What idea, then, is conveyed to the mind by the phrase "these Lines have the same direction"? If they were finite Lines, and the question concerned sameness of length, the process, of grasping this idea, would be a very simple one. We could either imagine one of the 2 Lines laid upon the other, and then apply the Axiom 'magnitudes which coincide are equal': or we could imagine a movable third Line, first applied to one of the 2 Lines and ascertained to have 'the same length' with it,[Pg 69] and then carried across the intervening space and applied to the other Line; and, on finding it to have 'the same length' with that also, we should pronounce the 2 Lines to have 'the same length,' in full confidence that our movable Line had preserved its length unchanged during the journey. Can we, then, use a similar process in grasping the idea of sameness of direction? That is, can we imagine a movable third Line, first applied to one of the 2 Lines and ascertained to have 'the same direction' with it, and then carried across the intervening space and applied to the second Line, to see if it has 'the same direction' with it also? But this process would convey to the mind no idea of 'sameness of direction,' unless we had some guarantee that the movable Line had preserved its direction unchanged during the journey. The only process, for securing this, that presents itself to my mind, is to imagine a transversal, cutting the 2 Lines, and thus bridging over the intervening space, and then to imagine the movable Line shifted along it, so as always to cut it at a constant angle. I have thought this matter out very carefully, and I feel convinced that this is the mental process by which we grasp the idea of 'sameness of direction,' when predicated of Lines that have no common Point, and therefore cannot be said to contain a zero-angle. Now this 'constant angle' is of course the angle at which the transversal cuts the first Line. Hence, the mental picture of a Line moving away from another, and yet maintaining 'sameness of direction' with it, is the picture of its so moving as that a certain transversal shall cut the two Lines at the same angle. And this is my answer to our first question, namely, "what idea is conveyed to the average human intellect by the phrase 'in the same direction,' when applied to non-coincident Lines?" If this be granted, our second question, "in accepting the Axiom that 'Lines, which have the same direction, make equal angles with all transversals,' to what other assertions are we[Pg 70] committing ourselves?", must be answered "we are consciously committing ourselves to the assertion that Lines, which are equally inclined to a certain transversal, are so to any transversal." We see, then, that, if this be so, the advocates of the Direction-Theory have not escaped the necessity of assuming, as axiomatic, the second Theorem enunciated in § 2. (See p. 63.)