Chapter III (part 2)

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. p. 36 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 p. 37 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.