Leonid Zhmud The Origin of the History of Science in Classical Antiquity

Met. 985b 27f.).
26
Frank, E.
Plato und die sogenannten Pythagoreer, Halle a. S. 1923, 260 n. 1; Bur-
kert.
L & S, 414f.
27
See Huffman.
Philolaus, 172ff.; Zhmud. Philolaus, 255ff.
28
Zhmud.
Wissenschaft, 320f.

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2. Aristoxenus:
 On Arithmetic
221
Aristoxenus speaks precisely of medicine, treated at length in his works on the
Pythagoreans,
29
Aristotle keeps complete silence upon this subject. The defini-
tions cited by Aristoxenus are of a purely mathematical character (§3), whereas
Aristotle explicitly states that, for the Pythagoreans, even and odd are stoice$a
toñ @riqmoñ (986a 16). Empty as it is of mathematical meaning, this state-
ment is of crucial importance for the numerical ontology he attributes to the Py-
thagoreans. As for Aristoxenus, he did not at all mean that a disease
is a
number, nor that it
consists of numbers, nor that the principles of number are
the same as the principles of disease. His example has to be understood as
pointing to a likeness between the odd numbers and the periods of an illness:
since of both them have a beginning, a culmination, and an end, it is on odd
days that changes in the course of illness occur.
30
Therefore, illness comes to
resemble number for the sake of prognosis (one needs to define the day of a
possible crisis), not for metaphysical
identification of things and numbers, nor
the principles of both. This is confirmed by the popularity the doctrine of criti-
cal days won in an empirically oriented Hippocratic medicine.
31
Hence, Aristoxenus associates the birth of arithmetic with Pythagoras and
then, having mentioned several versions of the origin of number, turns to the
‘principles’ of theoretical arithmetic (§ 3). Three of the four definitions he cites
(those of unit, odd number, and even number) differ from those given in Eu-
clid’s book VII:
Aristoxenus
Euclid
A unit is a beginning (@rc2) of
A unit is that by virtue of which each
number.
of the things that exists is called one
(def. 1).
A number is a multitude composed
A number is a multitude composed of
of units.
units (def. 2).
29
Fr. 21–22, 26–27; Iambl.
VP 163–164 = DK 58 D 1.6–16. This fact additionally sup-
ports the authenticity of the second part of Aristox. fr. 23.
30
See similar ideas in the Hippocratic corpus: “The odd days must be especially ob-
served, since on them patients tend to incline in one direction or the other.” (
De victu
in acutis [Appendix] 9 Littré; cf. Epid. I, 12; De sept. partu, 9). On the possible Py-
thagorean origin of the doctrine of critical days, see Zhmud.
Wissenschaft, 237f.
31
See e.g. Jouanna, J.
Hippocrate, Paris 1992, 475f. It is revealing that, in his natural-
scientific works, Aristotle himself was not averse to the ‘Pythagorean’ likening of
things and numbers. See
Mete. 372a 1ff., 374b 31f. and especially Cael. 268a 10f.:
“For, as the Pythagoreans say, the world and all that is in it is determined by the
number three, since beginning and middle and end give the number of an ‘all’, and
the number they give is the triad. And so, having taken these three from nature as (so
to speak) laws of it, we make further use of the number three in the worships of the
gods … And in this, as we have said, we do but follow the lead which nature gives.”
See also pánta tría kaì oÿdèn pléon 9 Élasson toútwn tõn triõn (Ion of
Chios, 36 B 1); Theon.
Exp
., 46.14f.: they say that three is the perfect number, for it
is the first to have beginning, middle, and end.

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Chapter 6: The history of arithmetic and the origin of number
222
Even numbers are those that are divi- An even number is that which is divisi-
sible into equal parts (eı~ Ísa).
ble into two parts (díca) (def. 6).
Odd numbers are those that are divi- An odd number is that which is not di-
sible into unequal parts (eı~ Ánisa)
visible into two parts (m3 diairoúme-
and have a middle.
no~ díca), or that which differs by a
unit from an even number (def. 7).
Since the definitions Aristoxenus gives also occur before him, for example
in Aristotle, they can go back either to one of the pre-Euclidean versions of