From every point in his world a line can be drawn parallel to AZ in the direction unknown to him. If we suppose the square in fig. 7 to be a geometrical square from every point of it, inside as well as on the contour, a straight line can be drawn parallel to AZ. The assemblage of these lines constitute a solid figure, of which the square in the plane is the base. If we consider the square to represent an object in the plane being’s world then we must attribute to it a very small thickness, for every real thing must possess all three dimensions. This thickness he does not perceive, but thinks of this real object as a geometrical square. He thinks of it as possessing area only, and no degree of solidity. The edges which project from the plane to a very small extent he thinks of as having merely length and no breadth—as being, in fact, geometrical lines. With the first step in the apprehension of a third dimension there would come to a plane being the conviction that he had previously formed a wrong conception of the nature of his material objects. He had conceived them as geometrical figures of two dimensions only. If a third dimension exists, such figures are incapable of real existence. Thus he would admit that all his real objects had a certain, though very small thickness in the unknown dimension, and that the conditions of his existence demanded the supposition of an extended sheet of matter, from contact with which in their motion his objects never diverge. Analogous conceptions must be formed by us on the supposition of a four-dimensional existence. We must suppose a direction in which we can never point extending from every point of our space. We must draw a distinction between a geometrical cube and a cube of real matter. The cube of real matter we must suppose to have an extension in an unknown direction, real, but so small as to be imperceptible by us. From every point of a cube, interior as well as exterior, we must imagine that it is possible to draw a line in the unknown direction. The assemblage of these lines would constitute a higher solid. The lines going off in the unknown direction from the face of a cube would constitute a cube starting from that face. Of this cube all that we should see in our space would be the face. Again, just as the plane being can represent any motion in his space by two axes, so we can represent any motion in our three-dimensional space by means of three axes. There is no point in our space to which we cannot move by some combination of movements on the directions marked out by these axes.