THE EXPANDIXG UNIVERSE — HUBBLE 131 



apparently smaller and smaller volumes of space as the distance 

 increased. 



Spatial curvature is an expected feature of an expanding universe, 

 and, together with the precise form of the law of red shifts, further 

 specifies a particular type of possible w^orld. Thus, if the measures 

 Avere reliable, we might conclude that tlio initial cosmological problem 

 had been solved; that now we knew the nature of the universe we 

 inhabit. But the situation is not so simple. Just as the departures 

 from linearity in the law of red shifts indicate a universe that is 

 strangely young, so the apparent departures from homogeneity indi- 

 cate a universe that is strangely small and dense. 



The sign of the curvature required to restore homogeneity is posi- 

 tive, hence the universe is "closed"; it has a finite volume although, 

 of course, there are no boundaries. The amount of curvature indicates 

 the volume of the universe : about four times the volume of the observ- 

 able region. Such a universe would contain perhaps 400 million 

 nebulae. The total mass, however, would be far greater than that 

 which can be attributed to the nebulae alone. 



CONCLUSION 



Thus the use of dimming corrections leads to a particular kind of 

 universe, but one which most students are likely to reject as highly 

 improbable. Furthermore, the strange features of this universe are 

 merely the dimming corrections expressed in different terms. Omit 

 the dimming factors, and the oddities vanish. We are left with the 

 simple, even familiar, concept of a sensibly infinite universe. All the 

 difHculties are transferred to the interpretation of red shifts which 

 cannot then be the familiar velocity shifts. 



Two further points may be mentioned. In the first place, the 

 reference of red shifts to some hitherto unknown principle does not 

 in any way destroy the validity of the theory of expanding universes. 

 It merely removes the theory from immediate contact with observa- 

 tions. We may still suppose that the universe is either expanding or 

 contracting, but at a rate so slow that it cannot now be disentangled 

 from the gross effects of the superposed red shifts. 



Secondly, the conclusions drawn from the empirical investigations 

 involve the assumptions that the measures are reliable and the data 

 are representative. These questions have been carefully reexamined 

 during the past few years. Various minor revisions have been made, 

 but the end results remain substantially unchanged. By the usual 

 criteria of probable errors, the data seem to be sufficiently consistent 

 for their purpose. Nevertheless, the operations are delicate, and the 

 most significant data are found near the limits of the greatest tele- 



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