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

El-
ements or, still more likely, to an elementary ‘introduction’ to arithmetic of a
Pythagorean origin. In Nicomachus, Theon, and Iamblichus, who have pre-
served some Pythagorean material not included in the arithmetical books of the
Elements, we find similar definitions of odd and even numbers: they also speak
of the division of even numbers into
equal parts (instead of by two, as in Euclid)
and of odd numbers as having a
middle.
32
Meanwhile, one can speak of a
number as having a middle only if it is represented by
pse¯phoi, counting stones,
as the early Pythagoreans used to do.
33
In Euclid, by contrast, where numbers
are represented by line segments, a ‘middle’ does not figure in the definition,
since the middle of a segment is a point, not another segment.
Let us recall again that pre-Euclidean arithmetic was not limited to the ma-
terial of books VII–IX of the
Elements, it contained various, even competing,
traditions. Thus, e.g., Euclid’s definition of unit bears some archaic features
and obviously shares its origin with other propositions of book VII, whereas
Aristoxenus’ much clearer definition comes from a different source.
34
The
existence of different traditions in arithmetic is also attested by the fact that
both definitions of an odd number cited above are found in Aristotle;
35
ob-
viously, they cannot come from the same arithmetical treatise.
In connection with the definitions of even and odd numbers given by Arist-
oxenus, it is worth noting that Philolaus also mentions the division of numbers
into even and odd,
36
whereas Plato repeatedly calls arithmetic the science of
32
Theon.
Exp., 21.22f.; Nicom. Intr. arith. I,7.2–3; Iambl. In Nicom., 12.11f.: even
numbers are divisible into equal parts, odd numbers are not. See Knorr.
Evolution, 53
n. 18. Cf. also a fragment from the pseudo-Pythagorean treatise
On Numbers by
Butheros: “The odd is more perfect than the even, for it has a beginning, a middle and
an end, while the even lacks a middle.” (Thesleff, H.
The Pythagorean texts of the
Hellenistic period, Åbo 1965, 59.10f.).
33
Arist.
Met. 1092b 10f.; Theophr. Met. 6a 15f. = 45 A 2–3.
34
Cf. Nicom.
Intr. arith. I,8.2: unit is the beginning of all numbers; Theon. Exp., 19.21:
unit is the beginning of numbers. There was another definition of unit, “a point with-
out position” (stigm3 Áqeto~, Arist. Met. 1084b 26), the opposite of the definition
of point as a unit having position (
Met. 1016b 24f.). Both these definitions go back
to the Academy, not to the Pythagoreans (Burkert.
L & S, 66f.).
35
Top. 142b 6f., 149a 29f.; SE 173b 8.
36
“Number, indeed, has two proper kinds, odd and even, and a third from both mixed
together, the even-odd.” (44 B 5), transl. by C. Huffman.

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2. Aristoxenus:
 On Arithmetic
223
even and odd.
37
Such a view of arithmetic undoubtedly has an
early Pythago-
rean origin, since arithmetic of the fourth century, as we find it in Euclid, is not
the science of even and odd in the least. In all the arithmetical books of the
El-
ements, the definitions of even and odd (VII, def. 6–7) occur only once –
namely, in the archaic theory of even and odd numbers that, as Becker has
shown, belongs to the earliest stratum of Pythagorean mathematics.
38
This the-
ory, consisting of propositions IX, 21–34 based on definitions VII, 6–11,
39
is of
an elementary character and stands in no logical connection with the material
of other arithmetical books of the