Page:Carroll - Euclid and His Modern Rivals.djvu/85

. II. § 5.] Min. That is so.

Euc. Then, for anything we know to the contrary, class  may be larger than class '.' Hence, if you assert anything of class ',' the logical effect is more extensive than if you assert it of class : for you assert it, not only of that portion of class  whitch is known to be included in class ',' but also of the unknown (but possibly existing) portion which is not'' so included.

Min. I see that now, and consider it a real and very strong reason for preferring your axiom.

But so far you have only answered Playfair. What do you say to the objection raised by Mr. Potts? 'A stronger objection appears to be that the converse of it forms Euc. I. 17; for both the assumed Axiom and its converse should be so obvious as not to require formal demonstration.'

Euc. Why, I say that I deny the general law which he lays down. (It is, of course, the technical converse that he means, not the logical one. 'All X is Y has for its technical converse 'All Y is X; for its logical, 'Some Y is X.') Let him try his law on the Axiom 'All right angles are equal,' and its technical converse 'All equal angles are right'!

Min. I withdraw the objection.

§ 6. The Principle of Superposition.

Min. The next subject is the principle of 'superposition.' You use it twice only (in Props. 4 and 8) in the First