Let A B C be a triangle, and G a line. If A B C moves round the end of the line, it can take up the position A′ B′ C′; but it cannot anyhow be made to take the position shown in Diagram V., A′ B′ C′. But if we move the triangle A B C out of the plane round the line G as axis, it will, in the course of its twisting round this axis, come into the position A′ B′ C′. It will come into this position when it has twisted half-way round. The point A, for instance, twists round in a circle lying in a plane which contains the direction A to A′, and the direction at right angles to the paper. Twisting half-way round in this circle, it becomes A′, and so on for the other points. Now a being who did not know what a direction was which lay out of the plane would not be able to conceive this twisting and turning movement. It would be as impossible for him to conceive the triangle A B C turned into the triangle A′ B′ C′, as it would be for us to suppose ourselves turned into the looking-glass image of ourselves by a simple twisting. Yet just as a thing inconceivable to the plane creature can be done, so we could be twisted round and turned into our image. But this only holds theoretically; our relation to the æther is such that we cannot be so turned, or any bodies of a magnitude appreciable to our senses. If we consider the case of a being limited to a plane, we see that he would have two directions marked out for him at every point of the rim of matter on which he must be conceived as standing. This is up and down, and forwards and backwards—the up being away from the attracting mass on which he is. Now, if he were to realize that he was in three-dimensional space, but confined to a plane surface in it, his first conclusion would be that there was a new direction starting from every point of matter, and that this new direction was not one of those which he knew. This new direction he could not represent in terms of the directions with which he was familiar, and he would have to invent new terms for it. And so we, when we conceive that from every particle of matter there is a new direction not connected with any of those which we know, but independent of all the paths we can draw in space, and at right angles to them all—we also must invent a new name for this new direction. And let us suppose a force acting in a definite way in this new direction. Let there be a force like gravitation. If there is such a direction, there will probably be a force acting in it; for in every known direction we find forces of some kind or another acting. Let us call away from this force by the Greek word ana, and towards the centre of this force kata. Then from every point in addition to the directions up and down, right and left, away from and towards us, is the new direction ana and kata.