I take as my opening text the kind of thing my colleagues — certainly the mathematicians and often the historians of mathematics — might say about the beginnings of Greek mathematics. Something like this:The early Pythagoreans based their theory of proportion on commensurable magnitudes (or on the rational numbers, or on common fractions ^m/_n), but their discovery of the phenomenon of incommensurability (or the irrationality of √2) showed that this was inadequate. This provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Elements V.