Page:Carroll - Euclid and His Modern Rivals.djvu/152

114 produced; but we ask you to believe their reality independently of that fact.

Min. But the only geometrical meaning I know of, as yet, for the phrase 'different directions,' refers to Lines known to have a common point. What geometrical meaning do you attach to the phrase when used of other Lines?

Nie. We cannot define it.

Min. Nor construct it? Nor test it?

Nie. No.

Min. You ask me, then, to believe in the reality of two classes of 'pairs of Lines,' each possessing a property that you can neither define, nor construct, nor test?

Nie. That is true. But surely you admit the reality of the second class? Why, intersectional Lines are a case in point.

Min. Certainly. And so much I am willing to grant you. I allow that some non-coincidental Lines, viz. intersectional Lines, have 'different directions.' But as to 'the same direction,' you have given me no reason whatever for believing that there are any non-coincidental Lines which possess that property.

Nie. But surely there are two real distinct classes of non-coincidental Lines, 'intersectional' and 'separational'?

Min. Yes. Thanks to Euc. I. 27, you may now assume the reality of both.

Nie. And you will hardly assert that the relationship of direction, which belongs to a Pair of intersectional Lines, is identical with that which belongs to a Pair of separational Lines?

Min. I do not assert it.