From the Beginning to Plato

288 FROM THE BEGINNING TO PLATO
24 See Heath [8.7] 1:128–37.
25 See, e.g., van der Waerden [8.13], 35–6. Cf. Neugebauer’s remark ([8.10], 91)
about astronomy: ‘Ancient science was the product of a very few men; and these
few happened not to be Egyptians.’
26 See Burkert [8.79], 380–2. For a defense of Heraclides of Pontus as Porphyry’s
source see During [8.78], 154–7.
27 Burkert ([8.79] 378–9), wishing to stress non-Pythagorean interest in music theory,
fastens on a corrupt text (Theon of Smyrna [8.92], 59.4–21, DK 18 A 13) in which
a physically impossible experiment involving the striking of vessels filled to various
heights with liquid, may be ascribed to Lasus of Hermione, a person from the last
half of the sixth century, who, according to the Suda ([8.87] 3:236.23–7), was the
first to write a book (logos) on music, to introduce dithyrambs into competition,
and to introduce eristic arguments (!)
28 See Burkert [8.79], 192–207.
29 Philolaus’ two accounts of the comma are equivalent since: tone−two dieses=(tone
−diesis)−diesis=apotome−diesis.
30 On the tetraktus see Delatte [8.80], 249–68.
31 I here ignore difficulties involved in treating 1 as a number. For Nicomachus ([8.55],
II.8.3) 1 is ‘potentially’ a triangular number.
32 On Euclid’s definition of ‘gnomon’ (Elements II, def. 2) and the origin of the word
itself, see Heath, [8.32] 1:370–2.
33 Burkert ([8.79], 433–4) accepts the figurate number material as early and admits
that ‘even a game may be regarded legitimately as a kind of mathematics’. But he
insists on the deductive character of even Thales’ geometry, in the context of which
‘Pythagorean arithmetic is an intrusive quasi-primitive element’.
34 Even contemporary defenders of the idea of an early Pythagorean mathematics are
usually willing to concede that attributions of scientific achievements to Pythagoras
are always subject to question and will settle for an attribution to the ‘early
Pythagoreans’, a somewhat vague locution which I take to refer to the period
before 450. In this essay I stress ancient attributions to Pythagoras because they
offer the greatest challenge to sceptics. I am, however, only interested in early
Pythagorean science, not in the science of Pythagoras.
35 For the sources see Heath [8.7] 1:144–5.
36 [8.74], 428.10–21. In the continuation Proclus attributes a parallel method to Plato.
37 For example,
38 Burkert also objects that Becker’s proof of IX.36 ‘requires an abundant use of
modern algebraic notation’. Here I think he points to a problem which cannot be
avoided in writing out an argument which turns on perceived spatial relations. The
argument I have given uses the fact there are no infinitely descending sequences of
integers, which is a form of what we know as the principle of mathematical
induction, but which is immediately obvious from the figurate representation of
numbers. On the other hand, Burkert is right to question whether the notion of
perfection involved in IX.36 could have coexisted with the notion of perfection
involved in calling 10, the sum of 1, 2, 3 and 4, perfect. But such a consideration
does not seem to me decisive.
39 I signal, but do not discuss, another passage ([8.74], 379.2–5) in which Proclus
says that Eudemus ascribed to the Pythagoreans the discovery and proof of the