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

§ 1. Pythagoras seems to have valued the science of numbers most of all and to
have advanced it, separating it from the merchants’ business and likening all
things to numbers. For number contains all else as well, and there is a ratio be-
tween all the numbers to each other <…>
20
§ 2. The Egyptians, for their part, believe numbers to be the invention of Hermes,
whom they call Thoth. And others derived numbers from the circular paths of the
divine luminaries.
§ 3. A unit is a beginning of number, and a number is a multitude consisting of
units. Of numbers, the even are those that are divisible into equal parts, and the
odd are those that are divisible into unequal parts and have a middle.
§ 4. It is considered, therefore, that crises and changes in illnesses relating to their
beginning, culmination, and end occur on odd days, since an odd number has a
beginning, a middle and an end.
Before we turn to the contents of the fragment, let us consider whether Arist-
oxenus is likely to have written a special treatise
On Arithmetic at all. In his
commentary, Wehrli wrote that fr. 23 “in its present form of an elementary in-
troduction to numerical notions” does not stem from Aristoxenus. On this
basis, Wehrli denied the existence of Aristoxenus’ treatise
On Arithmetic and
considered the part of this fragment that belongs to Aristoxenus to come from
one of his three treatises on the Pythagoreans. It is not clear, however, why
Aristoxenus’
On Arithmetic could not contain definitions he borrowed most
probably from an arithmetical treatise of Pythagorean origin. Wehrli does not
18
Elem. III, 408.12, 410.4. See below, 223 n. 40.
19
For greater convenience, I have broken the fragment into four paragraphs.
20
T3n dè perì toù~ @riqmoù~ pragmateían málista pántwn tim4sai doke$ Pu-
qagóra~ kaì proagage$n eı~ tò prósqen, @pagag§n @pò t4~ tõn ëmpórwn
creía~, pánta tà prágmata @peikázwn to$~ @riqmo$~. tá te gàr Álla
@riqmò~ Écei kaì lógo~ ëstì pántwn tõn @riqmõn prò~ @ll2lou~ … (Wehrli,
following Diels and Meineke, marked a lacuna here).

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2. Aristoxenus:
 On Arithmetic
219
seem to have real grounds for denying the existence of Aristoxenus’ work
On
Arithmetic and for relating fr. 23 to Aristoxenus’ book On Pythagoras and His
Disciples (fr. 11–25). Stobaeus never refers to this book; all of Aristoxenus’
fragments that he cites (except fr. 23) come from
Pythagorean Precepts
(fr. 34–37, 39–41), which he dutifully mentions quotation by quotation. Hence,
in citing this fragment, Stobaeus is all the more likely to indicate the right
source.
21
Since Pythagoreans are mentioned in many of Aristoxenus’ writings
(see, e.g. fr. 43, 90, 131), there is no particular reason to relate this fragment to
any of his three special works on this school.
Unlike Eudemus’
History of Arithmetic, Aristoxenus’ book bore the title On
Arithmetic, which fully accords with the wealth of historical, scientific, and
philosophical material we find in the fragment cited by Stobaeus. Having men-
tioned the founder of arithmetical science (§ 1), Aristoxenus adduces several
versions of the origin of number as such (i.e., of the art of calculation), which
go back to the Academy (§ 2), quotes the definitions of unit, number, and even
and odd numbers (§ 3), and proceeds to characterize the role of numbers in na-
ture (§ 4). Of course, we cannot be wholly certain that Stobaeus’ quotation
presents the text in its continuity rather than a set of separate fragments. The
passage, however, seems quite coherent and is more likely to come from a book
on arithmetic than from a work on the Pythagoreans.
22
Hence, the fact that fr. 23
goes back to Aristoxenus’ work