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

Elements. In Euclid the theorem I, 32 combines
two propositions: 1) for any triangle, the exterior angle formed by the continu-
ation of any of its sides is equal to the two inner alternate angles; 2) the three in-
terior angles of a triangle are equal to two right angles. The Pythagorean proof,
more simple and elegant, concerns the second proposition only and uses a dif-
ferent geometrical construction. It is noteworthy that its only premise is the
equality of the alternate angles (I, 29), one of the most evident corollaries of the
properties of parallels:
Let ABG be any triangle and let us draw through A a
line DE parallel to BG. Since BG and DE are parallel,
the alternate angles are equal. Then, the angle DAB is
equal to the angle ABG and the angle EAG to the
angle AGB. Let us add to each sum the common angle
BAG. Therefore, the angles DAB, BAG, GAE, i.e., the
angles DAB, BAE, i.e., two right angles, are equal to
the three angles of the triangle ABG. Therefore, the
three angles of a triangle are equal to two right angles.
Eudemus, who was acquainted with several different versions of the
Elements,
realized, naturally, that I, 32 could be proved in various ways. If he nevertheless
says “the Pythagoreans demonstrated this theorem as follows”, it means that
the text he cites must go back to the early Pythagorean compendium, rather
than to the
Elements by Leon or Theudius.
139
This is confirmed by the fact that
one of Aristotle’s references to this theorem implies the proof given in Euclid,
and not the Pythagorean one.
140
As Neuenschwander has shown, the theorem I,
32 antedated Hippocrates’
Elements: it is quoted verbatim in the fourth book,
which comes from the Pythagoreans (IV, 15), as well as in III, 22, used by Hip-
pocrates.
141
The theorem that the angles of the triangle equal two right angles is
138
See below, 219 f.
139
Cf. van der Waerden.
 Pythagoreer, 337f.
140
Met. 1051a 24. See Heiberg, I. L. Mathematisches zu Aristoteles, Leipzig 1904, 19;
Heath.
Elements 1, 320.
141
Neuenschwander. VB, 333, 375f.; van der Waerden. Postulate, 353f. Mueller, I. Re-
marks on Euclid’s
Elements I, 32 and the parallel postulate, Science in Context 16
(2003) 292, is ready to date the Pythagorean proof to the middle of the fifth century. –
Proposition III, 31 (an angle in a semicircle is a right angle), containing a reference
to I, 32, was also known to the early Pythagoreans. Pamphila (first century AD) as-
cribed it to Thales, while others, including Apollodorus the Calculator, related it to
Pythagoras (D. L. I, 24–25, cf. VIII, 12). This Apollodorus may be identified with
Apollodorus of Cyzicus, an author of the later half of the fourth century BC (Burkert.
L & S, 428). Since it is III, 31 that follows from I, 32, and not vice versa, its attribu-

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Chapter 5: The history of geometry
198
one of Aristotle’s favorite examples: he refers to it about ten times in the
Ana-
lytics alone.
142
That is probably why Eudemus paid particular attention to it and
cited its earliest demonstration.
Aristotle’s reference to I, 32 gave rise to a curious episode in the histori-
ography of Greek geometry. Geminus alleged that the ancients had investigated
this theorem for each individual type of triangle: first, for the equilateral, then
for the isosceles, and finally for the scalene, whereas the later geometers had
demonstrated the general theorem.
143
Since the Pythagorean proof quoted by
Eudemus refers to the general case, the earlier stages were associated with
Thales and even with his Egyptian teachers. Geminus’ evidence might seem to
undermine our conclusion that he did not use Eudemus and showed little inter-
est to the history of pre-Euclidean geometry.
144
In fact, three stages of the proof
of I, 32 pointed out by Geminus are wholly fictitious, since he relied not on the
real but on the hypothetical example, given by Aristotle (
APo 74a 25f.).
145
The incommensurability of the side of a square with its diagonal, along with
theorem I, 32, was one of Aristotle’s favorite mathematical examples.
146
The
indirect proof of this theorem, to which he several times alludes (
APr 41a 24f.,
50a 37), relies on the Pythagorean theory of the odd and even numbers.
147
In
Aristotle’s time, this proof was still a part of the