other than our common, every day three-space. Fortunately a comparison with lower dimensional geometries furnishes so many analogies that the subject can be very fully explained in a non-mathematical way. Only let me say just here that the geometry of the fourth dimension is a perfectly logical system of theorems and proofs entirely independent of these analogies.
We, the dwellers in -space, can best realize the reasonableness of conceiving of a fourth or higher dimensional space by considering as best we may what would take place in lower-dimensional space did such exist.
Consider a pipe of indefinite length with a bore of diameter as small as you please, and suppose that there dwell within this pipe "worms" of such diameter that they just fill the pipe. We can not conceive of anything with no breadth or thickness, but let us consider for sake of the illustration that this one-dimensional animal (which for brevity I shall call a unodim) has only length. Of course these unodims may vary in length according to age or family traits, perhaps. Now it is evident that a unodim can never turn around. He may move forward or backward, but one unodim can never pass another. If he possesses an eye in front or behind he can see a neighboring unodim as a mere point. His world is a very limited one.
Again, we might imagine a two-dimensional animal, taking hold
of a unodim, turning him around in his (two-dimensional) space and putting him back with his "tail" where his "head" was before. Evidently the unodim would be ignorant of the cause of his reversion, Fig. 2 for he has no knowledge of a two-dimensional space, and the two-dimensional animal is invisible to him. In other words, if and in the figure are equal in length but running in opposite directions, it is impossible to put in the place of , that is, where is and where is. To accomplish this, it would be necessary to take into -space and turn it around. While this would be an impossible feat for a unodim, a two-space animal could readily do it.
Now this one-dimensional space may not be "straight" (that is, of zero curvature); but it may be the space that we should get by bending the pipe around in the form of a circle, as in Fig. 2. In such a case, as his body would be constantly bent in the same direction and by the same degree, we may suppose that the unodim is totally unconscious of