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

 In Archim. De sphaer., 78.13–80.24. An anonymous solution via the intersec-
tion of two parabolas that Eutocius also cites does not belong to Menaechmus. See
Diocles. On burning mirrors, ed. and transl. by G.J. Toomer, New York 1976, 169f.;
Lasserre.
Léodamas, 552; Knorr. TS, 94f, 98.
198
Papp.
Coll. IV, 252.26ff. See Allman, op. cit., 180ff., Becker. Denken, 95; cf. Knorr.
AT, 80, 84f. Pappus took this construction from Sporus, whose ultimate source must
be Eudemus (Lasserre.
Léodamas, 561f.).
199
Geminus, while referring to Eratosthenes’ epigram (mhdè Menaicmeíou~ kwno-
tome$n triáda~), called Menaechmus the author of the theory of conic sections
(Procl.
In Eucl., 111.21f.; cf. Eutoc. In Archim. De sphaer., 96.17); the ‘triads’ are
usually taken to mean parabola, hyperbola, and ellipsis. Proclus too says that Me-
naechmus had solved the problem of doubling the cube by means of kwnikaì gram-
maí (
In Plat. Tim
., 34.1f.). See Schmidt. Fragmente; Allman,
op. cit.
, 166f.; Heath.
Apollonius
, xviiff; idem.
 History
1, 251f.; 2, 110; Becker.
Denken
, 82f. In the time
between Menaechmus and Euclid, Aristeas the Elder developed the theory of conic
sections (see above, 179). The names of the three curves (parabol2, ûperbol2,
Élleiyi~) go back to the Pythagorean application of areas (Eud. fr. 137).
200
Knorr.
AT, 61ff. See also Böker, op. cit., 1211f.
201
Knorr.
TS, 94ff.
202
Lasserre.
Léodamas, 550.
203
Procl.
In Eucl., 72.23–73.12 (on two meanings of the word ‘element’, with examples

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4. Early Greek geometry according to Eudemus
209
Nor, actually, is it certain that this Menaechmus is the same person as Eudoxus’
student. The
History of Geometry was concerned with mathematical dis-
coveries, not with discussions of differences between theorems and problems,
or the meanings of the word ‘element’. Besides, it is unlikely that Geminus
used this work of Eudemus at all.
204
The main source of the discussion related
by Geminus and figuring Amphinomus, Speusippus, Menaechmus, and Zeno-
dotus as its protagonists was undoubtedly Posidonius,
205
who had made use of
both classical and Hellenistic sources. The topics of this discussion were unlike
those treated in the treatises of professional mathematicians. Most likely, Posi-
donius relied here on the works of philosophers who took an interest in mathe-
matics (Speusippus, Posidonius, Geminus, and Proclus belonged to this cat-
egory) or on some Hellenistic introductory courses to mathematics. Though we
cannot rule out the possibility that Menaechmus wrote a popular treatise on
mathematics that reached Posidonius, the Stoic could well have meant a differ-
ent Menaechmus.
206
The group of Eudoxus’ students seems to have included the last author of
pre-Euclidean
Elements, Theudius of Magnesia, and his contemporary Athe-
naeus of Cyzicus.
207
All that is known about Theudius’
Elements is that they
were better arranged and gave a more general character to many partial theor-
ems (
In Eucl., 67.12f.).
208
Athenaeus of Cyzicus was “famous in mathematical
sciences, geometry in particular” (67.16f.), but we know nothing about his
specific discoveries. Hermotimus of Colophon extended the investigations of
Eudoxus and Theaetetus and discovered many new theorems of the
Elements
(67.20f.). Besides, he composed a writing on the so-called loci (tópoi), i.e.,
on the theory of geometrical places,
209
developed later by Aristeas the Elder
and Apollonius.
from Euclid!); 77.7–78.10 (all mathematical propositions are problems, the latter
being of two types), 253.16–244.5 (reversibility of theorems).
204
See above, 184f.
205
See above, 179 n. 54. This was admitted by Lasserre (