The Arithmetic of Incommensurability: Theaetetus and Archytas

Abstract

We have shown how Theodoras was able to apply the elements of early Pythagorean arithmetic for the proof of the incommensurability of certain line segments with the unit-line. These theorems treated in turn the special cases of the sides of the squares of area 3, of 5, and of other values associated with these. But the method failed for many other cases, as for 17. We turn now to the work of two of Plato’s contemporaries, Theaetetus and Archytas, who added to the results established by Theodoras. These additions include, as we will show, the determination of further classes of incommensurable lines and the organization of this material into a system based on a revised theory of proportion.