From the Beginning to Plato

268 FROM THE BEGINNING TO PLATO
structure for geometry, of having introduced proof into geometry. Indeed,
what is characteristic and absolutely new in Greek mathematics is the
advance by means of demonstration from theorem to theorem. Evidently,
Greek geometry has had this character from the beginning, and it is Thales
to whom it is due.
(2)
Harmonics in the Sixth and Fifth Centuries
In his commentary on Ptolemy’s Harmonics ([8.73], 30.1–9), Porphyry says:
And Heraclides writes these things about this subject in his Introduction to
Music:
As Xenocrates says, Pythagoras also discovered that musical intervals
do not come to be apart from number; for they are a comparison of
quantity with quantity. He therefore investigated under what conditions
there result concordant or discordant intervals and everything harmonious
or inharmonious. And turning to the generation of sound, he said that if
from an equality a concordance is to be heard, it is necessary that there be
some motion; but motion does not occur without number, and neither does
number without quantity.
The passage continues by developing an even more elaborate theory of the
relationship between movements and sound than the ones I mentioned earlier in
section 6. Scholars who are doubtful that Pythagoras was any kind of scientist
are happy to deny that Heraclides is Heraclides of Pontus, the student of Plato,
and to restrict the extent of the citation of Xenocrates to the first sentence. 26 Even
this sentence implies that Pythagoras discovered something about numbers and
concords, and I think everyone would agree that, if he discovered any such thing,
it was the association of the fundamental concords with the ratios 4:3, 3:2 and 2:
1. It is commonly thought that this association must have been known by people
familiar with musical instruments quite independently of theoretical
proclamations, but that ‘Pythagoras invested the applicability of these ratios to
musical intervals with enormous theoretical significance’ [KRS p.235]. Burkert
([8.79], 374–5) has pointed out how difficult it is to identify an early instrument
which would facilitate recognizing the correlation of pitch relations with
numerical ratios.
The traditional story of Pythagoras’ discovery of the ratios—for which our
earliest source is Nicomachus ([8.57], 6)—depends upon false assumptions
about the causal relation between pitches produced and the weights of hammers
striking a forge or weights suspended from plucked strings. However we find a
perfectly credible experiment, involving otherwise equal bronze discs with
thicknesses in the required ratios, associated with the early Pythagorean
Hippasus of Metapontum (Scholium on Phaedo 108d [8.64], 15, DK 18.12). 27