On their spherical
surface our flat beings would find for this ratio the value
eq. 27: file eq27.gif
i.e. a smaller value than p, the difference being the more
considerable, the greater is the radius of the circle in comparison
with the radius R of the " world-sphere." By means of this relation
the spherical beings can determine the radius of their universe ("
world "), even when only a relatively small part of their worldsphere
is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a
spherical " world " and not on a Euclidean plane, for a small part of
a spherical surface differs only slightly from a piece of a plane of
the same size.
Thus if the spherical surface beings are living on a planet of which
the solar system occupies only a negligibly small part of the
spherical universe, they have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of
universe " to which they have access is in both cases practically
plane, or Euclidean. It follows directly from this discussion, that
for our sphere-beings the circumference of a circle first increases
with the radius until the " circumference of the universe " is
reached, and that it thenceforward gradually decreases to zero for
still further increasing values of the radius.
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