It is easy to realise that the existence of such a plane being will be very limited indeed. He will be conscious of two directions only. One will be "up and down" that is to say, towards or away from the centre of his plane earth: the other will be "forwards and backwards" along its rim. Again any object, that projects beyond the rim of the disc on which he lives, will be for him an obstacle, which can only be passed by climbing over or burrowing under it. He cannot go round it, because that would mean coming out of the flat surface, which he is unable to do. Thus in Fig. 4, if the curved line AB represents a portion of the rim of the disc or "plane earth," and C a plane being, then he can only pass from A to B by "climbing over" any intervening object such as D, _i.e._:--by following the path indicated by the dotted line. Otherwise he would have to get out of the plane of the paper, which is impossible for him. [Illustration: _Fig. 4_] Now that I have described in outline the strict analogy of a race of plane beings inhabiting a smooth surface, I shall take the liberty, in the course of developing the idea more fully, of treating it in a slightly less rigid fashion. That is to say I shall assume that the reader has grasped the main idea and I shall not trouble about the "Plane earth" etc., unless it is desirable to do so for the sake of bringing out some special point; and I shall substitute for the foregoing somewhat elaborate representation the simpler one of a thin object free to slide on a smooth surface lying in front of us. But before doing so I would point out that already we begin to see our way a little. We can understand for instance that the fact of a Fourth dimension of space being unknown and inconceivable to us, is no proof that it does not exist. We have seen that a Third dimension would be equally unknown and inconceivable to a being limited in the manner described above; although we know that a third dimension does exist. We have only to suppose that analogous limitations obtain in our own case to see that a Fourth dimension might well exist of which we would still be unaware. We must, for instance, suppose that we have no sense organs facing that way and that we are prevented from moving in that direction by some circumstance analogous to the smooth sheet on which we supposed the plane being to live. The plane being would think that he could see all round his plane objects although we know that he could not really do so, and similarly our conviction that we can see all round our solid objects may be an illusion. Thus we are already in a position to appreciate the fact that our inability to perceive or imagine Four-dimensional space or objects in it, is no argument against its existence. There is, therefore, no 'a priori' reason for supposing that four dimensional space is not a reality. It is a point which must be settled by an appeal to the evidence.