364
DOC. 42 SPECIAL AND GENERAL RELATIVITY
THIRTY-ONE
The
Possibility
of
a
"Finite"
and
Yet
"Unbounded"
Universe
But
speculations on
the
structure
of the
universe also
move
in
quite
another direction.
The
development
of
non-Euclidean
geometry
led
to
the
recognition
of the
fact,
that
we can
cast
doubt
on
the
infiniteness
of
our
space
without
coming
into conflict
with
the
laws
of
thought
or
with
experience
(Riemann, Helmholtz).
These
questions
have al-
ready
been treated
in
detail and with
unsurpassable lucidity
by
Helmholtz and
Poincare,
whereas
I
can
only
touch
on
them
[74]
briefly
here.
In the
first
place,
we
imagine
an
existence in
two-
dimensional
space.
Flat
beings
with
flat
implements,
and
in
particular
flat
rigid
measuring-rods,
are
free
to
move
in
a
plane.
For them
nothing
exists outside of
this
plane:
that which
they
observe
to happen to
themselves and
to
their
flat "things" is
the all-inclusive
reality
of their
plane.
In
particular,
the
con-
structions of
plane
Euclidean
geometry
can
be carried
out by
means
of the
rods,
e.g.
the lattice
construction,
considered
in
[75]
Section
24.
In
contrast to
ours,
the universe
of
these
beings is
two-dimensional; but,
like
ours,
it extends
to
infinity.
In their
122