51. These last considerations show that, while it is quite safe to attribute the substance of the First Book of Euclid to Pythagoras, the arithmetic of Books VII.-IX., and the “geometrical algebra” of Book II. are certainly not his. They operate with lines or with areas instead of with units, and the relations which they establish therefore hold good whether they are capable of numerical expression or not. 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” 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,[248] we find that 12 : 6 is the octave, 12 : 8 the fifth, and 8 : 6 the fourth, and it can hardly be doubted that it was Pythagoras himself who discovered these intervals. The stories which have come down to us about his observing the harmonic intervals in a smithy, and then weighing the hammers that produced them, or of his 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.[249] For our purpose their absurdity is their chief merit. They are not stories which any Greek mathematician or musician could possibly have invented, but genuine popular tales bearing witness to the existence of a real tradition that Pythagoras was the author of this momentous discovery. Footnote 248: Plato (_Tim._ 36 a 3) defines the harmonic mean as 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. Footnote 249: For these stories and a criticism of them, see Max C. P. Schmidt, _Kulturhistorische Beiträge_, i. pp. 78 sqq. The smith’s hammers belong to the region of _Märchen_, and it is not true either that the notes would be determined by the weight of the hammers, or that, if they were, the weights hung to equal strings would produce the notes. These inaccuracies were pointed out by Montucla (Martin, _Études sur le Timée_, i. p. 391). [Sidenote: Things are numbers.]