ARISTOTLE
339
elementary as I. 5, although one would say that the assump
tions are no more obvious than the proposition to be proved ;
indeed some kind of proof, e. g. by superposition, would
doubtless be considered necessary to justify the assumptions.
It is a natural inference that Euclid’s proof of I. 5 was his
own, and it would appear that his innovations as regards
order of propositions and methods of proof began at the very
threshold of the subject.
There are two passages 1 in Aristotle bearing on the theory
of parallels which seem to show that the theorems of Eucl.
I. 27, 28 are pre-Euclidean ; but another passage 2 appears to
indicate that there was some vicious circle in the theory of
parallels then current, for Aristotle alludes to a petitio prin-
cipii committed by ‘ those who think that they draw parallels ’
(or ‘ establish the theory of parallels toly napaXXijXovs
ypd(f)€Lv), and, as I have tried to show elsewhere,^ a note of
Philoponus makes it possible that Aristotle is criticizing a
direction-theory of parallels such as has been adopted so
often in modern text-books. It would seem, therefore, to have
been Euclid who first got rid of the petitio principii in earlier
text-books by formulating the famous Postulate 5 and basing
I. 29 upon it.
A difference of method is again indicated in regard to the
theorem of Eucl. III. 31 that the angle in a semicircle is right.
Two passages of Aristotle taken together 4 show that before
Euclid the proposition was proved by means of the radius
drawn to the middle point of the
arc of the semicircle. Joining the
extremity of this radius to the ex
tremities of the diameter respec
tively, we have two isosceles right-
angled triangles, and the two angles,
one in each triangle, which are at the middle point of the arc,
being both of them halves of right angles, make the angle in
the semicircle at that paint a right angle. The proof of the
theorem must have been completed by means of the theorem
1 Anal. Post. i. 5. 74 a 18-16 ; Anal. Prior, ii. 17. 66 a 11-15.
2 Anal. Prior, ii. 16. 65 a 4.
3 See The Thirteen Books of Euclid's Elements, vol. i, pp. 191-2 (cf.
pp. 308-9).
4 Anal. Post. ii. 11. 94 a 28; Metaph. 6. 9. 1051 a 26.