On the other hand, the mathematicians, looking at it as a purely speculative idea, have endeavored to arrive at definite conclusions in regard to what would be the condition of things if the universe really exists in a fourth, or even in some higher dimension. Professor W. W. R. Ball tells us that
"the conception of a world of more than three dimensions is facilitated by the fact that there is no difficulty in imagining a world confined to only two dimensions—which we may take for simplicity to be plane—though equally well it might be a spherical or other surface. We may picture the inhabitants of flatland as moving either on the surface of a plane or between two parallel and adjacent planes. They could move in any direction along the plane, but they could not move perpendicularly to it, and would have no consciousness that such a motion was possible. We may suppose them to have no thickness, in which case they would be mere geometrical abstractions; or we may think of them as having a small but uniform thickness, in which case they would be realities."
"If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will; could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable to them."
"Our conscious life is in three dimensions, and naturally the idea occurs whether there may not be a fourth dimension. No inhabitant of flatland could realize what life in three dimensions would mean, though, if he evolved an analytical geometry applicable to the world in which he lived, he might be able to extend it so as to obtain results true of that world in three dimensions which would be to him unknown and inconceivable. Similarly we cannot realize what life in four dimensions is like, though we can use analytical geometry to obtain results true of that world, or even of worlds of higher dimensions. Moreover, the analogy of our position to the inhabitants of flatland enables us to form some idea of how inhabitants of space of four dimensions would regard us."
"If a finite solid was passed slowly through flatland, the inhabitants would be conscious only of that part of it which was in their plane. Thus they would see the shape of the object gradually change and ultimately vanish. In the same way, if a body of four dimensions was passed through our space, we should be conscious of it only as a solid body (namely, the section of the body by our space) whose form and appearance gradually changed and perhaps ultimately vanished. It has been suggested that the birth, growth, life, and death of animals, may be explained thus as the passage of finite four-dimensional bodies through our three-dimensional space."
Attempts have been made to construct drawings and models showing a four-dimensional body. The success of such attempts has not been very encouraging.
Investigators of this class look upon the actuality of a fourth dimension as an unsolved question, but they hold that, provided we could see our way clear to adopt it, it would open up wondrous possibilities in the way of explaining abstruse and hitherto inexplicable physical conditions and phenomena.