analogy, namely, the three-dimensional spherical space which was
discovered by Riemann. its points are likewise all equivalent. It
possesses a finite volume, which is determined by its radius
(2 2_R_3). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an 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 _r_ with a
measuring-rod. All the free end-points of these lengths lie on a
spherical surface. We can specially measure up the area (_F_) of this
surface by means of a square made up of measuring-rods. If the universe
is Euclidean, then _F_ = 4 _r_2; if it is spherical, then _F_ is always
less than 4 _r_2. With increasing values of _r, F_ increases from zero
up to a maximum value which is determined by the world-radius, but
for still further increasing values of _r_, 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
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