Page:Carroll - Euclid and His Modern Rivals.djvu/168

130 You assert, as axiomatic, that different Lines exist, whose relationship of direction is identical with that which exists between coincidental Lines.

Nie. Yes.

Min. Now, does the phrase 'the same direction,' when used of two Lines not known to have a common point, convey to your mind a clear geometrical conception?

Nie. Yes, we can form a clear idea of it, though we cannot define it.

Min. And is that idea (this is the crucial question) independent of all subsequent knowledge of the properties of Parallels?

Nie. We believe so.

Min. Let us make sure that there is no self-deception in this. You feel certain you are not unconsciously picturing the Lines to yourself as being equidistant, for instance?

Nie. No, they suggest no such idea to us. We introduce the idea of equidistance later on in the book, but we do not feel that our first conception of 'the same direction' includes it at all.

Min. I think you are right, though Mr. Cuthbertson, in his 'Euclidian Geometry,' says (Pref. p. vi.) 'the conception of a parallelogram is not that of a figure whose opposite sides will never meet…, but rather that of a figure whose opposite sides are equidistant.' But do you feel equally certain that you are not unconsciously using your subsequent knowledge that Lines exist which make equal angles with all transversals?

Nie. We are not so clear about that. It is, of course, extremely difficult to divest one's mind of all later