But from this face there extends equally a cube in the unknown dimension. We can think of the higher cube, then, by taking the set of sections A, B, C, D, etc., and considering that from each of them there runs a cube. These cubes have nothing in common with each other, and of each of them in its actual position all that we can have in our space is an isolated square. It is obvious that we can take our series of sections in any manner we please. We can take them parallel, for instance, to any one of the three isolated faces shown in the figure. Corresponding to the three series of sections at right angles to each other, which we can make of the cube in space, we must conceive of the higher cube, as composed of cubes starting from squares parallel to the faces of the cube, and of these cubes all that exist in our space are the isolated squares from which they start. CHAPTER III THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE Having now obtained the conception of a four-dimensional space, and having formed the analogy which, without any further geometrical difficulties, enables us to enquire into its properties, I will refer the reader, whose interest is principally in the mechanical aspect, to Chapters VI. and VII. In the present chapter I will deal with the general significance of the enquiry, and in the next with the historical origin of the idea. First, with regard to the question of whether there is any evidence that we are really in four-dimensional space, I will go back to the analogy of the plane world. A being in a plane world could not have any experience of three-dimensional shapes, but he could have an experience of three-dimensional movements. We have seen that his matter must be supposed to have an extension, though a very small one, in the third dimension. And thus, in the small particles of his matter, three-dimensional movements may well be conceived to take place. Of these movements he would only perceive the resultants. Since all movements of an observable size in the plane world are two-dimensional, he would only perceive the resultants in two dimensions of the small three-dimensional movements. Thus, there would be phenomena which he could not explain by his theory of mechanics—motions would take place which he could not explain by his theory of motion. Hence, to determine if we are in a four-dimensional world, we must examine the phenomena of motion in our space. If movements occur which are not explicable on the suppositions of our three-dimensional mechanics, we should have an indication of a possible four-dimensional motion, and if, moreover, it could be shown that such movements would be a consequence of a four-dimensional motion in the minute particles of bodies or of the ether, we should have a strong presumption in favour of the reality of the fourth dimension.