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 ${\displaystyle (s,\ell ,d)}$ that satisfies the Pythagorean theorem, ${\displaystyle s^{2}+\ell ^{2}=d^{2}}$, 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 ${\displaystyle (s,\ell ,d)}$ in which the longer leg, ${\displaystyle \ell }$, (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 ${\displaystyle {\tfrac {s}{\ell }}}$ and ${\displaystyle {\tfrac {d}{\ell }}}$ 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 ratio, ${\displaystyle {\tfrac {d^{2}}{\ell ^{2}}}}$, and is in descending order, starting with a number close to 2, the value for the isosceles right triangle with angles ${\displaystyle 45^{\circ }}$, ${\displaystyle 45^{\circ }}$, ${\displaystyle 90^{\circ }}$, and ending with the ratio for a triangle with angles roughly ${\displaystyle 32^{\circ }}$, ${\displaystyle 58^{\circ }}$, ${\displaystyle 90^{\circ }}$. The Babylonians, however, are believed not to have made use of the concept of measured angle. Columns 2 and 3 are most commonly interpreted as containing the short side and hypotenuse. Due to some errors in the table and damage to the tablet, variant interpretations, still related to right triangles, are possible.

Neugebauer and Sachs saw Plimpton 322 as a study of solutions to the Pythagorean equation in whole numbers, and suggested a number-theoretic motivation. They proposed that the table was compiled by means of a rule similar to the one used by Euclid in Elements. Many later scholars have favored a different proposal, in which a number ${\displaystyle x}$, greater than 1, with regular numerator and denominator, is used to form the quantity ${\displaystyle {\tfrac {1}{2}}\left(x+{\tfrac {1}{x}}\right)}$. This quantity has a finite sexagesimal representation and has the key property that if it is squared and 1 subtracted, the result has a rational square root also with a finite sexagesimal representation. This square root, in fact, equals ${\displaystyle {\tfrac {1}{2}}\left(x-{\tfrac {1}{x}}\right)}$. The result is that

${\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right),1,{\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)}$

is a rational Pythagorean triple, from which an integer Pythagorean triple can be obtained by rescaling. The column headings on the tablet, as well as the existence of tablets YBC 6967, MS 3052, and MS 3971 that contain related calculations, provide support for this proposal.

The purpose of Plimpton 322 is not known. Most current scholars consider a number-theoretic motivation to be anachronistic, given what is known of Babylonian mathematics as a whole. The proposal that Plimpton 322 is a trigonometric table is ruled out for similar reasons, given that the Babylonians appear not to have had the concept of angle measure. Various proposals have been made, including that the tablet had some practical purpose in architecture or surveying, that it was geometrical investigation motivated by mathematical interest, or that it was compilation of parameters to enable a teacher to set problems for students. With regard to the latter proposal, Creighton Buck, reporting on never-published work of D. L. Voils, raises the possibility that the tablet may have only an incidental relation to right triangles, its primary purpose being to help set problems relating to reciprocal pairs, akin to modern day quadratic-equation problems. Other scholars, such as Jöran Friberg and Eleanor Robson, who also favor the teacher's aid interpretation, state that the intended problems probably did relate to right triangles.