Page:Carroll - Euclid and His Modern Rivals.djvu/148

110 Min. Well then, can you test whether a given Pair of Lines have this property? I mean, if I give you a certain diagram, and tell you its geometrical history, can you pronounce, on a certain Pair of finite Lines, which have no visible common point, as to whether they have this property?

Nie. We cannot undertake it.

Min. You ask me, then, to believe in the reality of a class of 'Pairs of Lines' possessing a property which you can neither define, nor construct, nor test?

Nie. We can do none of these things, we admit: but yet the class is not quite so indefinite as you think. We can give you a geometrical description of it.

Min. I shall be delighted to hear it.

Nie. We have agreed that a Pair of coincidental finite Lines have a certain relationship of direction, which we call 'the same direction,' and which you allow to be an intelligible geometrical relation?

Min. Certainly.

Nie. Well, all we assert of this new class is that their relationship of direction is identical with that which belongs to coincidental Lines.

Min. It cannot be identical in all respects, for it certainly differs in this, that we cannot reach the conception of it by the same route. I can form a conception of 'the same direction,' when the phrase is used of two Lines which have a common point, but it is only by considering that one 'falls on' the other—that they have all other points common—that they coincide. When you ask me to form a conception of this relationship of direction,