216 DOC. 52 GEOMETRY AND EXPERIENCE
240
CONTRIBUTIONS TO SCIENCE
that
part
of
the
universe which
is
accessible
to
our
observation.
This
hope
is
illusory.
The distribution of the visible
stars
is
extremely irregular,
so
that
we on no
account may venture to
[p.
130]
set
the
average density
of
star-matter
in the universe
equal
to,
let
us
say,
the
average density
in the
Galaxy.
In
any
case,
how-
ever
great
the
space
examined
may
be,
we
could
not
feel
con-
vinced that
there
were
any
more stars
beyond
that
space.
So
it
seems
impossible to
estimate the
average density.
But there
is
another
road,
which
seems
to
me more prac-
ticable, although
it
also
presents
great
difficulties. For if
we
inquire
into
the deviations of the
consequences
of the
general
theory
of
relativity
which
are
accessible
to
experience,
from the
consequences
of the Newtonian
theory,
we
first of
all find
a
deviation
which manifests itself in close
proximity to
gravitat-
ing
mass,
and
has been confirmed
in
the
case
of
the
planet
Mer-
cury.
But if the universe
is
spatially
finite,
there
is
a
second
deviation
from the Newtonian
theory,
which,
in
the
language
of
the Newtonian
theory, may
be
expressed
thus: the
gravita-
tional
field
is
such
as
if it
were
produced, not
only by
the
ponderable
masses,
but in
addition
by
a
mass-density
of
negative
sign,
distributed
uniformly throughout
space.
Since this ficti-
tious
mass-density
would have
to
be
extremely
small,
it would
be
noticeable
only
in
very
extensive
gravitating
systems.
Assuming
that
we
know,
let
us say,
the statistical
distribution
and
the
masses
of the
stars
in the
Galaxy,
then
by
Newton’s law
we can
calculate the
gravitational
field
and the
average
velocities
which
the
stars must
have,
so
that the
Galaxy
should
not
col-
lapse
under the
mutual attraction
of its
stars,
but should
main-
tain
its actual
extent.
Now if the actual velocities of the
stars-
which
can
be
measured-were smaller than the calculated
velocities,
we
should
have
a
proof
that the
actual attractions
at
great
distances
are
smaller than
by
Newton’s law. From such
a
deviation it
could be
proved
indirectly
that the universe
is
finite. It would
even
be
possible to
estimate its
spatial
dimen-
sions.
[31]
Can
we
visualize
a
three-dimensional universe
which is
finite,
yet
unbounded?
[30]
[29]