Imagine a large disk on the table before you, and a being, such as the one described, proceeding round it. Let there be small movable particles surrounding him, which move out of his way as he goes along, and let these serve him for respiration; let them constitute an atmosphere.
Forwards and backwards would be to such a being direction along the rim—the direction in which he was proceeding and its reverse.
Then up and down would evidently be the direction away from the disk’s centre and towards it. Thus backwards and forwards, up and down, would both lie in the plane in which he was.
And he would have no other liberty of movement except these. Thus the words right and left would have no meaning to him. All the directions in which he could move, or could conceive movement possible, would be exhausted when he had thought of the directions along the rim and at right angles to it, both in the plane.
What he would call solid bodies, would be groups of the atoms of his world cohering together. Such a mass of atoms would, we know, have a slight thickness; namely, the thickness of a single atom. But of this he would know nothing. He would say, “A solid body has two dimensions—height (by how much it goes away from the rim) and thickness (by how much it lies along the rim).” Thus a solid would be a two-dimensional body, and a solid would be bounded by lines. Lines would be all that he could see of a solid body.
Thus one of the results of the limitations under which he exists would be, that he would say, “There are only two dimensions in real things.”
In order for his world to be permanent, we must suppose the surface on which he is to be very compact, compared to the particles of his matter; to be very rigid; and, if he is not to observe it by the friction of matter moving on it, to be very smooth. And if it is very compact with regard to his matter, the vibrations of the surface must have the effect of disturbing the portions of his matter, and of separating compound bodies up into simpler ones.
Fig. 1.
