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

particular mathematical material. It is important to make clear what is meant by
‘geometry’ here. The section of the
Collection on ‘famous men’ could have
said that Thales, Mamercus, and Pythagoras became famous for their studies of
geometry, and that among the ‘barbarians’ the Egyptians excelled in geometri-
cal knowledge. It could even have contained references to some particular
problems studied by these mathematicians. What seems to me highly improb-
able is that Hippias quoted geometrical propositions at length, including their
proofs, e.g., Thales’ demonstration of the equality of the angles at the base of
the isosceles triangle. Eudemus clearly knew substantially more about the
proofs offered by Thales and by the other early geometers than could be in-
cluded in Hippias’ book, focused on looking for similarities between the ideas
of Greek and ‘barbarian’ poets and sages.
123
Eudemus’ information on the Pythagoreans is especially rich, which corre-
sponds to the role of this school in the development of early Greek geometry.
Eudemus ascribes the “transformation of geometry into the form of a liberal
education” and the discovery of the first three mean proportionals to Pythago-
ras himself, the study of proportions and (probably) the discovery of irra-
tionals to his student Hippasus.
124
The following discoveries are related to the
Pythagorean school as a whole: 1) the theorem that the sum of the interior
angles of a triangle is equal to two right angles (I, 32); 2) the theorem, omitted
from the
Elements, that the space around a point can only be filled up with six
triangles, four squares or three hexagons; 3) the theory of the application of
the areas set forth mainly in books I and II of the
Elements; 4) the entire book
120
See Becker.
Denken, 37f.
121
Panchenko. ˙Omoio~ and ômoióth~, 42f., believes that Hippocrates of Chios could
have mentioned Thales, but this remains a conjecture.
122
Patzer,
op. cit., 106f.
123
Eudemus certainly used the
Collection in his History of Theology (fr. 150).
124
Procl.
In Eucl., 65.15f.; Iambl. In Nicom., 100.19–101.9, 113.16f., 116.1f.,
118.23f.; Papp.
Comm., 63f.; Schol. In Eucl., 415.7, 416.13, 417.12f.

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4. Early Greek geometry according to Eudemus
195
IV of the
Elements; 5) the construction of three regular solids (cube, pyramid,
and dodecahedron); and 6) the beginnings of the theory of irrationals.
125
Be-
sides, the
Catalogue mentions two other Pythagorean mathematicians, Theo-
dorus and Archytas, who should be discussed separately. The abundance and
variety of this information shows that Eudemus used various sources. Since
Archytas approvingly refers to his Pythagorean predecessors in mathematics
(oî perì maq2mata), retelling their views (47 B 1 and A 17) and, elsewhere, a
theory by Philolaus’ student Eurytus (A 13), his writings might have contained
evidence on earlier geometers. Eudemus’ other possible source was Glaucus
of Rhegium, the author of the book on the history of music and poetry.
126
In
this field, Glaucus was a predecessor of Eudemus, and his book, used by Her-
aclides Ponticus and Aristoxenus,
127
must have been known to Eudemus as
well. It is revealing that Glaucus and Eudemus organize their material in a
similar way: both of them proceed chronologically, from one
pro¯tos heurete¯s
to another. Glaucus’ origin in Rhegium in Southern Italy might point to his Py-
thagorean connections, so it does not seem odd that in his work he mentions
Empedocles and the Pythagorean teachers of Democritus (fr. 5–6 Lanata).
Glaucus’ name is also attested in Aristoxenus’ description of Hippasus’ acous-
tical experiment:
Hippasus made four bronze discs in such a way that, while their diameters were
equal, the first disc was one-third as thick as the second (4:3), a half as thick as the
third (3:2), and twice as thick as the fourth (2:1). When struck, the discs sounded
in a certain consonance. It is also said that when Glaucus heard the notes pro-
duced by the discs, he was the first to master the art of playing on them (fr. 90).
Hippasus, as we noted,
128
made the discs in accordance with the musical pro-
portion (12:9 = 8:6), which was probably known to Pythagoras, and arrived at
the same intervals as the latter: the octave, the fifth, and the fourth. It is very
likely that Glaucus is the source of other two testimonies: 1) on Hippasus’ con-
temporary Lasus of Hermione, musician and theoretician of music; and 2) on
acoustical experiments carried out by Lasus and Hippasus.
129
Thus, Glaucus’
book could have provided Eudemus with information not only on Hippasus, but
also on the early theory of proportions, which sprang from harmonics and re-
mained closely connected with it for quite a long time.
130
It seems that Eudemus’ main source on the early Pythagorean mathematics
was a mathematical compendium that preceded Hippocrates’