“Rational Diameters” and the Discovery of Incommensurability

Abstract

1. The study of incommensurables and irrationals constituted a prominent part of early Greek geometry, stimulating efforts from the Pythagoreans, the key geometers Theaetetus and Eudoxus at the time of Plato, and extending to Euclid. [4] The Tenth Book of the Elements, by far the lengthiest and most imposing of the thirteen books, stands as the culmination of this inquiry. But before Euclid, we have only a few testimonies of the earliest work in this field, so that any attempt to describe the first discovery of the incommensurables is necessarily speculative. For Aristotle the existence of incommensurable magnitudes is already a familiar phenomenon, suitable as an example in his discussions of topics in logic and science. [1, pp. 295-297] In Prior Analytics I 23, for instance, he takes as his principal specimen of an indirect proof that of the incommensurability of the side and diameter* of the square and indicates its key feature: the hypothesis of commensurability entails that odd numbers equal even numbers. [3, pp. 22-23], [4, chap. 2] This manner of proof, which is still commonly used in elementary mathematics (see Section 8), has thus been supposed to represent the reasoning first adopted by the Pythagoreans. [2, Vol. III, p. 2]