great difficulties. For if we inquire into the deviations shown
by the consequences of the general theory of relativity which are
accessible to experience, when these are compared with the consequences
of the Newtonian theory, we first of all find a deviation which
shows itself in close proximity to gravitating mass, and has been
confirmed in the case of the planet Mercury. 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 gravitational field is in its nature such as
if it were produced, not only by the ponderable masses, but also by
a mass-density of negative sign, distributed uniformly throughout
space. Since this factitious mass-density would have to be enormously
small, it could make its presence felt only in gravitating systems
of very great extent.

Assuming that we know, let us say, the statistical distribution
of the stars in the Milky Way, as well as their masses, then by
Newton's law we can calculate the gravitational field and the mean
velocities which the stars must have, so that the Milky Way should
not collapse under the mutual attraction of its stars, but should
maintain its actual extent. Now if the actual velocities of the stars,
which can, of course, 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 magnitude.

Can we picture to ourselves a three-dimensional universe which is
finite, yet unbounded?