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

On Means (Perì mesot2twn); it is Eratosthenes who discovered the last
four proportions (van der Waerden.
EW, 385; Wolfer, op. cit., 20ff.). The same the-
ory can be found in Pappus (
Coll. III, 70.16f., 84.1f.), who mentioned On Means
several times (ibid., 636.24, 672.5, cf. 662.15). When citing Nicomachus, Pappus re-
peats his short historical reference, but omitting the names. According to his evi-
dence, the ancients discovered the first three proportions, as well as the second three,
while the moderns discovered the last four (84.1f.). Finally, Theon of Smyrna, who
also used Eratosthenes, relates the first six (not the first three!) proportions to the Py-
thagoreans in general (
Exp., 116.3f.). It follows that On Means was a purely math-
ematical treatise and contained no (or almost no) historical information, otherwise it
is hard to understand why Pappus used Nicomachus’ evidence, rather than Eratos-
thenes’.

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Chapter 5: The history of geometry
174
this number, i.e., six, of means could be set up; but the moderns have found four
more in addition …
29
It is clear that we have a fragment of the history of mathematics before us,
taken by Iamblichus from some reliable and well-informed source. It contains
the names of Hippasus, Archytas, and Eudoxus that were missing in Nicoma-
chus, along with much additional information on the early history of propor-
tions. This information perfectly matches the fragment of Archytas quoted by
Porphyry: there are three means in music, the arithmetic, the geometric, and
the subcontrary, “which we call harmonic”.
30
Since Eudemus ascribed the ap-
plication of the first three proportions to Theaetetus and the discovery of the
three others to Eudoxus, the information provided by Iamblichus likewise has
to be related to the Peripatetic. Specifically, he seemed to consider not only
Eudoxus but also Pythagoras (in connection with the discovery of the first three
means), as well as Hippasus and Archytas (as his followers).
31
Interestingly,
Iamblichus turns two more times to the history of proportions, saying again
that the first three come from Hippasus and Archytas, whereas the first six were
used from the times of Plato till Eratosthenes.
32
His immediate source here
was, in all probability, Porphyry’s commentary on Euclid’s
Elements, which
traced the history of means from Pythagoras to Eratosthenes, relying mainly on
Eudemus.
33
29
In Nicom., 100.19–101.9 = 18 A 15, transl. by M. L. D’Ooge.
30
In Ptol. Harm., 93.13 = 47 B 2. Since Philolaus also called this mean harmonic
(Nicom.
Intr. arith., 135.10f. = 44 A 24), it had to be renamed before Archytas. Tan-
nery (Sur l’arithmétique pythagoricienne,
Mémoires scientifiques, T. 2, Toulouse
1912, 190) believed that Archytas quoted Hippasus. Even if this was not the case,
Archytas surely could have mentioned his name. Cf. Huffman,
Philolaus, 167 ff.
31
Hippasus made an acoustical experiment with four bronze discs (Aristox. fr. 90),
using the so-called musical proportion that includes the arithmetic and harmonic
means (12:9 = 8:6). This implies that the first three means were known to Pythago-
ras, whose experiments Hippasus followed (Zhmud.
Wissenschaft, 162ff.). Iambli-
chus associates the musical proportion with Pythagoras and Philolaus (
In Nicom.,
118.23f. = 44 A 24). Eudemus’ and Aristoxenus’ source might have been Glaucus of
Rhegium (see below, 195). On the Pythagorean theory of proportions, see Heath.
History 1, 85f.
32
In Nicom., 113.16f., 116.1f. In the last passage Iamblichus ascribes the last four
means not to Eratosthenes, but to completely unknown (and very probably fictitious)
Pythagoreans, Myonides and Euphranor (116.5).
33
See below, 186ff. Though Lasserre.
Eudoxos, 175, also connected Iamblichus’ pas-
sage with Eudemus, he did not see an intermediary in Porphyry, but in Eratosthenes.
Wolfer,
op. cit., 24, however, rightly pointed out that the Platonicus discusses only
the first three proportions known to Plato, while Iamblichus mentions six and even
ten proportions. Besides, in Iamblichus’ passage Plato is missing, whereas Nico-
machus, Theon of Smyrna, and Pappus, who knew the material of the
Platonicus, do
not mention Eudoxus in connection with the discovery of proportions.

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2.
The History of Geometry: on a quest for new evidence
175
The name of the author of the first
Elements, Hippocrates of Chios, occurs
both in the
Catalogue (Procl. In Eucl., 66.4) and in two of Eudemus’ fragments
(fr. 139–140). As follows from his detailed account on the quadrature of lunes
(fr. 140), Eudemus was well acquainted with Hippocrates’ work and estimated
his contribution to mathematics highly. Eratosthenes’ letter devoted to the
problem of the duplication of the cube says:
Hippocrates of Chios was the first to conceive (prõto~ ëpenóhsen) that if, for
two given lines, two mean proportionals were found in continued proportion, the
cube will be doubled. Whence he turned his puzzle (@pórhma) into another no
less puzzling.
34
This evidence almost certainly goes back to Eudemus, on whose material Era-
tosthenes relied heavily.
35
In Proclus we find a similar account of Hippocrates.
Giving a definition of the @pagwg2, i.e., of the reduction of a complicated
problem to another that, if known or constructed, will make the original prop-
osition evident, he adds: so, for example, the problem of doubling the cube was
reduced to the finding of two means in continuous proportion between two
given straight lines.
They say that the first to effect reduction (@pagwg2) of difficult constructions
(tõn @porouménwn diagrammátwn) was Hippocrates of Chios, who also
squared the lune and made many other discoveries in geometry, being a man of
genius when it came to construction if there ever was one (
In Eucl., 213.7–11,
transl. by G. Morrow).
As follows from Proclus’ fasi, we have here a reference to a source that seems
very close to Eratosthenes’ evidence. In both cases, Hippocrates is called the
pro¯tos heurete¯s of the problem of doubling the cube, and this was one of Eude-
mus’ standard methods of describing mathematical and astronomical dis-
coveries.
36
In both cases, @pórhma, a difficult geometric construction, is men-
tioned, as well as a problem to which this puzzle was further reduced. It is note-
worthy that, for the first time, @pagwg2 occurs in Aristotle (APo 69a 20 f.),
who brings as an example of its application the problem of squaring the circle
with the help of the lunes (69a 30–34), i.e., the famous problem of Hippo-
crates.
37
It is only natural that Aristotle’s student also applied the term
@pagwg2 to Hippocrates’ method. This is a further proof that Proclus’ note de-
rives from Eudemus’
History of Geometry. The words of admiration for Hippo-
crates’ talent, much more suitable for the classical than for the later author, are
probably Eudemus’ as well.
34
Eutoc.
In Archim. De sphaer. III, 88.18–23 = 42 A 4, transl. by W. Knorr (cf. above,
85).
35
Eud. Fr. 141, com. ad loc.; Knorr
AT, 21.
36
See above, 149.
37
Cf. Arist.
SE 171b 12f. = 42A3; see below, 177 n.45. The same method, which Plato
calls ëx ûpoqésew~, is said to be generally accepted in geometry (Men. 86e–87c).
See Knorr.
AT, 71f.

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Chapter 5: The history of geometry
176
In Diogenes Laertius’ biography of Archytas, there is an interesting passage
where Archytas is called prõto~ twice:
He was the first to make mechanics into a system by applying mathematical prin-
ciples;
38
he also first employed mechanical motion (kínhsi~ örganik2) in a geo-
metrical construction, namely, when he tried, by means of a section of a half-cyl-
inder, to find two mean proportionals in order to duplicate the cube (VIII, 83,
transl. by R. Hicks).
Diogenes’ immediate source is likely to have been Favorinus, who was very in-
terested in various eûr2mata and mentioned elsewhere Archytas’ studies in
mechanics.
39
Favorinus, in turn, could have relied on Eratosthenes, who cer-
tainly used Eudemus.
40
The second part of Diogenes’ passage closely matches
both Eratosthenes’ letter that also mentions half-cylinders
41
and Archytas’ so-
lution to the problem of doubling the cube, known from Eudemus (fr. 140).
Does the first part of the passage that refers to Archytas’ pioneering work in
mechanics come from Eudemus too? An obvious symmetry between the two