If, in the course of our investigation, we find that there are in our space phenomena, which closely resemble those which would in "two space" indicate the existence of a third dimension, then we shall be entitled to say that these phenomena indicate the probable existence of a fourth dimension. We can now proceed with our consideration of a two dimensional world, remembering that,-- Shapes and events in four space bear to shapes and events in three space, the same relation that those in three space bear to those in two space. [Illustration: _Fig. 5^{[a=]}_] [Illustration: _Fig. 5^{[b=]}_] The very small three-dimensional thickness which we have supposed to exist in all the objects of our plane world is imperceptible to the plane beings which inhabit it and the objects which they perceive they will accordingly think of as geometrical figures and of their boundaries as geometrical lines, having length but no breadth. A circle will appear to a plane being as a completely closed space. He will, as he thinks, be able to go all round it without being able to find any opening in its bounding line. It will in fact be to him what a sphere is to us. A two space room will be a thing like the figure shown in Fig. 5_a_. He will be able to get into or out of it by the gap in the wall which is shown and which corresponds to the door. But he will not be able to conceive of any other mode of entry or exit, although we can see that from the direction of the third dimension it is not closed at all. Similarly, if Fig. 5_b_ represents a closed two-dimensional box, we see that this is absolutely open to us, who are three dimensional beings, though appearing to be closed on all sides to a plane being. If we took advantage of this fact we could play all sorts of tricks on him for we could put things into the box or take them out of it, by way of the third dimension, while to the plane being the box would appear to be tightly closed the whole time. It will be noticed that as the path of an object in transference would lie wholly outside the plane being's space he would not be able to form any conception of the nature of the process involved. If he tried to understand it at all he would probably imagine that the object has been disintegrated into particles inside the box, passed in this condition through the minute interstices which he might suppose to exist in its walls, and reintegrated on the other side. Whereas the true explanation is far simpler. The very great importance of this will become apparent when we come to consider the question of the positive evidences for the existence of a fourth dimension. It is because of this importance that I have dwelt on a point which to many readers will have been obvious as soon as stated. Similarly we could make things appear "from nowhere" and disappear equally mysteriously simply by putting them down on to his flat surface and picking them up again.