Early Greek Philosophy by John Burnet, with Burnet's notes
49. Geometry and Harmonics 51. Proportion and Harmony

From Chapter II., Science and Religion

50. Incommensurability
One great disappointment, however, awaited him. 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 arithmetical 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 may be 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.86 However that may be, it is certain that Pythagoras did not care to pursue the subject any further. He may have 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 irrationals.87 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.88

Burnet's Notes


86. Arist. An. Pr. A, 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.

87. Plato, Theaet. 147 d 3 sqq.

88. 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). The excommunicated Hippasos is the enfant terrible of Pythagoreanism, and the traditions about him are full of instruction. See p. 94, n. 2.

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