366 DOC. 42
SPECIAL AND GENERAL RELATIVITY
124
Relativity
(arcs
of
circles
as
judged
in
three-dimensional
space)
of
equal
length
in all
directions.
They will call
the line
joining
the free
ends of these lines
a
"circle." For
a
plane surface,
the
ratio
of
the circumference of
a
circle
to
its
diameter,
both
lengths
being
measured with the
same
rod,
is,
according to
Euclidean
geometry
of the
plane, equal to
a
constant
value
n,
which
is
independent
of the diameter of the
circle.
On their
spherical
surface
our
flat
beings
would
find for
this
ratio
the value
sin(r/R),
n=(v/R)
i.e.
a
smaller value than
n,
the difference
being
the
more
considerable,
the
greater is
the radius of the circle
in
compar-
ison
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
world-sphere
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,