Chapter III (part 1)

p. 34 CHAPTER III What may we learn about the fourth dimension by a study of the geometrical relations within our space? What should be the relation between a three-dimensional body and one of four dimensions? The four-dimensional body as the tracing of the movement of a three-dimensional body in the direction which is not confined within it. A four-dimensional body as containing an infinite number of three-dimensional bodies. A three-dimensional body as a section of a four-dimensional one. Parts of bodies and entire bodies in three and in four dimensions. The incommensurability of a three-dimensional and a four-dimensional body. A material atom as a section of a four-dimensional line. IF we consider the very great difference between the point and the line, between the line and the surface—surface and solid, i.e., the difference between the laws to which line and plane, plane and surface, etc., are subjected, and the difference of phenomena possible in point, in line, in surface, we shall indeed come to understand how much of the new and inconceivable the fourth dimension holds for us. As in the point it is impossible to imagine the line and the laws of the line; as in the line it is impossible to imagine the surface and the laws of the surface; as in the surface it is impossible to imagine the solid and the laws of the solid; so in our space it is impossible to imagine the body having more than three dimensions, and impossible to understand the laws of the existence of such a body. But studying the mutual relations between the point, the line, the surface, the solid, we begin to learn something about the fourth dimension, i.e., of four-dimensional space. We begin to learn what it can be in comparison with our three-dimensional space, and what it cannot be . This last we learn first of all. And it is especially important, because it saves us from many deeply inculcated illusions, which are very detrimental to right knowledge. p. 35 We learn what cannot be in four-dimensional space, and this permits us to set forth what can be there . In his book, The Fourth Dimension , Hinton makes an interesting statement concerning the method by which we may approach the problem of higher dimensions. He says: Our space itself bears within it relations through which we can establish relations to other (higher) spaces. For within space are given the conception of point and line, line and plane, which really involve the relation of space to a higher space. Let us consider these relations within our space, and see what conclusions we can derive from their investigation. We know that our geometry regards the line as a tracing of the movement of a point; the surface as a tracing of the movement of a line; and the solid as a tracing of the movement of a surface. On these premises we put to ourselves this question: Is it not possible to regard the "four-dimensional body" as a tracing of the movement of a three-dimensional body? But what is this movement, and in what direction? The point , moving in space, and leaving the tracing of its movement, a line, moves in a direction not contained in it, because in a point there is no direction whatsoever. The line , moving in space, and leaving the tracing of its movement, the surface, moves in a direction not contained in it because, moving in a direction contained in it, a line will continue to be a line. The surface , moving in space, and leaving a tracing of its movement, the solid, moves also in a direction not contained in it. If it should move otherwise, it would remain always the surface. In order to leave a tracing of itself as a "solid," or three-dimensional figure, it must set off from itself , move in a direction which in itself it has not.