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

Particularly interesting is the fragment of Archytas’ work Perì maqh-
mátwn (47 B 3), where he formulates in a concise and aphoristic manner the
main notions and ideas of the contemporary theory of técnh. The first part of
the fragment deals with the scientific method, or, to be more precise, with the
cognitive method as such; the second dwells upon the usefulness of arithmetic
and on the importance of its discovery for social life and morality. Let us turn to
the title of the work first. Initially, máqhma, a passive derivative from the verb
manqánw, denoted the result (‘what has been learned’), or the subject of study
and could refer to different fields of knowledge.
75
It is in this initially large
sense, that the title of Protagoras’ Perì tõn maqhmátwn (D. L. IX, 55) is to be
understood: what is meant here is not mathematics, but various branches of
learning.
76
In Archytas, the word maq2mata acquires terminological character
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Cf. @riqmhtik3 técnh (Pl. Ion. 531e 3, Gorg. 451b 1, Tht. 198a 5, Pol. 258d 4, Phil.
55e 1); gewmetrik3 técnh (Charm. 165e 6).
71
t3n dè sofían Én te ta$~ técnai~ to$~ @kribestátoi~ tà~ técna~ @podídomen
(Arist.
EN 1141a 9).
72
“The analysis of certain classes of problems in geometry, e.g. the construction of ir-
rational lines, can only be completed by means of arithmetical principles.” (Knorr,
W. R.
The evolution of the Euclidean Elements, Dordrecht 1979, 311). For Aristotle,
too, arithmetic is more exact than geometry (
APo 87a 34f., Met. 982a 26f.). Philo-
laus, on the contrary, singled out geometry as the ‘source’ and the ‘mother-city’ of
all mathematical sciences: 44 A 7a; Huffman, C. A.
Philolaus of Croton. Pythago-
rean and Presocratic, Cambridge 1993, 193f.
73
Cf. above, 57 n. 52 and saf4 diágnwsi~ in Archytas (47 B 1).
74
See
Bus. 37 (cf. Hel. 61, Antid. 243) and Antid. 118, 273. With regard to the gods,
eıdénai tò safé~ is impossible (Nic. 26), but here too there is shme$on, allowing us
to form judgments. Cf. Alcmaeon (24 B 1);
De arte 12.
75
Snell,
op. cit., 76.
76
Burkert.
L & S, 207 n. 80. maq2mata was used later with the same meaning: Isoc.
Antid. 10, 267; Pl. Lach. 108c, Soph. 224c, Leg. 820b. For Hellenistic inscriptions

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3. Archytas and Isocrates
63
and designates a particular group of sciences including arithmetic, geometry,
astronomy, and harmonics, all of which he regards as akin (B 1 and 4, cf. Pl.
Res. 530d). Affixing the term maq2mata to the sciences of the mathematical
quadrivium would have been impossible, had it not been preceded, first, by set-
ting them apart as a special group and, second, by turning them into the subjects
of learning.
The origin of the quadrivium has been a subject of long discussion. Some
scholars date it to the time of the Sophists, others to the time of Plato, and still
others relate it directly to Plato.
77
Indeed, the particular importance that he ac-
corded to mathematical sciences is manifest already in the relatively early
 Re-
public: the ten years dedicated to the study of mathe¯mata were to prepare the
future guardians of the ideal state to master the main science, dialectic. Still,
Plato, though an expert in mathematical sciences and their advocate, hardly
taught them himself.
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Besides, he never claimed for himself the honor of being
the discoverer of the quadrivium. On the contrary, he mentions repeatedly that
geometry, arithmetic, astronomy, and harmonics were taught by Hippias of Elis
and Theodorus of Cyrene.
79
There are no reasons to doubt that Hippias, a remarkable polymath for his
time, taught all the disciplines of the quadrivium. Since it has become clear,
however, that the mathematician Hippias who discovered the curve called
quadratrix is not to be identified with the Sophist Hippias of Elis,
80
one can
hardly ascribe to the latter any discoveries in mathematics. Meanwhile, the
uniting of the four sciences into a special group can only have been effected by
someone who directly took up not only mathematics, but (mathematical) as-
tronomy and harmonics as well, since the intrinsic relationship between the
latter two sciences is far from evident to a layman. It is not among the Sophists,
who picked up and developed the already existing tradition,
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but among the
Pythagorean mathematicians that the origin of the quadrivium is to be sought.
82
relating to school teaching, see Grassberger, L.