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

Eratosthenes von Kyrene, Munich
2002, 175ff. Vitruvius wrote on the Delian problem without naming Plato (IX,1.
13–14).
8
Knorr.
TS, 11ff., offers more than ten solutions. To be sure, unlike Fermat’s theorem,
this one was already solved in the generation after Hippocrates.
9
Cf. oî parà tŒ Plátwni ën ^Akadhmí+ gewmétrai (Eutoc. In Archim. De sphaer.,
90.3).
10
Wolfer, E. P.
Eratosthenes von Kyrene als Mathematiker und Philosoph, Groningen
1954, 4ff.; Riginos,
op. cit., 141; Knorr. AT, 17ff., 49ff. Unlike Plutarch, Theon di-
rectly referred to this work (
Exp., 2.3). A recent work on Eratosthenes attempts to re-
fute – unsuccessfully, it seems to me – the idea that the
Platonicus was a dialogue
(Geus,
op. cit., 141–194, esp. 192). Even more difficult is to agree with the author’s
tendency to regard the
Platonicus as the only source for Eratosthenes’ mathematics,
denying, e.g., the existence of his work
On Means, attested by Pappus (Coll. VII,
636.24, 672.5, cf. 662.16).

---

1. Plato as architect of mathematical sciences?
85
the task of finding two mean proportionals between two given lines,
11
and the
brilliant solution to this last problem was first found by Archytas. Eudemus
gives a detailed account of it (fr. 141), so that the evidence on Eudoxus’ and
Menaechmus’ solutions that we find in Eratosthenes, as well as his mention of
Hippocrates, must go back to the same source.
12
Eudemus, however, does not
even mention Plato. To whom does the legend about three great mathematicians
of three subsequent generations (Eudoxus was a pupil of Archytas, and Me-
naechmus a pupil of Eudoxus), all working under Plato’s supervision, belong?
Was Eratosthenes its author or does it date back to an earlier time?
The answer is made more complicated, since Eratosthenes’ letter to King
Ptolemy III, preserved by Eutocius (
In Archim. De sphaer., 88.3–96.9), gives
an entirely different ending to the story. It states that Archytas, Eudoxus, and
Menaechmus proposed too
abstract solutions to the problem and therefore did
not deal with the problem in a practical and useful way, with the exception of
Menaechmus, though even he met practical criteria only to a very small degree
and with difficulty.
13
Knorr, who analyzed this text in great detail, convincing-
ly showed that the letter is not a later forgery (as Wilamowitz thought)
14
and
that it belongs to Eratosthenes.
15
Eratosthenes also studied the problem of du-
plicating the cube, and it is noteworthy that his own solution was mechanical.
He manufactured a device for drawing lines, the
mesolabe, and dedicated a
bronze model of it to King Ptolemy, accompanied with a letter and an epigram.
Eratosthenes’ solution correlates much better with the ‘mechanical’ ending of
the story than with the ‘anti-mechanical’ one presented by Plutarch, all the
more so because the epigram that is widely recognized as authentic also says
that Archytas’ solution was badly adapted to practice.
16
Hence Knorr con-
cludes that Eratosthenes had two versions: one more historically accurate, in
the letter to Ptolemy, and another, more literary version, recorded in the
Pla-
tonicus and carried down to us by Theon and Plutarch.
17
Knorr considers the
11
Eratosthenes, by the way, was well aware of this fact (Eutoc.
In Archim. De sphaer.,
88.18f.). For more details, see below, 175f.
12
Cf. Eud. fr. 139–140. See below, 175f., 207.
13
Eutoc.