of describing the geometrical behaviour of bodies which are large
as compared with the molecule. Success alone can decide as to the
justification of such an attempt, which postulates physical reality
for the fundamental principles of Riemann's geometry outside of the
domain of their physical definitions. It might possibly turn out
that this extrapolation has no better warrant than the extrapolation
of the idea of temperature to parts of a body of molecular order
of magnitude.

It appears less problematical to extend the ideas of practical
geometry to spaces of cosmic order of magnitude. It might, of course,
be objected that a construction composed of solid rods departs more
and more from ideal rigidity in proportion as its spatial extent
becomes greater. But it will hardly be possible, I think, to assign
fundamental significance to this objection. Therefore the question
whether the universe is spatially finite or not seems to me
decidedly a pregnant question in the sense of practical geometry.
I do not even consider it impossible that this question will be
answered before long by astronomy. Let us call to mind what the
general theory of relativity teaches in this respect. It offers
two possibilities:--

1. The universe is spatially infinite. This can be so only if the
average spatial density of the matter in universal space, concentrated
in the stars, vanishes, i.e. if the ratio of the total mass of the
stars to the magnitude of the space through which they are scattered
approximates indefinitely to the value zero when the spaces taken
into consideration are constantly greater and greater.

2. The universe is spatially finite. This must be so, if there is