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

particular to the general, from the humble empirical origins in Egypt, which
were born of practical necessity, to the perfection of theoretical geometry in
Eudemus’ own times.
226
ëpeì dè cr3 tà~ @rcà~ kaì tõn tecnõn kaì tõn ëpisthmõn prò~ t3n paroñsan
períodon skope$n, légomen … (64.16f.). Heath. History 1, 120, attributed these
words to Proclus.
227
See above, 48ff.
228
Weil. Aristotle’s view, 203.

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Chapter 6
The history of arithmetic and the origin of number
1. The fragment of Eudemus’
History of Arithmetic
The only surviving fragment of Eudemus’
History of Arithmetic comes from
Porphyry’s commentary to Ptolemy’s
Harmonics. Its content is related, ac-
cordingly, to the mathematical theory of music, rather than to arithmetic. The
broad context of Porphyry’s commentary is the following. Discussing the Py-
thagorean mathematical theory of concordant intervals, Ptolemy on the whole
agrees with it, while criticizing some of its propositions.
1
For his starting point
he takes the Pythagorean method of associating equal numbers with tones of
equal pitch and unequal numbers with tones of unequal pitch (
Harm., 11.8ff.).
Further, the Pythagoreans divide tones of unequal pitch into two classes, con-
cordant and discordant intervals, and associate the first class with epimoric and
multiple ratios, and the second with epimeric ratios. The reason for this is that
as concordant intervals are ‘finer’ than discordant, so epimoric and multiple ra-
tios are ‘better’ than epimeric, because of the simplicity of the comparison.
2
Ptolemy acknowledges that concordant intervals differ from each other by the
degree of their proximity to absolute equality: the less the difference between
the terms of their ratios, the better. Thus, the octave is the finest of the concord-
ant intervals, since its ratio (2:1) “alone makes an excess equal to that which is
exceeded” (i.e., the difference between its terms is equal to the smaller term of
the ratio); it is immediately followed by the fifth (3:2) and the fourth (4:3).
The principle of comparing the tones of unequal pitch by their closeness to
equality is further elaborated in Ptol.
Harm. I, 7 (15.18f.). In his commentary to
this passage, Porphyry emphasizes that the method comes from the Pythago-
reans and adds: many of them took their starting point not in equality alone, but
in the so-called
pythmenes, or the ‘first numbers’, as well, i.e., the ratios of the
concordant intervals expressed in their lowest terms.
3
Earlier, Porphyry defined
1
Harm. I, 5–7. See Barker, A. Scientific method in Ptolemy’s Harmonics, Cambridge
2000, 54ff.
2
For details, see Düring, I.
Ptolemaios und Porphyrios über die Musik, Göteborg
1934, 176f.; Barker.
GMW II, 284f.; idem. Scientific method, 61f.
3
In Ptol. Harm., 114.23ff. puqm2n, ‘base’, is the first in a series of equal ratios that is
expressed in the lowest terms. Thus, the ratio 2:3 is a
pythm
e¯
n
in the series 2:3, 4:6,
8:12, etc. With this meaning,
pythm
e¯
n
is found already in Plato (
Res
. 546c 1);
cf. Speus. fr. 28 Tarán. See Nicomachus of Gerasa
. Introduction to arithmetic
,
transl. by M. L. D’Ooge, New York, 1926, 216.

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1. The fragment of Eudemus’
History of Arithmetic
215
the first numbers or
pythmenes as the smallest numbers in which concordant in-
tervals are produced.
4
Now he says:
For Eudemus makes clear in the first book of the
History of Arithmetic that they
demonstrated the ratios of concordant intervals through the
pythmenes, saying
about the Pythagoreans the following word for word: “(They said) moreover that
it turned out that the ratios of the three concords, of the fourth, the fifth and the oc-
tave, taken in the first numbers (ën prøtoi~), belong to the number nine. For 2
and 3 and 4 are nine.” (fr. 142)
The ratios 2:1, 3:2, and 4:3 are expressed in the ‘first numbers’ indeed. It is
not clear, however, what precisely Eudemus had in mind and why it was im-
portant for the Pythagoreans that the numerators of these three ratios make up
nine. Unfortunately, the quotation is too fragmentary to conclusively suggest
any immediate context. Besides, it is different from other fragments of Eude-
mus’ works on the history of science, which usually deal with particular dis-
coveries. Let us nevertheless make the most of the scant information we find in
Porphyry.
It follows from the quoted passage that Eudemus’
History of Arithmetic
comprised at least two books, i.e., was of about the same length as his
History
of Astronomy. If Eudemus treated the history of arithmetic in as much detail as
he did the history of astronomy, he must have had enough material for it at
hand. Euclid’s arithmetical books (VII–IX) are known to have preserved only