Babylonian Number Theory and Trigonometric Functions: Trigonometric Table and Pythagorean Triples in the Mathematical Tablet Plimpton 322

Abstract

The mathematical cuneiform tablet Plimpton 322 is one of the most important source materials in history of mathematics. It lists fifteen Pythagorean triples together with a certain table which suggests that one of the angles of a right-angled triangle decreases from $45^\circ\ \text{to}\ 31^\circ,$ as O. Neugebauer explained in detail. However, we do not know the principle of constructing the fifteen triples and the true purpose of the table.

In this paper I shall clarify the mathematical meanings of a few technical terms which occur in the headings of the four columns of the tablet, and also the constructing principle of the listed numbers by analyzing Babylonian calculation methods. As a result, we can conclude that the Babylonian scribe of our tablet calculated the fifteen Pythagorean triples using the trigonometric table of the first column which was made by a kind of linear-interpolation.