From Chapter II., Science and Religion
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=1212/3 = 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).
