Page:Carroll - Euclid and His Modern Rivals.djvu/149

. § 1.] when asserted of other Lines, you know that none of these considerations will help me, and you do not furnish me with any substitutes for them. To me the relationship does not seem to be identical: I should prefer saying that separational Lines have 'collateral,' or 'corresponding,' or 'separational' directions, to using the phrase 'the same direction' over again. It is, of course, true that 'collateral' directions produce the same results, as to angles made with a transversal, as 'identical' directions; but this seems to me to be a Theorem, not an Axiom.

Nie. You say that the relationship does not seem to you to be identical. I should like to know where you think you perceive any difference?

Min. I will try to make my meaning clearer by an illustration.

Suppose that I and several companions are walking along a railway, which will take us to a place we wish to visit. Some amuse themselves by walking on one of the rails; some on another; others wander along the line, crossing and recrossing. Now as we are all bound for the same place, we may say, roughly speaking, that we are all moving 'in the same direction': but that is speaking very roughly indeed. We make our language more exact, if we exclude the wanderers, and say that those who are walking along the rails are so moving. But it seems to me that our phrase becomes still more exact, if we limit it to those who are walking on one and the same rail.

As a second illustration, suppose two forces, acting on a certain body; and let them be equal in amount and opposite in direction. Now, if they are acting along the same