Page:Carroll - Euclid and His Modern Rivals.djvu/159

. § 1.]

P. 11. 'From this Definition, and the Axioms above given, the following results are immediately deduced:

(1) That parallel—I beg your pardon—that 'sepcodal' Lines would not meet however far they were produced. For if they met'

Min. You need not trouble yourself to prove it. I grant that, if such Lines existed, they would not meet. Your assertion is simply the Contranominal of Ax. 7 (p. 115), and therefore is necessarily true if the subject be real.

But remember that, though I have granted to you that, if we are given a Line and a point not on it, we can draw, through the point, a certain Line separational from the given Line, we do not yet know that it is the only such Line. That would take us into Table II. With our present knowledge, we must allow for the possibility of drawing any number of Lines through the given point, all separational from the given Line: and all I grant you is, that your ideal 'sepcodal' Line will, if it exist at all, be one of this group.

(2) 'That Lines which are sepcodal with the same Line are sepcodal with each other. For'

Min. Wait a moment. I observe that you say that such Lines are sepcodal with each other. Might they not be 'compuncto-codirectional'?

Nie. Certainly they might: but we do not wish to include that case in our predicate.