Page:Carroll - Euclid and His Modern Rivals.djvu/153

. § 1.] Nie. And you allow that intersectional Lines have 'different directions'?

Min. Yes. Are you going to argue, from that, that separational Lines must have 'the same direction'? Why may I not say that intersectional Lines have one kind of 'different directions' and that separational Lines have another kind?

Nie. But do you say it?

Min. Certainly not. There is no evidence, at present, one way or the other. For anything we know, Pairs of separational Lines may always have 'the same direction,' or they may always have 'different directions,' or there may be Pairs of each kind. I fear I must decline to grant the first part of your Axiom altogether, and the second part in the sense of referring to Lines not known to have a common point. You may now proceed.

P. 11. Ax. 7. 'Two different straight Lines which meet one another have different directions.'

Min. That I grant you, heartily. It is, in fact, a Definition for the phrase 'different directions,' when used of Lines which have a common point.

P. 11. Ax. 8. 'Two straight Lines which have different directions would meet if prolonged indefinitely.'

Min. Am I to understand that, if we have before us a Pair of finite Lines which are not known to have a common point, but of which we do know that they have