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But calculation shows that in a quasi-Euclidean universe the

average density of matter would necessarily be nil. Thus such a

universe could not be inhabited by matter everywhere; it would

present to us that unsatisfactory picture which we portrayed in

Section XXX.

If we are to have in the universe an average density of matter

which differs from zero, however small may be that difference,

then the universe cannot be quasi-Euclidean. On the contrary, the

results of calculation indicate that if matter be distributed

uniformly, the universe would necessarily be spherical (or

elliptical). Since in reality the detailed distribution of matter

is not uniform, the real universe will deviate in individual

parts from the spherical, i.e. the universe will be

quasi-spherical. But it will be necessarily finite. In fact, the

theory supplies us with a simple connection[25] between the

space-expanse of the universe and the average density of matter

in it.