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

History of Geometry. Eudemus, as we know, num-
bered Hippocrates among the earliest geometers
177
and treated the squaring of
lunes in book II of the
History of Geometry. Most of the mathematicians men-
tioned in this work belong to the period after Hippocrates, so that if their dis-
coveries were treated at comparable length, one may suppose that the
History
of Geometry comprised at least four books.
178
The first book might have con-
174
Ibid., 59.19–24, transl. by I. Bulmer-Thomas.
175
“That in Eudemus’ text not everything said is proved, corresponds to the character of
Eudemus as a historian.” (Becker. Zur Textgestaltung, 415). At any rate, Simplicius’
interventions in Eudemus’ text are more or less equally distributed among all four of
Hippocrates’ demonstrations, no matter how fully they were reported by Eudemus.
176
Rudio, F.
Der Bericht des Simplicius über die Quadraturen des Antiphon und des
Hippokrates, Leipzig 1907 (summarizes the earlier works of Allman, Tannery, and
Heiberg); Björnbo. Hippokrates; Heath.
History 1, 183ff.; Becker. Zur Textgestal-
tung, 411–419; ibid.
Denken, 58f.; Böker, R. Würfelverdoppelung, RE 9 A (1961)
1198f.; Bulmer-Thomas, I. Hippokrates of Chios,
DSB 6 (1972) 410–418; Knorr.
AT, 26ff. Recently, Reviel Netz (Eudemus of Rhodes, Hippocrates of Chios and the
earliest form of a Greek mathematical text,
Centaurus 46 [2004] 243–286) offered a
very ingenious but admittedly speculative account of how Eudemus’ report relates to
Hippocrates’ text. Cf. Federspiel, M. Sur la locution ëf’ o0 / ëf’ fl servant à de-
signer des êtres géometriques par des lettres,
Mathématiques dans l’Antiquité, ed. by
J.-Y. Guillaumin, Saint-Étienne 1992, 9–25.
177
In Phys.,
69.23f.). Netz (Eudemus) seemed to overlook this evidence, though it could rein-
force his case, viz. that Hippocrates was in a sense
the first Greek geometer. Now,
Eudemus might have considered Hippocrates to be one of the earliest mathematical
writers whose works were available in the late fourth century, but hardly the founder
of Greek geometry as it was known to him.
178
This corresponds to the number of books given in the list of Theophrastus’ works (fr.
264 No. 3 FHSG). See above, 166 n. 2–3.

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4. Early Greek geometry according to Eudemus
205
sidered discoveries by Thales and the Pythagoreans, the second geometers of
the second part of the fifth century, and the last two geometry of the fourth cen-
tury.
Hippocrates’
 Elements is the first of a number of mathematical treatises
under the same title that ultimately led to Euclid’s collection. The title itself,
Stoice$a, i.e., the basic, fundamental elements, indicated its task: to organize
interrelated mathematical propositions in their logical sequence, starting from
the very basic ones.
179
Apart from the axioms and definitions, Hippocrates’
Elements might have contained the first three construction postulates,
180
al-
though the last point is disputable.
181
At any rate, by the last decades of the
fifth century, geometry acquired features of a truly scientific axiomatico-de-
ductive system.
Theodorus of Cyrene, a contemporary of Hippocrates, appears in the
Cata-
logue probably owing to his contribution to the theory of irrationals. According
to Plato (
Tht. 147d), Theodorus had proved the irrationality of magnitudes
from
√
3 to
√
17. This is practically all we know about his discoveries in ge-
ometry. Even if Eudemus provided further information on Theodorus, it did not
attract the attention of the later commentators. The historian attributes the cre-
ation of the general theory of irrationals (fr. 141.I), set forth in book X of the