DOC. 52 GEOMETRY AND EXPERIENCE 219
GEOMETRY AND EXPERIENCE
243
seems
to promise
success
at
the
outset,
and
the
smaller the
radius
of the disc
in
proportion to
that
of the
sphere,
the
more promis-
ing
it
seems.
But
as
the
construction
progresses
it
becomes
more
and
more
patent
that
the
arrangement
of the
discs
in the
manner
indicated,
without
interruption,
is
not
possible, as
it
should
be
possible by
the
Euclidean
geometry
of
the
plane.
In
this
way creatures
which
cannot
leave the
spherical
surface,
and
cannot
even peep
out
from the
spherical
surface
into
three-
dimensional
space,
might discover,
merely by
experimenting
with
discs,
that
their
two-dimensional
“space”
is not
Euclidean,
but
spherical space.
From the latest results of the
theory
of
relativity
it
is probable
[33]
that
our
three-dimensional
space
is
also
approximately
spherical,
that
is,
that the
laws
of
disposition
of
rigid
bodies in
it
are
not
given by
Euclidean
geometry,
but
approximately
by spherical
geometry,
if
only
we
consider
parts
of
space
which
are
suffi-
ciently
extended. Now this
is
the
place
where the reader’s
imagination
boggles. “Nobody
can
imagine
this
thing,”
he cries
indignantly.
“It
can
be
said,
but
cannot
be
thought.
I
can
imagine
a
spherical
surface well
enough,
but
nothing
analogous
to
it
in
three
dimensions.”
We
must try
to surmount
this
barrier in
the mind, and the
patient
reader will
see
that it
is by
no means
a
particularly diffi-
cult
task. For this
purpose
we
will first
give our
attention
once
more
to
the
geometry
of two-dimensional
spherical
surfaces.
In the
adjoining
figure
let
K be the
spherical
surface,
touched
at
S
by a plane,
E,
which,
for
facility
of
presentation,
is
shown
in the
drawing
as a
bounded
surface. Let L be
a
disc
on
the
spherical
surface. Now
let
us imagine
that
at
the
point
N of the
FIG. 2