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

particular mathematical or ‘numerological’ sense.
Though Didymus must have referred to Archytas,
16
the theory he sets forth
cannot belong to the latter; it relates, most likely, to the earlier stage of the Py-
thagorean harmonics.
17
One can suggest that Eudemus’ Pythagoreans belong to
the same period, the more so because his histories of geometry and astronomy
generally place ‘the Pythagoreans’ in the first part of the fifth century. Archy-
tas, too, constituted ratios of the tetrachords in their lowest terms (ën prøtoi~
@riqmo$~, 47 A 16), but his theory has nothing to do with calculations described
above. In the arithmetical books of the
Elements, the term prõtoi (@riqmoí)
was consistently replaced by the more technical one, oî ëlácistoi @riqmoì
tõn aÿtòn lógon ëcóntwn, the least numbers of those that have the same
ratio – probably to avoid confusion with the prime numbers. But Euclidean
‘numbers prime to one another’ (prõtoi prò~ @ll2lou~ @riqmoí) are the
same as the Pythagorean ‘first numbers’ (
Elem. VII, 21). Revealingly, the
his turn, did not fail to mention his sources. The fact that he refers to Eudemus only
once means either that he found nothing worthy of notice in him, or that he quotes
him second-hand.
14
Porph.
In Ptol. Harm., 107.15ff. = Archytas A 17.
15
For interpretations of this method, see Barker.
GMW II, 35 n. 29; idem. Scientific
method, 71f.; Huffman, C. A. Archytas of Tarentum: Pythagorean, philosopher and
mathematician king, Cambridge 2005, 428ff.
16
Düring.
Ptolemaios, 157; Barker. GMW II, 34f.
17
Barker.
GMW II, 34 n. 25; Huffman. Archytas, 432f.

---

Chapter 6: The history of arithmetic and the origin of number
218
ancient Pythagorean demonstration of the incommensurability of the square’s
diagonal with its side uses the notion of the lowest numbers having the same
ratio (ëlácistoi tõn tòn aÿtòn lógon ëcóntwn), i.e., of pythmenes.
18
The
‘first numbers’ obviously proved to be quite a useful tool in the Pythagorean
arithmetic.
2. Aristoxenus:
On Arithmetic
Unfortunately, no other references to the
History of Arithmetic have survived in
the ancient tradition. For later authors, this subject was obviously less attractive
than the history of geometry and astronomy. It is revealing that we also have
only one fragment (cited in Stobaeus) from Aristoxenus’ Perì @riqmhtik4~
(fr. 23), which is luckily more informative.
From Aristoxenus’ books
On Arithmetic.
19