The departures from uniformity are positive; the numbers of nebulae increase faster than the volume of space through which they are scattered. Thus the density of the nebular distribution increases outwards, symmetrically in all directions, leaving the observer in a unique position. Such a favoured position, of course, is intolerable; moreover, it represents a discrepancy with the theory, because the theory postulates homogeneity. Therefore, in order to restore homogeneity, and to escape the horror of a unique position, the departures from uniformity, which are introduced by the recession factors, must be compensated by the second term representing effects of spatial curvature.

There seems to be no other escape. Observations demonstrate that

Relativistic cosmology requires that

Therefore,

The curvature of space is demonstrated and measured by the postulated recession of the nebulae. To the observer the procedure seems artificial. He has counted the nebulae to various limits, applied only the corrections that are necessarily required (energy-corrections), and derived the quite plausible result of uniform distribution. Now, in testing the relativistic theory, he introduces a new postulate, namely, recession of the nebulae, and it leads to discrepancies. Therefore, he adds still another postulate, namely, spatial curvature, in order to compensate the discrepancies introduced by the first. The accumulation of assumptions is uneconomical, and the justification must be sought in the general background of knowledge. The outstanding argument is the fact that velocity-shifts remain the only permissible interpretation of red-shifts that is known at the present time.

Well, perhaps the interpretation is correct and we do
inhabit a rapidly expanding universe. In that case the
surveys indicate the nature and amount of the spatial
curvature. It must be such that the effects on the
nebular distribution are just equal to the recession
factors at the limits of the various surveys: Actually, no
curvature can be found which exactly compensates for
the apparent departures from uniformity in each of the
surveys. Nevertheless, if we admit the presence of
rather considerable systematic errors in the observations, it is
possible to select a curvature which will more
or less restore the homogeneity. Hidden errors of the
necessary dimensions are by no means impossible in
the very delicate investigations near the limits of a
great telescope.
^{6} They may be improbable,
but they are not impossible. Therefore, the expanding universe
can be saved by introducing a sufficient amount of
spatial curvature. The plausible values are narrowly
limited, and they indicate a radius of curvature that is
positive and comparatively small. In fact the radius,
about 470 million light-years, is a trifle less than the
range of the 100-inch reflector for normal nebulae.
Thus the second of the three arbitrary elements in the
description of the relativistic universe is determined.
If the universe is expanding, the spatial curvature is real and positive.

The nature of the curvature has rather grave implications. Since the
curvature is positive, the universe is
closed. Space is closed as the surface of a sphere is
closed. The universe has a definite, finite volume
although it has no boundaries in three-dimensional
space. The remarkably small numerical value of the
radius of curvature is a complete surprise. It implies
that a large fraction of the universe, perhaps a quarter,
can be explored with existing telescopes.
^{7} The small
volume of the universe is another strange and dubious
conclusion. The familiar interpretation of red-shifts as
velocity-shifts very seriously restricts not only the time
scale, the age of the universe, but the spatial dimensions
as well. On the other hand, the alternative possible
interpretation, that red-shifts are not velocity-shifts,
avoids both difficulties, and presents the observable
region as an insignificant sample of a universe that extends
indefinitely in space and in time.

^{6} A systematic error of the order of 0.35
mag., in the limiting
magnitudes of the surveys, as compared with the magnitudes of the
nebulae used in the formulation of the law of red-shifts, would be
required in order to fully satisfy the relation *C*_{v} =
*d* /
.
Back.

^{7} The volume of this universe would be
2^{2}
*R*^{3}, where *R*
is the radius of curvature, or about 2 × 10^{27} cubic
light-years. The universe would contain about 400 million nebulae.
Back.