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

Catalogue (In Eucl., 66.6). Plato as-
cribes to him a proof of irrationality of the magnitudes from
√
3 to
√
17.
25
Further, it seems very improbable that Eudemus, who left a detailed account of
the authorship of many elementary theorems, should have neglected to say
something about the discovery of irrationality as such, which was made
before
Theodorus. In any case, it would be very untypical for Eudemus to refer to
Theaetetus (and Theodorus) without saying a word about the
pro¯tos heurete¯s of
irrationality, Hippasus.
26
Although Hippasus’ name is not attested in Eudemus’
fragments, the probability that he figured in the
History of Geometry is rather
high.
The same passage in Pappus says that Theaetetus classified the irrational
lines in accordance with the different means, the geometric, the arithmetic, and
the harmonic, whereas in the
Catalogue we read that Eudoxus added to the
three known mean proportionals three new ones (67.2f.).
27
If the latter in-
later addition (Neuenschwander. VB, 374). How can we reconcile the Pythagorean
origin of book IV with Oenopides’ authorship of IV, 16? Since the context of Eude-
mus’ remarks is unknown, we can only suppose that he wrote about the Pythagorean
origin of all the theorems of book IV except the last.
23
The commentary of Pappus on book X of Euclid’s Elements, transl. by G. Junge, W.
Thomson, London 1930, 63–64; cf. Burkert.
L & S, 440 n. 182.
24
Following Pappus, the scholia to book X also call the Pythagoreans the originators of
the theory of irrationals (
Schol. In Eucl., 415.7, 416.13, 417.12f.). Burkert’s position
in this question is inconsistent. He states that: 1) Eudemus, quoted by Pappus, does
not mention the Pythagoreans; 2) the scholia to book X are mostly from Pappus;
3) the same scholia, ascribing the discovery of the irrationality to one of the Pytha-
goreans, are based ultimately on Eudemus (
L & S, 450 n. 13, 457, 458 n. 57, 462
n. 72–73). The contradiction is easily removed by suggesting that Pappus’ reference
to the Pythagoreans also goes back to Eudemus.
25
Tht. 147d = 43 A 4; Papp. Comm., 72–74.
26
Cf. his remark on the discovery of the regular solids by the Pythagoreans and
Theaetetus (
Schol. in Eucl. 654.3f.) and below, 177 n. 47. On Hippasus, see below,
189f.
27
Since the majority of Greek authors used the terms mesóth~ (a mean proportional)
and @nalogía (a proportion) interchangeably (Archytas 47 B 2; Papp. Coll. III,
70.16f.; Heath.
History 2, 292f.; Wolfer, op. cit., 23f.), we shall follow this usage.

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2.
The History of Geometry: on a quest for new evidence
173
formation comes from Eudemus (and there seem to be no grounds to doubt it),
we can surmise that he also mentioned the person who discovered the first three
means. In this connection, I would like to draw attention to the reports of Ni-
comachus and Iamblichus on the discovery of the proportionals. According to
Nicomachus,
There are the first proportions that are acknowledged by all the ancients – Pytha-
goras, Plato, and Aristotle. The very first three are the arithmetic, the geometric,
and the harmonic; the other three subcontrary to them have no proper names and
are called more generally the fourth, the fifth, and the sixth means. After them the
later mathematicians discovered the other four proportions … (
Intr. arith.,
122.11f.).
Having considered the first six means, he summarizes:
These are then the first six means generally known by the ancients: three proto-
types that came down to Plato and Aristotle from Pythagoras, and the other three
subcontrary to them, which came into use with later writers and followers (ibid.,
142.21f.).
Thus, it comes out that the first three proportions were discovered by Pythago-
ras and the second three by contemporaries of Plato and Aristotle. Nicoma-
chus’ evidence is correct, but it lacks details that would allow us to connect it
with Eudemus.
28
We find such details, however, in Iamblichus’ commentary to
Nicomachus:
Of old there were but three means in the days of Pythagoras and the mathema-
ticians of his times, the arithmetic, the geometric, and the third in order, which
once was called the subcontrary, but had its own name changed forthwith to har-
monic by Archytas and Hippasus, because it seemed to embrace the ratios that
govern the harmonized and tuneful. And it was formerly called subcontrary be-
cause its character was somehow subcontrary to the arithmetic … After this name
has been changed, those who came later, Eudoxus and his school, invented three
more means, and called the fourth properly subcontrary because its properties
were subcontrary to the harmonic … and the other two they named simply from
their order, the fifth and the sixth. The ancients and their successors thought that
28
Stated in Nicomachus (II, 22–28), the theory of ten proportions goes back to Eratos-
thenes’