220 orna harari
Proclus’ reason for regarding this Euclidean proof as based on signs rather
than on causes concerns the relationship between the auxiliary construc-
tion employed in this proof and the triangle. According to Proclus, the
extension of the triangle’s base is merely a sign and not a cause of the equal-
ity of the triangle’s angles to two right angles because ‘there is a triangle
even if its side is not extended’. Th e exact force of this statement is clarifi ed
in Proclus’ discussion of the employment of this auxiliary construction in
another Euclidean proof – the proof that the sum of any two interior angles
of a triangle is less than two right angles ( Elements i .17). In this discussion,
Proclus claims that the extension of the triangle’s base cannot be considered
the cause of the conclusion since it is contingent: the base of a triangle may
be extended or not, whereas the conclusion that the sum of any two inte-
rior angles of a triangle is less than two right angles is necessary.^31 H e n c e ,
in questioning the conformity of certain Euclidean proofs to Aristotelian
demonstrations, Proclus raises the two questions that Philoponus ignores
in the case of mathematical demonstrations. Unlike Philoponus, Proclus
asks whether the middle term in Euclid’s proofs is the cause of the conclu-
sion and whether it is essentially related to the triangle.
Furthermore, Proclus’ attempt to accommodate Euclid’s proofs of the
equality of the sum of the interior angle of a triangle to two right angles
with Aristotle’s requirement that demonstrations should establish essential
relations indicates that he shares with Philoponus the assumption that
demonstrations regarding material entities require an appeal to causal con-
siderations. In concluding his lengthy discussion of Euclid’s proof that the
sum of the interior angles of a triangle is equal to two right angles, Proclus
says:
We should also say with regard to this proof that the attribute of having its interior
angles equal to two right angles holds for a triangle as such and in itself. For this
reason, Aristotle in his treatise on demonstration uses it as an example in discuss-
ing essential attributes ... For if we think of a straight line and of lines standing in
right angles at its extremities, then if they incline so that they generate a triangle we
would see that in proportion to their inclination, so they reduce the right angles,
which they made with the straight line; the same amount that they subtracted from
these [angles] is added through the inclination to the angle at the vertex, so of
necessity they make the three angles equal to two right angles. 32
Th e procedure described in the passage, in which a triangle is generated
from two perpendiculars to a straight line that rotate towards each other31 311.15–21, Friedlein.
32 384.5–21, Friedlein.