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

Erziehung und Unterricht im klas-
sischen Altertum, Würzburg 1881, 315, 437.
77
See e.g. Merlan, P.
From Platonism to Neoplatonism, The Hague 1960, 88; Burkert.
L & S, 422f.
78
See below, 101f.
79
Prot. 316d–e, Hipp. Mai. 285b, Hipp. Min. 366c, 368e; Tht. 145c.
80
Knorr.
AT, 82f.
81
It is unlikely that any of the Sophists, except for Hippias, taught geometry, let alone
arithmetic and harmonics (see above, 45 n. 2). In Aristophanes (
Nub. 200f.), the en-
trance into Socrates’ ‘thinking shop’ is flanked by the statues of geometry and as-
tronomy, which does not mean, however, that all the four
mathe¯mata were integrated
into the Sophists’ educational curriculum before the end of the fifth century (so Bur-
kert,
L & S, 421). In Athens there were other experts in astronomy and geometry:
Meton (Ar.
Av. 997) and Euctemon, Hippocrates of Chios (42A 2, 5) and Theodorus
of Cyrene (43 A 3–5).
82
Heath, T. L.
 A history of Greek mathematics, Vol. 1, Oxford 1921, 10f. The scientific
meaning of the word maq2mata goes back to the Pythagoreans (Snell, op. cit., 77f.).

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Chapter 2: Science as técnh: theory and history
64
Soon after Pythagoras added arithmetic and harmonics to Ionian astronomy
and geometry,
83
the
mathe¯mata began to form a special group. As clearly fol-
lows from Philolaus’ fragments, he was very familiar with all the four sciences
of the Pythagorean quadrivium. If we combine this with a tradition according to
which Pythagoras took up all the four
mathe¯mata and Hippasus at least three of
them (geometry, arithmetic, and harmonics, 18 A 4, 12–15), then we come to
the conclusion that as a young man, i.e., before his escape to Thebes (ca. 450
BC), Philolaus was brought up within the framework of the Pythagorean math-
ematical quadrivium. Distinct traces of this sort of education are also found
among his contemporaries, the Pythagorean Theodorus (43 A 2–5) and Demo-
critus, who had Pythagorean teachers (68 A 1, 38).
84
On the other hand, there is
no evidence that the Ionian mathematicians Oenopides and Hippocrates of
Chios took up arithmetic and harmonics; the very word maq2mata was hardly
ever used in the Ionian dialect.
85
Let us now return to the text of the fragment, which by all appearances
comes from the introductory part of Archytas’ work. Like many other works
belonging to the genre perì técnh~, the introduction to Perì maqhmátwn lays
out the major methodological principles of the sciences considered in it and
points out their characteristic features, most of which coincide with those of
other técnai. To judge from the fact that Archytas does not bother to explain
and demonstrate them in detail, by the time when Perì maqhmátwn was
brought out (presumably, at the close of the fifth century), the Sophistic theory
of técnh was more or less well-known.
86
The first part of the fragment dis-
cusses, in succession, different ways of acquiring knowledge:
de$ gàr 9 maqónta par^ Állw 9 aÿtòn ëxeurónta, %n @nepistámwn 7sqa,
ëpistámona genésqai. tò mèn %n maqèn par^ Állw kaì @llotríai, tò dè ëx-
eurèn di^ aÚtauton kaì ıdíai. ëxeure$n dè m3 zatoñnta Áporon kaì spá-
nion, zatoñnta dè eÚporon kaì ®áidion, m3 ëpistámenon dè zhte$n @dúna-
ton.
According to Anatolius (an Aristotelian of the third century AD renowned for his
mathematical learning), the Pythagoreans gave the name maqhmatik2 to arithmetic
and geometry, which heretofore had not been referred to by a single term (Ps.-Heron.
Def., 160.23–162.5).
83
Isocrates presents Pythagoras as a disciple of Egyptian priests, among whose occu-
pations he mentions astronomy, arithmetic, and geometry (