independent of the diameter of the circle. On their spherical surface
our flat beings would find for this ratio the value
image036
_i.e._ a smaller value than , 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. During this process the area of the
circle continues to increase more and more, until finally it becomes
equal to the total area of the whole world-sphere.
Perhaps the reader will wonder why we have placed our beings on a
sphere rather than on another closed surface. But this choice has its
justification in the fact that, of all closed surfaces, the sphere is
unique in possessing the property that all points on it are equivalent.
I admit that the ratio of the circumference _c_ of a circle to its
radius _r_ depends on _r_, but for a given value of _r_ it is the same
for all points of the worldsphere ; in other words, the world-sphere
is a surface of constant curvature.
To this two-dimensional sphere-universe there is a three-dimensional
76