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

History 1, 175f.
159
Knorr.
AT, 15f.
160
He shared also some astronomical views of the Pythagoreans (cf. 41 A 10 and 58 B
37c).
161
Szabó, Á., Maula, E. EGKLIMA. Untersuchungen zur Frühgeschichte der antiken
griechischen Astronomie, Geographie und Sehnentafeln, Athens 1982, 118f. It is
worth noting that IV, 16 differs somewhat from other propositions of this book. In all
of them, the construction of a polygon is followed by a proof that the constructed fig-
ure does possess the required qualities. Such proof is lacking in the case of IV, 16
(Neuenschwander. VB, 374).

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Chapter 5: The history of geometry
202
fr. 6 Lanata), to whom he probably owed his vast knowledge in mathematics.
162
Democritus, who asserted that “nobody excelled him in drawing lines with
proofs”, was the author of about ten mathematical works.
163
It is difficult, then,
to find any reason why Eudemus should have omitted him from his
History of
Geometry. If my hypothesis is true, he considered Democritus to be the author
of propositions that the cone is equal to one-third of the cylinder and the pyra-
mid to one-third of the prism with the same base and height (Eucl. XII, 7 and
10), while noting at the same time that the scientific demonstration of these the-
orems was given by Eudoxus.
164
The level of geometry in ca. 440–430 is better appreciated, not from the
scant information about Oenopides and Democritus, but rather from the non-
trivial problems of duplicating the cube and squaring the circle studied by
Hippocrates of Chios, a younger contemporary and probably a student of Oe-
nopides. These problems found responses outside the circle of geometers as
well. In the
Republic we find some hints at duplicating the cube, which later
gave rise to a legend about Plato’s part in solving this problem (3.4). Whereas
Aristotle remained indifferent to it, Eudemus attests the achievements of Hip-
pocrates and cites the solutions offered by Archytas, Eudoxus, and his stu-
dents. The problem of squaring the circle aroused still greater interest in wide
circles: Aristophanes mentioned it (
Av. 1004–1010), it preoccupied Antiphon
and Bryson, and Aristotle often referred to it.
165
The latter circumstance at-
tracted to the problem the attention of Aristotle’s commentators Alexander
and Simplicius, who brought to us Eudemus’ evidence on Antiphon and Hip-
pocrates. In the first case, Eudemus follows Aristotle’s judgment: Antiphon
does not admit the basic principles of geometry, in particular, that geometrical
magnitudes are infinitely divisible.
166
Unlike Aristotle, however, who accused
Hippocrates of having committed a logical mistake by squaring the circle with
the help of lunes (
SE 171b 12f.; Phys. 185a 14f.), Eudemus found no fault
with him.
167
162
Zhmud.
Wissenschaft, 40 n. 69. One of Democritus’ works is related to irrational
lines – the problem that only the Pythagoreans had treated before him. Democritus
wrote a book on Pythagoras (14 A 6 = fr. 154 Luria).
163
68 B 299 = fr. 137 Luria. The list of mathematical writings by Democritus includes
works on geometry, arithmetic, astronomy, and the theory of perspective (D. L. IX,
47–48).
164
See above, 177, and 68 B 155 on the cutting of a cone by parallel planes. Cf. fr. 125
Luria with commentary; Waschkies,
op. cit., 267ff.
165
Cat. 7b 31, APr 69a 30–34, APo 75b 40f., SE 171b 13–172a 8, Phys. 185a 16f., EE
1226a 29.
166
Fr. 140, p. 59, 9–12 Wehrli. For a modern appraisal of Antiphon’s approach, see
Heath.