XXX.
COSMOLOGICAL DIFFICULTIES OF NEWTON S THEORY
Part from the difficulty discussed in Section XXI, there is a second
fundamental difficulty attending classical celestial mechanics, which,
to the best of my knowledge, was first discussed in detail by the
astronomer Seeliger. If we ponder over the question as to how the
universe, considered as a whole, is to be regarded, the first answer
that suggests itself to us is surely this: As regards space (and time)
the universe is infinite. There are stars everywhere, so that the
density of matter, although very variable in detail, is nevertheless on
the average everywhere the same. In other words: However far we might
travel through space, we should find everywhere an attenuated swarm of
fixed stars of approrimately the same kind and density.
This view is not in harmony with the theory of Newton. The latter
theory rather requires that the universe should have a kind of centre
in which the density of the stars is a maximum, and that as we proceed
outwards from this centre the group-density of the stars should
diminish, until finally, at great distances, it is succeeded by an
infinite region of emptiness. The stellar universe ought to be a finite
island in the infinite ocean of space.[24]
[24] _Proof_ According to the theory of Newton, the number of lines
of force which come from infinity and terminate in a mass m is
proportional to the mass _m_. If, on the average, the mass density 0
is constant throughout the universe, then a sphere of volume _V_ will
enclose the average mass 0_V_. Thus the number of lines of force
passing through the surface _F_ of the sphere into its interior is
proportional to 0_V_. For unit area of the surface of the sphere the
number of lines of force which enters the sphere is thus proportional
to 0_V/F_ or to 0_R_. Hence the intensity of the field at the
surface would ultimately become infinite with increasing radius _R_ of
the sphere, which is impossible.
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