Page:Carroll - Euclid and His Modern Rivals.djvu/150

112 Line, we know that they neutralise each other, and that the body remains at rest. But if one be shifted ever so little to one side, so that they act along parallel Lines, then, though still equal in amount and (according to the 'direction' theory) opposite in direction, they no longer neutralise each other, but form a 'couple'.

As a third illustration, take two points on a certain Plane. We may, first, draw a Line through them and cause them to move along that Line: they are then undoubtedly moving 'in the same direction.' We may, secondly, draw two Lines through them, which meet or at least would meet if produced, and cause them to move along those Lines: they are then undoubtedly moving 'in different directions.' We may, thirdly, draw two parallel Lines through them, and cause them to move along those Lines. Surely this is a new relationship of motion, not absolutely identical with either of the former two? But if this new relationship be not absolutely identical with that named 'in the same direction,' it must belong to the class named 'in different directions.'

Still, though this new relationship of direction is not identical with the former in all respects, it is in some: only, to prove this, we must use some disputed Axiom, as it will take us into Table II. For instance, they are identical as to angles made with transversals: this fact is embodied in Tab. II. 4. (See p. 34). Would you like to adopt that as your Axiom?

Nie. No. We are trying to dispense with Table II altogether.

Min. It is a vain attempt.