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.
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.
In analogy with all this, the solid, in order to leave as the tracing of its movement, the four-dimensional figure (hypersolid) shall move in a direction not confined in it; or in other words it shall come out of itself, set off from itself, move in a direction which is not present in it. Later on it will be shown in what manner we shall understand this.
But for the present we can say that the direction of the movement in the fourth dimension lies out of all those directions which are possible in a three-dimensional figure.
We consider the line as an infinite number of points; the surface as an infinite number of lines; the solid as an infinite number of surfaces.
In analogy with this it is possible to consider that it is necessary to regard a four-dimensional body as an infinite number of three-dimensional bodies, and four-dimensional space as an infinite number of three-dimensional spaces.
Moreover, we know that the line is limited by points, that the surface is limited by lines, that the solid is limited by surfaces.
It is possible that a four-dimensional body is limited by three-dimensional bodies.
Or it is possible to say that the line is the distance between two points; the surface the distance between two lines; the solid—between two surfaces.
Or again, that the line separates two points or several points from one another (for a straight line is the shortest distance between two points); that the surface separates two or several lines from each other; that the solid separates several surfaces one from another; as the cube separates six flat surfaces one from another—its faces.
The line binds several separate points into a certain whole (the straight,, the curved, the broken line); the surface binds several lines into a certain whole (the quadrilateral, the triangle); the solid binds several surfaces into a certain whole (the cube, the pyramid).
It is possible that four-dimensional space is the distance between a group of solids, separating these solids, yet at the same time binding them into some to us inconceivable whole, even though they seem to be separate from one another.
Moreover, we regard the point as a section of a line; the line as a section of a surface; the surface as a section of a solid.
By analogy, it is possible to regard the solid (the cube, sphere, pyramid) as a section of a four-dimensional body, and our entire three-dimensional space as a section of a four-dimensional space. If every three-dimensional body is the section of a four-dimensional one, then every point of a three-dimensional body is the section of a four-dimensional line. It is possible to regard an "atom" of a physical body, not as something material, but as an intersection of a four-dimensional line by the plane of our consciousness.
The view of a three-dimensional body as the section of a four-dimensional one leads to the thought that many (for us) separate bodies may be the sections of parts of one four-dimensional body.
A simple example will clarify this thought. If we imagine a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then upon this plane the sections of branches will seem separate, and not bound to one another. Yet in our space, from our standpoint, these are sections of branches of one tree, comprising together one top, nourished from one root, casting one shadow.
Or here is another interesting example expressing the same idea, given by Mr. Leadbeater, the theosophical writer, in one of his books. If we touch the surface of a table with our finger tips, then upon the surface will be just five circles, and from this plane presentment it is impossible to construe any idea of the hand, and of the man to whom this hand belongs. Upon the table's surface will be five separate circles. How from them is it possible to imagine a man, with all the richness of his physical and spiritual life? It is impossible. Our relation to the four-dimensional world will be analogous to the relation of that consciousness which sees five circles upon the table to a man. We see just "finger tips"—to us the fourth dimension is inconceivable.
We know that it is possible to represent a three-dimensional body upon a plane, that it is possible to draw a cube, a polyhedron or a sphere. This will not be a real cube or a real sphere, but the projection of a cube or of a sphere on a plane. We may conceive of the three-dimensional bodies of our space somewhat in the nature of images in our space of to us incomprehensible four-dimensional bodies.
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