From the Beginning to Plato

266 FROM THE BEGINNING TO PLATO
The question whether Archytas’ tuning is mathematical manipulation without
musical significance must be considered moot. Barker ([8.14], 46–52) has argued
that musical practice may have played a much more significant role in Archytas’
theorizing than is usually allowed, but he leaves no doubt that a kind of
mathematical a priorism, particularly the faith in the consonance of epimorics,
played a central role in Archytas’ musical thought. Moreover, a passage in
Porphyry ([8.73], 107.15–108.21, DK 47 A 17) suggests that Archytas used very
arbitrary numerical manipulations to determine the relative concordance of octave,
fifth, and fourth, subtracting 1 from each term of the corresponding ratios, adding
the results for each interval and taking lower sums to mean greater concord;
since
, he declared the octave to be more concordant
than the fifth, which, in turn is more concordant than the fourth.
This mixture of mathematical reasoning and mathematical mystification
makes it difficult for us to classify the musical work of Archytas (and even of
Euclid) as either science or numerology. 22 It is difficult to believe that Archytas
did not know the truth of VIII.7 and 8, at least for the case of one mean
proportional. But it is hard to see how he could even begin to think about such
results without a well-developed idea of arithmetical reasoning and proof. 23
PART TWO:
THE SIXTH AND FIFTH CENTURIES
(1)
Thales and Early Greek Geometry
In addition to his general remarks about Thales quoted in the introduction
Proclus ([8.74]) records four of Thales’ mathematical achievements, twice citing
Eudemus as authority. I quote the passages:
(a) The famous Thales is said to have been the first to prove that the circle is
bisected by the diameter. (157.10–11)
(b) We are indebted to the ancient Thales for the discovery of this theorem
[asserting the equality of the base angles of an isosceles triangle] and many
others. For he, it is said, was the first to recognize and assert that the angles
at the base of any isosceles triangle are equal, although he expressed himself
more archaically and called the equal angles similar. (250.20–251.2)
(c) According to Eudemus, this theorem [asserting the equality of the nonadjacent
angles made by two intersecting straight lines]…was first
discovered by Thales. (299.1–4)
(d) In his history of geometry Eudemus attributes to Thales this theorem
[asserting the congruence of triangles with two sides and one angle equal].
He says that the method by which Thales is said to have determined the