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

Elements. In their entirety, the frag-
ments do not cover even one-tenth of the material that – judging by the
Cata-
logue of geometers in Proclus
12
– was presented in the
History of Geometry. Of
the twenty mathematicians mentioned in the
Catalogue,
13
we find only six in
the fragments, including Antiphon, who is omitted from the
Catalogue.
14
In reconstructing the original scope of the
History of Geometry, we can rely
on these fragments as solid ground, yet we cannot confine ourselves solely to
them. It is well known that, for the late authors, Eudemus was one of the main
sources, if not
the main source of information on pre-Euclidean geometry. This
does not mean, of course, that any anonymous evidence concerning early Greek
mathematics goes back to Eudemus. Nevertheless, there are many cases in
which his authorship seems firmly established. Proclus, for example, informs
us about two of Thales’ theorems with a reference to Eudemus (fr. 134–135)
and about two others without mentioning his name (
In Eucl., 157.10f.,
250.20f.). It was suggested long ago that the latter two pieces of evidence are
also based on Eudemus’ authority,
15
which seems to me rather obvious. The
same conclusion can be reached about two of Oenopides’ discoveries, one of
which Proclus mentions with a reference to Eudemus (fr. 138) and the other
without it (
In Eucl., 283.7f.).
16
It is also very possible that Eutocius, who cites
Archytas’ solution to the problem of doubling the cube with reference to Eude-
mus (fr. 141), ultimately owes his information about the solutions of Eudoxus
and Menaechmus to the same source.
17
Here is another example: who was the authority for Proclus’ information
that the Pythagoreans knew the theorem that only the following polygons can
fill up the space around a point: six equilateral triangles, four squares, and
three equilateral equiangular hexagons (
In Eucl., 304.11f.)? There is no such
theorem in Euclid, but his older contemporary Eudemus could have referred to
it, since it follows immediately from the theorem on the equality of the angles
of the triangle to two right angles (I, 32), which he ascribes to the Pythago-
11
See also Eudemus’ reference to Hippocrates, omitted by Wehrli: EÚdhmon ën to$~ palaiotéroi~ aÿtòn @riqme$n (Simpl. In Phys., 69.23f.).
12
Procl.
In Eucl., 64.16–68.23 = Eud. fr. 133.
13
I hesitantly include in this number Philip of Opus, but not Plato; see above, 3.2 and
below, 5.3. Hippias of Elis is mentioned here only as a source, not as a mathema-
tician.
14
The
Catalogue considers those who contributed to the progress of geometry,
whereas Antiphon is only known for his unsuccessful attempt to square the circle.
15
Pesch, J.G. van.
De Procli fontibus (Diss.), Leiden 1900, 78f.; Heath. Elements I, 36.
16
Van Pesch, ibid.; Heath, ibid.
17
Wehrli, com. ad loc.; Knorr
AT, 21. Probably through Eratosthenes, who derives
from Eudemus his knowledge of the solutions of Archytas, Eudoxus, and Menaech-
mus (3.1).

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2.
The History of Geometry: on a quest for new evidence
171
reans.
18
It seems very likely that two testimonies from the scholia to Euclid can
also be attributed to Eudemus: first, that book IV of the
Elements belongs to
the Pythagoreans, and second, that they constructed three of the five regular
solids (pyramid, cube and dodecahedron), to which Theaetetus added the oc-
tahedron and icosahedron.
19
Eudemus, as we know, wrote both on the Pytha-
goreans and on Theaetetus; besides, this version contradicts the later erroneous
tradition, which ascribed to Pythagoras the construction of all five regular
solids (Procl.