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.