Philoponus and Aristotelian demonstrations 209
are two right angles? We know it from the defi nition of right angles, [stating] that
when a straight line set up on a straight line makes the adjacent angles equal to each
other, the two equal angles are right. Well, having brought [the conclusion] back to
the defi nition and the principles of geometry, we no longer inquire further, but we
have the triangle proved from geometrical principles. 5
In showing that Euclid’s proof conforms to the Aristotelian model of dem-
onstration, Philoponus focuses on two issues: (1) he presents Euclid’s proofs
in a syllogistic form, and (2) he grounds the proved proposition in the
defi nition of right angle. Th e notion of fi rst principles, on which Philoponus’
account is based, includes only one of the characteristics of Aristotelian fi rst
principles – namely, their being proper to the discipline. In Philoponus’
view, the dependence of Euclid’s geometrical proof on geometrical fi rst
principles, rather than on principles common to or proper to other disci-
plines, is suffi cient to establish that this proof conforms to the Aristotelian
model. Two other characteristics of Aristotelian fi rst principles are not taken
into account in this passage. First, Philoponus does not raise the question
whether the middle term employed in this proof is related essentially to the
subject of this proof; that is, he does not consider the question whether a
proposition regarding an essential attribute of adjacent angles can by any
means serve to establish the conclusion that this attribute holds essentially
for triangles. 6 Nor does he express any reservations concerning the auxiliary
construction, in which the base is extended and two adjacent angles are
produced. Second, Philoponus does not mention Aristotle’s requirement
that the fi rst principles should be explanatory or causal; he does not raise
the question whether the middle term in his syllogistic reformulation of
Euclid’s proof has a causal or explanatory relation to the conclusion. Th us
Philoponus’ account of the conformity of Euclid’s proofs to Aristotelian
demonstrations raises two questions: (1) why Philoponus ignores the ques-
tion whether mathematical propositions state essential relations; and (2)
why the causal role of the principles of demonstration is not taken into
account. Th e following two sections answer these questions respectively.
Essential predications
Philoponus addresses the question whether mathematical proofs establish
essential predications in his comments on the Posterior Analytics i .22. He
(^5) 116. 7–22, Wallies. All translations are mine.
(^6) For Philoponus’ syllogistic reformulation to be a genuine Aristotelian demonstration, one has
to assume that adjacent angles and triangles are related to each other as genera and species.
Th is assumption is patently false.