224 orna harari
Demonstration (I speak of the most perfect type of demonstration) must depend
upon those things which are per se and proper to that which is demonstrated;
indeed, those things which are accidental and in common are excluded from
perfect demonstrations ... Th e geometer proves that the triangle has three angles
equal to two right ones on account of the fact that the external angle which results
from extending the side of that triangle is equal to two angles of the same triangle
which are opposed to it. Who does not see that this medium is not the cause of the
property which is demonstrated? . . . Besides, such a medium is related in an alto-
gether accidental way to that property. Indeed, whether the side is produced and
the external angle is formed or not, or rather even if we imagine that the production
of the one side and the bringing about of the external angle is impossible, nonethe-
less that property will belong to the triangle; but what else is the defi nition of an
accident than what may belong or not belong to the thing without its corruption? 39
Pereyra’s argument for the inconformity of mathematical proofs to
Aristotelian demonstrations is similar to Proclus’ argument. Like Proclus,
Pereyra focuses on the question whether mathematical proofs meet the
non-formal requirements of the theory of demonstration. More specifi cally,
he raises the two questions that were at the centre of Proclus’ discussion of
this issue: (1) Do the premises of mathematical proofs state essential or
accidental relations? (2) Are Euclid’s proofs, which are based on auxiliary
constructions, explanatory? Th ese questions are viewed in this passage as
interrelated; real explanations are provided when the relation between a
mathematical entity and its property is proved to be essential. Th is require-
ment is met if the premises on which the mathematical proof is based state
essential relations. Th e only allusion to the syllogistic form of inference
made in this passage is to the middle term in syllogistic demonstrations.
However, like Proclus, Pereyra considers the middle term only in its role
as the cause of the conclusion. Its formal characteristics, such as its posi-
tion, are not discussed here. Th us, pre-modern and modern discussions
of the conformity of mathematical proofs to Aristotelian demonstrations
concern diff erent facets of the theory of demonstration. Whereas the
modern discussions focus on the formal structure of Aristotelian demon-
strations, pre-modern discussions concern its non-formal requirements.
Accordingly, the questions asked in these discussions are diff erent. Th e
modern question is whether syllogistic inferences can accommodate rela-
tional terms whereas the pre-modern question is whether mathematical
proofs establish essential relations.39 Th e translation is based on Mancosu 1996 : 13. Th e complete Latin text appears on p. 214, n. 12
of Mancosu’s book.