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

Elements; later it was excluded
by Euclid, who employed Theaetetus’ general theory of the irrational magni-
tudes. Since Eudemus obviously took account of the mathematical examples
Aristotle often referred to, we can assume that in considering the Pythagorean
origin of the theory of irrationals he cited as an example the very indirect proof
that might be related to the early Pythagorean mathematical compendium.
Another proposition going back to this compendium is the theorem that only
three regular polygons – the triangle, the square, and the hexagon – can fill the
space around a point. It was not included in the
Elements, but Aristotle refers to
it (
Cael. 306b 5f.), adding that the same is true only for two regular solids, the
cube and the tetrahedron. The Pythagorean theorem Eudemus refers to is linked
tion to Thales remains highly doubtful (Heath.
History 1, 133f.; idem. Elements 1,
319). In any case, theorem III, 31, featuring mixed angles, belongs to the oldest stra-
tum of Greek geometry. Aristotle refers to it several times (
APo 94a 24f., Met. 1051a
26f.), implying a proof different from that of Euclid (Heath.
Mathematics, 71f.).
142
APr 66a 13, 67a 13–20; APo 71a 17, 71a 27, 73b 31, 74a 16–b 4, 85b 5f., b 11f.,
86b 25. For further references, see Heiberg.
Mathematisches zu Aristoteles, 19f.
143
Eutoc.
In Apollon. Con. comm. II, 170.4–8.
144
See above, 184 f.
145
Heiberg.
Mathematisches zu Aristoteles, 20; Heath. History 1, 135f.; idem. Elements
1, 319f.; idem.
Mathematics, 43ff.; Kullmann, W. Die Funktion der mathematischen
Beispiele in Aristoteles’
Analytica Posteriora, Aristotle on science, 255f. Cf.
Becker.
Denken, 39; Neuenschwander, E. Beiträge zur Frühgeschichte der grie-
chischen Geometrie I,
AHES 11 (1973) 127–133.
146
Heiberg.
Mathematisches zu Aristoteles, 24, gives more than 15 references to this
theorem.
147
Ibid., 24. See below, 223 n. 40.

---

4. Early Greek geometry according to Eudemus
199
with both I, 32 and Thales’ theorem on the equality of the alternate angles (I,
15). The latter has a corollary that the space around a point is divided into
angles whose sum equals four right angles. It follows from I, 32 that the angle
of an equilateral triangle is equal to
2
⁄
3
of the right angle, which means that the
sum of six such triangles is equal to four right angles. Accordingly, the angle of
a hexagon equals 1
1
⁄
3
of the right angle, and the angle of a square is right, so that
the area around a point can be filled by either three hexagons or four squares.
All other regular polygons, as Proclus notes (
In Eucl., 304.11f.), give in sum
either more or less than four right angles.
The entire book IV of the
Elements on the relations between the regular
polygons and the circle, which Eudemus ascribes to the Pythagoreans, must be
a part of the early Pythagorean compendium, as well. It is evident, at any rate,
that it was written before Hippocrates, who used it in his attempt to square the
lunes, and underwent only insignificant revision later.
148
The theory of the application of areas with excess and deficiency (6 te
parabol3 tõn cwríwn kaì 1 ûperbol3 kaì 1 Élleiyi~), which Eudemus,
pointing out its antiquity (Ésti mèn @rca$a), attributed to the ‘Pythagorean
muse’ (fr. 137), relates to the transformation of areas into equivalent areas of
different shape.
149
The propositions of this theory, comprising theorems I,
44–45, the entire book II of the