Early Greek Philosophy by John Burnet, with Burnet's notes
50. Incommensurability 52. Things Are Numbers

From Chapter II., Science and Religion

51. Proportion and Harmony
These last considerations show that, while it is quite safe to attribute the substance of the early books of Euclid to the early Pythagoreans, his arithmetical method is certainly not theirs. It operates with lines instead of with units, and it can therefore be applied to relations which are not capable of being expressed as equations between rational numbers. That is doubtless why arithmetic is not treated in Euclid till after plane geometry, a complete inversion of the original order. For the same reason, the doctrine of proportion which we find in Euclid cannot be Pythagorean, and is indeed the work of Eudoxos. Yet it is clear that the early Pythagoreans, and probably Pythagoras himself, studied proportion in their own way, and that the three "medieties" (μεσότητες) in particular go back to the founder, especially as the most complicated of them, the "harmonic," stands in close relation to his discovery of the octave. If we take the harmonic proportion 12 : 8 : 6,89 we find that 12 : 6 is the octave, 12 : 8 the fifth, and 8 : 6 the fourth, and it can hardly be doubted that Pythagoras himself discovered these intervals. The stories about his observing the harmonic intervals in a smithy, and then weighing the hammers that produced them, or suspending weights corresponding to those of the hammers to equal strings, are, indeed, impossible and absurd; but it is sheer waste of time to rationalise them.90 For our purpose their absurdity is their chief merit. They are not stories which any Greek mathematician could possibly have invented, but popular tales bearing witness to the existence of a real tradition that Pythagoras was the author of this momentous discovery. On the other hand, the statement that he discovered the "consonances" by measuring the lengths corresponding to them on the monochord is quite credible and involves no error in acoustics.

Burnet's Notes


89. The harmonic mean is thus defined by Archytas (fr. 2, Diels) ἁ δὲ ὑπεναντία (μεσότας), ἃν καλοῦμεν ἁρμονικάν, ὅκκα ἔωντι <τοῖοι (sc. οἱ ὅροι) · ᾧ> ὁ πρῶτος ὅρος ὑπερέχει τοῦ δευτέρου αὐταύτου μέρει, τωὐτῷ ὁ μέσος τοῦ τρίτου ὑπερέχει τοῦ τρίτου μέρει.. Cf. Plato, Tim. 36 a 3, τὴν . . . ταὐτῷ μέρει τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχομένην. The harmonic mean of 12 and 6 is, therefore, 8; for 8=12-12/3 = 6+6/3.

90. The smith's hammers belong to the region of Märchen, and it is not true that the notes would correspond to the weight of the hammers, or that, if they did, the weights hung to equal strings would produce the notes. The number of vibrations really varies with the square root of the weights. These inaccuracies were pointed out by Montucla (Martin, Études sur le Timée, i. p. 391).

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