[Sidenote: Incommensurability.] 50. One great disappointment, however, awaited Pythagoras. It follows at once from the Pythagorean proposition that the square on the diagonal of a square is double the square on its side, and this ought surely to be capable of numerical expression. As a matter of fact, however, there is no square number which can be divided into two equal square numbers, and so the problem cannot be solved. In this sense, it is doubtless true that Pythagoras discovered the incommensurability of the diagonal and the side of a square, and the proof mentioned by Aristotle, namely, that, if they were commensurable, we should have to say that an even number was equal to an odd number, is distinctly Pythagorean in character.[244] However that may be, it is certain that Pythagoras did not care to pursue the subject any further. He had, as it were, stumbled on the fact that the square root of two is a surd, but we know that it was left for Plato’s friends, Theodoros of Kyrene and Theaitetos, to give a complete theory of the matter.[245] The fact is that the discovery of the Pythagorean proposition, by giving birth to geometry, had really superseded the old view of quantity as a sum of units; but it was not till Plato’s time that the full consequences of this were seen.[246] For the present, the incommensurability of the diagonal and the square remained, as has been said, a “scandalous exception.” Our tradition says that Hippasos of Metapontion was drowned at sea for revealing this skeleton in the cupboard.[247] Footnote 244: Arist. _An. Pr._ Α, 23. 41 a 26, ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γίγνεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης. The proofs given at the end of Euclid’s Tenth Book (vol. iii. pp. 408 sqq., Heiberg) turn on this very point. They are not Euclidean, and may be substantially Pythagorean. Cf. Milhaud, _Philosophes géomètres_, p. 94. Footnote 245: Plato, _Theaet._ 147 d 3 sqq. Footnote 246: How novel these consequences were, is shown by the fact that in _Laws_, 819 d 5, the Athenian Stranger says that he had only realised them late in life. Footnote 247: This version of the tradition is mentioned in Iamblichos, _V. Pyth._ 247, and looks older than the other, which we shall come to later (§ 148). Hippasos is the _enfant terrible_ of Pythagoreanism, and the traditions about him are full of instruction. [Sidenote: Proportion and harmony.]