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 = 4pR2 ; if it is spherical, then F is
always less than 4pR2. 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).
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