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

Elements and
125
Fr. 136–137; Procl.
In Eucl., 301.11f.; Schol. In Eucl., 273.3–13, 654.3; Papp.
Comm., 63f.
126
See above, 49 n. 18.
127
Aristox. fr. 90–91. See also Wehrli’s commentary on Heracl. Pont. fr. 157–163.
128
See above,174 n. 31.
129
1) Ps.-Plut.
De mus. 1141 C (through Heraclides Ponticus or Aristoxenus); 2) Theon.
Exp., 59.4f. = 18 A 13 (through Aristoxenus?).
130
See above,174 n. 30–31, cf. 47 B 2 and Eud. fr. 142.

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Chapter 5: The history of geometry
196
contained the basis of the first four books of Euclid.
131
In this Pythagorean com-
pendium, Eudemus could have found the information about specific proofs as
well as about entire theories or books, such as the theory of the application of
areas or book IV of the (future)
Elements. Apart from planimetrical proposi-
tions, the compendium must have included a number of stereometrical ones, at
least those that concerned the three regular solids discovered by the Pythago-
reans. And, still more important, the compendium contained the first explicitly
formulated definitions and axioms of geometry, laying the basis for geometrical
demonstration. Van der Waerden, in particular, ascribes to the Pythagoreans the
formulation of axioms 1–3 and 7–8.
132
Like the treatises of the Hippocratic corpus, the compendium, representing
the achievements of the school as a whole, did not contain the names of its auth-
ors. That is why most of the information we find in Eudemus on the Pythago-
reans does not concern the individual representatives of the school, but the Py-
thagoreans as a whole, by which we should understand the mathematicians of
the late sixth and the first half of the fifth century. For the same reason, Eude-
mus’ information on Pythagoras is very general, exactly like our notions of
Hippocrates of Cos as a doctor. Apart from the theory of proportions, closely
linked with Pythagoras’ acoustical experiments,
133
the historian mentions only
one, though very important, achievement of his: the transformation of ge-
ometry into the form of a liberal education (sc4ma paideía~ ëleuqérou),
which aimed at acquiring knowledge, rather than serving practical needs.
134
The last testimony implies that Eudemus knew about the role of Pythagoras’
school in the formation of the mathematical quadrivium.
135
The tradition on Py-
thagoras as an advocate of the
vita contemplativa is familiar to us through Ar-
istotle’s
Protrepticus;
136
elsewhere he testifies that “Pythagoras devoted him-
self to the study of mathematics, in particular of numbers”.
137
According to Ar-
istoxenus, Pythagoras was the first to turn arithmetic into a theoretical science
131
For its convincing reconstruction, see Waerden, B. L. van der. Die Postulate und
Konstruktionen in der frühgriechischen Geometrie,
AHES 18 (1978) 354ff. Van der
Waerden relied on Neuenschwander’s historical analysis of the first four books of the
Elements (Neuenschwander. VB).
132
Van der Waerden.
 Pythagoreer, 360f. According to Favorinus, Pythagoras was the
first to give definitions in geometry (D. L. VIII, 48). On the axiomatico-deductive
character of Pythagorean arithmetic, see below, 221 ff.
133
See Zhmud.
Wissenschaft, 187ff. Xenocrates credits Pythagoras with the discovery
of the numerical expression of musical intervals (fr. 87 Isnardi Parente).
134
Cf. Aristotle’s characterization of philosophy, mónhn ëleuqéran tõn ëpisthmõn
(
Met. 982b 27): philosophy like a free person exists for its own sake.
135
See above, 63 f. Already Isocrates had associated Pythagoras’ name with geometry,
arithmetic, and astronomy (
Bus. 23, 28).
136
The following fragment is particularly revealing: kalõ~ Ára katá ge toñton tòn
lógon Puqagóra~ eÍrhken !~ ëpì tò gnõnaí te kaì qewr4sai pã~ Ánqrwpo~
ûpò toñ qeoñ sunésthken (fr. 11 Ross = fr. 20 Düring, cf. fr. 18 Düring).
137
Puqagóra~ … diepone$to perì tà maq2mata kaì toù~ @riqmoú~ (fr. 191 Rose).

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4. Early Greek geometry according to Eudemus
197
(fr. 23), which gives us a further proof that both Eudemus and Aristotle relied
on a common tradition.
138
Ascribing to the Pythagoreans the discovery of the theorem about the sum of
the interior angles of the triangle, Eudemus quotes their proof (fr. 136), which is
different from that given in the