Page:Carroll - Euclid and His Modern Rivals.djvu/84

46 '.' Lines, which have property ',' meet if produced; for, if not, there would be two Lines both separational from the same Line, which is absurd. Hence Lines, which have property ',' meet if produced.

Min. I see now that those who grant Playfair's Axiom have no right to object to yours: and yours is certainly the more simple one.

Euc. To make assurance doubly sure, let me give you two additional reasons for preferring my Axiom.

In the first place, Playfair's Axiom (or rather the Contranominal of it which I have been using, that 'a Line which intersects one of two separational Lines will also meet the other') does not tell us which way we are to expect the Lines to meet. But this is a very important matter in constructing a diagram.

Min. We might obviate that objection by re-wording it thus:—'If a Line intersect one of two separational Lines, that portion of it which falls between them will, if produced, meet the other.'

Euc. We might: and therefore I lay little stress on that objection.

Euc. In the second place, Playfair's Axiom asserts more than mine does: and all the additional assertion is superfluous, and a needless strain on the faith of the learner.

Min. I do not see that in the least.

Euc. It is rather an obscure point, but I think I can make it clear. We know that all Pairs of Lines, which have property ',' have also property ; but we do not'' know as yet (till we have proved I. 29) that all, which have property ',’ have also property '.'