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

part of his predecessors’ theories, the rest remained outside the
Elements
5
and
in part was subsequently included in compendia by Nicomachus, Theon of
Smyrna, Iamblichus, and other later authors. In the early fourth century, arith-
metic already enjoyed the status of an exact science; some believed its demon-
strations to be more conclusive than even geometrical ones.
6
No wonder then
that Eudemus considered arithmetic to be worth a special historical treatise.
The fact that the only quotation from it relates to the mathematical har-
monics of the Pythagoreans can hardly be fully explained by its coming from
commentary to Ptolemy’s
Harmonics. Theoretical arithmetic, created by the
early Pythagoreans, was closely connected to harmonics,
7
especially as con-
4
prøtou~ labónte~ @riqmoú~, oŸ~ ëkáloun puqména~ … toutéstin ën o‰~
ëlacístoi~ @riqmo$~ sumfwníai @poteloñntai (In Ptol. Harm., 107.18f.). Cf.
Theon of Smyrna’s definition: “Of all the ratios … those that are expressed in the
smallest numbers and prime to each other are called firsts among those having the
same ratio or
pythmenes of the same species.” (Exp., 80.15).
5
E.g., the arithmetical part of the theory of irrationality is not retained in book X
(Knorr.
Evolution, 311). Speusippus analyzed linear and polygonal numbers (fr. 28
Tarán), about which Euclid remains silent. See also Philip of Opus’
On the Poly-
gonal Numbers (Lasserre. Léodamas, 20 T 1).
6
Archytas (47 B 4), Aristotle (
APo 87a 34–7, Met. 982a 26f.). On the role of arith-
metic in Archytas’ and Plato’s classification of sciences, see Knorr.
 Evolution, 58
n. 71, 90f., 311.
7
Van der Waerden.
Pythagoreer, 364ff., 406ff.

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Chapter 6: The history of arithmetic and the origin of number
216
cerns the theory of means. It is highly possible that the development of arith-
metic in its application to music could also have been reflected in the
History of
Arithmetic, for example, in problems that occupied Archytas (47 A 17, 19,
B 2).
8
Further considerations lead in the same direction. Of the quadrivium of
mathematical sciences, harmonics was the only one to which Eudemus did not
devote a special treatise. Assuming that the Peripatetic project supposed the
description of all fields of knowledge that had a long enough history behind
them, the absence of the history of harmonics may, indeed, appear strange.
That Eudemus could have treated the problems of mathematical harmonics in
his
History of Arithmetic provides a possible explanation for this.
9
This hy-
pothesis seems the more probable in that the Aristotelian classification of
sciences links arithmetic and harmonics with each other as a main and a subor-
dinate discipline: first, they are based on common principles and, second, har-
monics depends on arithmetic inasmuch as it uses the latter’s methods of dem-
onstration.
10
Let us return to the text of the fragment. To judge from the verbatim quo-
tation and the reference to the first book of the
History of Arithmetic, Porphyry
must have cited this work first-hand. Since ‘the Pythagoreans’ are not men-
tioned in the quotation, they must have been dealt with outside the quoted pas-
sage, too. On the other hand, Porphyry never refers to the
History of Arithmetic
elsewhere. Hence, he could have borrowed this passage from an intermediary
source that quoted Eudemus
verbatim and more amply. This source may well
have been the book
On the Differences Between the Theories of Aristoxenians
and the Pythagoreans by Didymus of Alexandria, a musicologist of the early
first century AD.
11
Both Porphyry and Ptolemy used this book, particularly as a
source on Pythagorean harmonics.
12
According to Porphyry, Ptolemy bor-
rowed much from Didymus, though not always referring to him.
13
Indeed, in
8
See above, 173f. In his treatise on harmonics, Archytas discussed, in particular, the
mean proportionals (47 B 2). His approach to harmonics is taken up in Euclid’s
Sec-
tio canonis, where a number of arithmetical theorems are proved: van der Waerden.
Pythagoreer, 364ff., 406ff.; Barker. GMW II, 42f., 48f. Further, Theaetetus and
Eudoxus also worked on the theory of means (see above, 173f.).
9
Cf. above, 129.
10
APo 75a 38–75b 17, 76a 9–15. Geometry and optics relate to each other in the same
way.
11
See Barker.
GMW II, 230, 241f.; idem. Greek musicologists in the Roman empire,
Apeiron 27 (1994) 53–74, esp. 64ff.; West, M.L. Ancient Greek music, Oxford 1992,
169f., 239f.
12
Didymus was Porphyry’s source for Xenocrates’ fragment: “Pythagoras discovered
also that musical intervals do not come into being apart from numbers, for they are
an interrelation of one quantity with another.” (
In Ptol. Harm., 30.1f. = fr. 87 Isnardi
Parente). See Düring.
Ptolemaios, 155f.
13
In Ptol. Harm., 5.12f. Ptolemy mentions Didymus only in book II of Harmonics,
doing so, as a rule, in connection with Archytas and other early theoreticians. But he
used Didymus’ material in book I as well (Düring.
Ptolemaios, 139f.). Porphyry, in

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1. The fragment of Eudemus’
History of Arithmetic
217
Harmonics I, 6, where Ptolemy criticizes another Pythagorean method and
mentions “the first numbers that make up the ratios” (i.e., the
pythmenes),
Didymus does not figure among his sources, whereas Porphyry adduces the
same arithmetical method in a fuller version, referring to both Didymus and
Archytas.
14
According to Didymus, to determine the most concordant intervals, the Py-
thagoreans proceeded as follows. Taking the ‘first numbers’, which they called
pythmenes (i.e., 2:1, 3:2, 4:3), and assigning them to concordant intervals, they
subtracted a unit from each of the terms of the ratio and compared the re-
mainders. Thus, subtracting a unit from both terms of the octave (2:1) they ob-
tained one, from the fourth (4:3) five and from the fifth (3:2) three. The smaller
the remainder, the more concordant was the given interval considered. The
whole procedure hardly seems convincing, from either a mathematical or musi-
cal point of view.
15
No wonder Ptolemy calls it ‘utterly ludicrous’ (
Harm.,
14.6). He notes, in particular, that a ratio remains the same whether it is ex-
pressed by
pythmenes or not, while the Pythagorean method is valid for pyth-
menes only. Porphyry on the whole agrees with him (In Ptol. Harm., 109.1ff.),
repeatedly pointing out, however, that it is precisely on the lowest terms that the
Pythagorean theory was based. This was precisely his reason for quoting Eude-
mus’ passage dealing with ratios taken ën prøtoi~. It remains unclear whether
the number nine, being the sum of the numerators of the three
pythmenes, had a