DOC. 52 GEOMETRY AND EXPERIENCE 221
GEOMETRY
AND
EXPERIENCE
245
unable
to compare
the disc-shadows
with Euclidean
rigid
bodies
which
can
be moved
about
on
the
plane
E. In
respect
of the
laws of
disposition
of the shadows
L',
the
point S
has
no special
privileges
on
the
plane any
more
than
on
the
spherical
surface.
The
representation given
above of
spherical
geometry
on
the
plane is important
for
us,
because
it
readily
allows itself
to
be
transferred
to
the three-dimensional
case.
Let
us imagine a
point S
of
our space,
and
a
great
number
of small
spheres,
L', which
can
all be
brought
to
coincide
with
one
another. But these
spheres are
not to
be
rigid
in the
sense
of Euclidean
geometry;
their
radius
is to
increase
(in
the
sense
of Euclidean
geometry)
when
they
are
moved
away
from
S to-
ward
infinity;
it
is to
increase
according to
the
same
law
as
the
radii
of the disc-shadows L'
on
the
plane.
After
having
gained
a
vivid mental
image
of the
geometrical
behavior
of
our
L'
spheres,
let
us assume
that in
our
space
there
are no
rigid
bodies
at
all
in
the
sense
of Euclidean
geometry,
but
only
bodies
having
the
behavior
of
our
L'
spheres.
Then
we
shall have
a
clear
picture
of
three-dimensional
spherical space,
or,
rather
of
three-dimensional
spherical geometry.
Here
our
spheres must
be called
“rigid”
spheres.
Their
increase in
size
as
they depart
from
S
is
not to
be detected
by
measuring
with
[34]
measuring-rods, any more
than
in the
case
of the disc-shadows
on
E,
because the standards
of
measurement
behave in the
same
way
as
the
spheres. Space
is homogeneous,
that
is
to
say,
the
same spherical configurations are possible
in
the
neighborhood
of
every point.*
Our
space
is finite, because,
in
consequence
of
the
“growth”
of the
spheres, only a
finite
number
of them
can
find
room
in
space.
In this
way,
by using
as a
crutch the
practice
in
thinking
and
visualization
which Euclidean
geometry gives us,
we
have
ac-
quired
a
mental
picture
of
spherical
geometry.
We
may
without
difficulty
impart
more
depth
and
vigor
to
these ideas
by
carry-
ing
out special imaginary
constructions. Nor would
it be
diffi-
cult
to represent
the
case
of what
is
called
elliptical geometry
in
[35]
*
This
is
intelligible
without calculation-but
only
for the two-dimensional
case-if
we
revert once
more
to
the
case
of the disc
on
the surface
of
the
sphere.