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

Elements, to Theaetetus, a student of Theodorus. It was also Theaetetus who,
adding the icosahedron and the octahedron to the three regular solids known to
the Pythagoreans (
Schol. in Eucl., 654.3), developed a general theory of regular
solids (book XIII).
The discoveries of Theaetetus’ contemporary Leodamas remain unknown to
us. The stories in which Leodamas figures as the receiver of the method of
analysis developed by Plato are unlikely to be true and can hardly go back to
Eudemus.
182
The
Catalogue does not mention the discovery of analysis. Its ap-
plication is associated with Eudoxus, not Leodamas, though the former is not
named as its inventor. This implies that Eudemus related analysis to an earlier
period, but it remains unclear whether he associated its discovery with Leoda-
mas or Hippocrates. The author of the Academic work quoted by Philodemus,
as we remember, dated the discovery of analysis to Plato’s time (3.1).
Archytas, who proceeded from the results of Hippocrates’ research, was the
first to solve the problem of doubling the cube. His remarkable stereometrical
construction, which for the first time introduces movement into geometry, em-
ployed the intersection of the cone, the torus, and the half-cylinder, which pro-
179
Burkert, W. STOICEION. Eine semasiologische Studie, Philologus 103 (1959)
167–197.
180
Van der Waerden.
 Pythagoreer, 361f.
181
See Mueller. Remarks, 293f.: Aristotle does not discuss any of Euclid’s postulates.
182
D. L. III, 24 = Favor. fr. 25 Mensching; Procl.
In Eucl. 211.18f. See Mensching, op.
cit., 103f. On the application of analysis in the fifth century, see Allman, op. cit., 41
n. 62, 97f.; Heath.
History 1, 291; Cherniss. Plato as mathematician, 418 f. See
above, 92 f.

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Chapter 5: The history of geometry
206
duced the necessary curve.
183
Eutocius cites Archytas’ solution with reference
to Eudemus (fr. 141); this text, however, unlike the verbatim quotation in Sim-
plicius, bears obvious traces of a later revision.
184
Diogenes Laertius’ evidence
that Archytas was “the first to employ mechanical motion (kínhsi~ örganik2)
in a geometrical construction, when he tried, by means of a section of a half-
cylinder, to find two mean proportionals in order to duplicate the cube” (VIII,
83) might also go back, through Favorinus and Eratosthenes, to Eudemus, as
might the related reference to Archytas as the founder of mechanics.
185
It is also
very probable that Archytas figured in the Eudemian outline of the history of
proportions, which is preserved in Iamblichus.
186
We know nothing about the discoveries of Neoclides, who follows Archytas
in the
Catalogue. Eudemus ascribes to Leon, his disciple: 1) the authorship of
new
Elements exceeding the older ones “both in number and in the utility of
propositions proved in it”, and 2) the discovery of the method of diorism,
allowing one “to determine when a problem under investigation is capable of
solution and when it is not” (
In Eucl., 66.22f.). A particular interest in com-
pilers of
Elements manifested in the Catalogue can only partly be explained by
the fact that this text formed part of Porphyry’s commentary on the
Elements.
Eudemus obviously regards the appearance of new and improved
Elements as
progress in geometry. Some historians of mathematics believe that such basic
principles of mathematics as axiom, postulate, hypothesis had already been ac-
knowledged and defined in Leon’s