Plimpton 322

Plimpton 322 is a Babylonian clay tablet, believed to have been written around 1800 BC, that contains a mathematical table written in cuneiform script. Each row of the table relates to a Pythagorean triple, that is, a triple of integers ( s , l , d ) {\displaystyle (s,l,d)} that satisfies the Pythagorean theorem, s 2 + l 2 = d 2 {\displaystyle s^{2}+l^{2}=d^{2}} , the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse. The era in which Plimpton 322 was written was roughly 13 to 15 centuries prior to the era in which the major Greek discoveries in geometry were made. At the time that Otto Neugebauer and Abraham Sachs first realized the mathematical significance of the tablet in the 1940s, a few Old Babylonian tablets making use of the Pythagorean rule were already known. In addition to providing further evidence that Mesopotamian scribes knew and used the rule, Plimpton 322 strongly suggested that they had a systematic method for generating Pythagorean triples as some of the triples are very large and unlikely to have been discovered by ad hoc methods. Row 4 of the table, for example, relates to the triple (12709,13500,18541). The table exclusively lists triples ( s , l , d ) {\displaystyle (s,l,d)} in which the longer leg, l {\displaystyle l} , (which is not given on the tablet) is a regular number, that is a number whose prime factors are 2, 3, or 5. As a consequence, the ratios s / l {\displaystyle s/l} and d / l {\displaystyle d/l} of the other two sides to the long leg have exact, terminating representations in the Mesopotamians' sexagesimal (base-60) number system. The first column most likely contains the square of the latter...