called Si.427, dates from 1900 to 1600 BC. It was excavated in Baghdad in 1894, but was reported lost until Australian researcher Daniel Mansfield located 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 advance of mathematics in Egypt and Mesopotamia: land redistribution. It is a kind of property registry, containing legal and geometric information about a piece of land that was divided up so that half of it could be sold.
To make this division precisely, it is important to know how to draw perpendiculars to a given line. This is where the two documents connect: a practical method of obtaining perpendicularity is to construct triangles whose sides have lengths given by some Pythagorean triple, just as my carpenter did.
Thus, the practical problems posed by Si.427 can be solved using the “theory” contained in Plimpton 322. Mansfield suggests that, instead of a list of didactic material, the latter would in fact be a kind of ‘glue’ for a real estate agent.
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