It is of course inconceivable that they should be truly "plane" beings in the mathematical sense and possess no thickness at all. But if we suppose that their thickness is of the same order as the diameter of a chemical "Atom"--that they are "one atom thick" so to speak,--the conditions laid down as to their limitation will be fulfilled. Now we have supposed the flat surface in our analogy to be _perfectly_ smooth in the true sense of the word. That is to say of such a nature as to offer no resistance whatever to the passage of objects over it. This means that plane beings will not be sensible of any opposition to their movement as far as the surface is concerned. Also, as we have supposed that they have no nerve endings facing it, it follows that they cannot feel any pressure from it. In short they will be totally unaware of its existence. But for the purpose of strict analogy this is insufficient, because a being placed on such a surface would be as incapable of movement as we should be if we were freely suspended in infinite space, remote from all the material objects we know. There would be nothing, in any direction known to him, from which he could "push off." We must therefore further suppose that the force of gravity operates in his world in a manner similar to that which we know,--every particle of matter attracting every other particle. This will mean two things; first, that every particle on the surface will be held against that surface and that plane beings will, therefore, never be able to move away from it; and, second, that matter on the surface will tend to collect together in a manner precisely analogous to what we observe in our space. Finally, we may suppose that these hypothetical beings whom we are considering live on the rim of a very large disc of plane matter, which has collected and is held together by the action of gravity, just as we live on the surface of a very large sphere of solid matter. They will be kept up against the rim of the disc by the force of gravity, which will attract them towards its centre, in the same way that we are kept against the surface of the earth.