220
DOC. 52 GEOMETRY AND EXPERIENCE
244
CONTRIBUTIONS TO SCIENCE
spherical
surface, diametrically
opposite to
S,
there
is
a
lumi-
nous
point,
throwing
a
shadow L' of the disc L
upon
the
plane
E.
Every
point
on
the
sphere
has its shadow
on
the
plane.
If
the disc
on
the
sphere
K
is moved,
its shadow
L'
on
the
plane
E
also
moves.
When the disc L
is
at S,
it
almost
exactly
coincides
with its shadow. If
it
moves on
the
spherical
surface
away
from
S
upwards,
the disc shadow
L'
on
the
plane
also
moves
away
from
S
on
the
plane
outwards,
growing
bigger
and
bigger.
As
the disc L
approaches
the luminous
point
N,
the shadow
moves
off
to infinity,
and becomes
infinitely great.
Now
we
put
the
question:
what
are
the
laws
of
disposition
of the disc-shadows L'
on
the
plane
E?
Evidently they
are ex-
actly
the
same
as
the
laws
of
disposition
of the
discs
L
on
the
spherical
surface. For
to
each
original
figure on
K there
is
a
corresponding
shadow
figure
on
E. If
two
discs
on
K
are
touch-
ing,
their shadows
on
E
also
touch. The
shadow-geometry on
the
plane agrees
with the
disc-geometry on
the
sphere.
If
we
call
the disc-shadows
rigid
figures,
then
spherical geometry
holds
good
on
the
plane
E with
respect to
these
rigid
figures.
In
par-
ticular, the
plane is
finite with
respect
to
the
disc-shadows,
since
only
a
finite number of the shadows
can
find
room on
the
plane.
At this
point
somebody
will
say,
“That
is
nonsense.
The
disc-
shadows
are
not rigid
figures.
We have
only to
move
a
two-foot
rule about
on
the
plane
E
to
convince ourselves
that
the
shadows
constantly
increase in
size
as they move away
from
S
on
the
plane
toward
infinity.”
But what if the two-foot
rule
were
to
behave
on
the
plane
E in the
same
way
as
the
disc-shadows L'?
It would then be
impossible to
show that the shadows increase
in
size
as
they
move away
from
S;
such
an
assertion would then
no longer
have
any meaning
whatever.
In
fact the
only
objec-
tive assertion that
can
be made
about
the disc-shadows
is
just
this,
that
they are
related in
exactly
the
same
way as are
the
rigid
discs
on
the
spherical
surface in the
sense
of Euclidean
geome-
try.
We
must
carefully
bear in mind that
our statement as
to
the
growth
of the
disc-shadows,
as
they
move
away
from
S
toward
infinity,
has in
itself
no objective meaning,
as
long
as we are
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Extracted Text (may have errors)


220
DOC. 52 GEOMETRY AND EXPERIENCE
244
CONTRIBUTIONS TO SCIENCE
spherical
surface, diametrically
opposite to
S,
there
is
a
lumi-
nous
point,
throwing
a
shadow L' of the disc L
upon
the
plane
E.
Every
point
on
the
sphere
has its shadow
on
the
plane.
If
the disc
on
the
sphere
K
is moved,
its shadow
L'
on
the
plane
E
also
moves.
When the disc L
is
at S,
it
almost
exactly
coincides
with its shadow. If
it
moves on
the
spherical
surface
away
from
S
upwards,
the disc shadow
L'
on
the
plane
also
moves
away
from
S
on
the
plane
outwards,
growing
bigger
and
bigger.
As
the disc L
approaches
the luminous
point
N,
the shadow
moves
off
to infinity,
and becomes
infinitely great.
Now
we
put
the
question:
what
are
the
laws
of
disposition
of the disc-shadows L'
on
the
plane
E?
Evidently they
are ex-
actly
the
same
as
the
laws
of
disposition
of the
discs
L
on
the
spherical
surface. For
to
each
original
figure on
K there
is
a
corresponding
shadow
figure
on
E. If
two
discs
on
K
are
touch-
ing,
their shadows
on
E
also
touch. The
shadow-geometry on
the
plane agrees
with the
disc-geometry on
the
sphere.
If
we
call
the disc-shadows
rigid
figures,
then
spherical geometry
holds
good
on
the
plane
E with
respect to
these
rigid
figures.
In
par-
ticular, the
plane is
finite with
respect
to
the
disc-shadows,
since
only
a
finite number of the shadows
can
find
room on
the
plane.
At this
point
somebody
will
say,
“That
is
nonsense.
The
disc-
shadows
are
not rigid
figures.
We have
only to
move
a
two-foot
rule about
on
the
plane
E
to
convince ourselves
that
the
shadows
constantly
increase in
size
as they move away
from
S
on
the
plane
toward
infinity.”
But what if the two-foot
rule
were
to
behave
on
the
plane
E in the
same
way
as
the
disc-shadows L'?
It would then be
impossible to
show that the shadows increase
in
size
as
they
move away
from
S;
such
an
assertion would then
no longer
have
any meaning
whatever.
In
fact the
only
objec-
tive assertion that
can
be made
about
the disc-shadows
is
just
this,
that
they are
related in
exactly
the
same
way as are
the
rigid
discs
on
the
spherical
surface in the
sense
of Euclidean
geome-
try.
We
must
carefully
bear in mind that
our statement as
to
the
growth
of the
disc-shadows,
as
they
move
away
from
S
toward
infinity,
has in
itself
no objective meaning,
as
long
as we are

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