epitome of our "space" experience, i.e. of experience that we can have in the movement of "rigid" bodies. In this sense we can imagine a spherical space.
Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of these the distance with a measuring-rod. All the free end-points
of these lengths lie on a spherical surface. We
can specially measure up the area () of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then ; if it is spherical,
then is always less than . With increasing values of , increases from zero up to a maximum value which is determined by the "world-radius," but
for still further increasing values of , the area gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a "counter-point" to the starting point. Under such
conditions they have traversed the whole spherical space. It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface. It is finite (i.e. of finite volume), and
has no bounds.
It may be mentioned that there is yet another kind of curved space: "elliptical space." It can be regarded
as a curved space in which the two "counter-points" are identical (indistinguishable from each other). An elliptical universe can thus be considered to some
extent as a curved universe possessing central symmetry.
It follows from what has been said, that closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. As a result of this discussion, a most interesting question arises for astronomers and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe. Our experience is far from being sufficient to enable us to answer this question. But the general theory of relativity permits of our answering it with a moderate degree of certainty, and in this connection the difficulty mentioned in [chapter:XXX]Section XXX finds its solution.