132 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1936 



We must not, however, go on in this way forever. It would take 

 an infinity of time to reduce space to a point, but this is unimportant, 

 for obviously we must not reduce space to a point; we must stop 

 somewhere before we reach that stage. Detailed mathematical in- 

 vestigations, too complicated even to summarize here, seem to suggest 

 that space cannot have been expanding for more than about 100,000 

 million years. This figure is, however, very uncertain, and in any 

 case provides no conclusive evidence as to the age of the universe. 

 1^'or mathematical investigation also shows that the present period 

 of expansion may well have been preceded by a period of contrac- 

 tion. Indeed the mathematical equations admit of solutions of two 

 difl'erent types. In one, the present epoch of expansion is preceded 

 by an earlier epoch of contraction, and no limits can be s?t to the 

 possible duration of this. In the other, the present epoch of expan- 

 sion is only one of a great number of epochs of regularly alternating 

 contractions and expansions, and no limit can be set to the possible 

 number of these epochs. This line of discussion, then, can tell us 

 nothing definite as to the age of the universe. 



It may, of course, be argued that, even though definite evidence is 

 lacking, considerations of probability fix a probable limit to the age 

 of the universe. The general line of argument would be that in the 

 last 1,000 million years, the dimensions of space have changed very 

 appreciably — probably by about 60 percent — so that the time-scale of 

 change is one of thousands of millions of years, and it is likely that 

 the total age must also be measured in terms of thousands of millions 

 of years. To take a parallel instance, if a zoologist captured a new 

 kind of animal, of entirely unknown species, and found that its 

 weight increased by 60 percent in a month, he would probably con- 

 clude, rightly or wrongly, that the creature was not many months 

 old. If a tree increases its height by 60 percent in a year, the bot- 

 anist may be fairly sure that it is not many years old. The argu- 

 ment undeniably carries some weight, but we must be careful not to 

 overrate it. In brief, we must remember that the universe is neither 

 an animal nor a tree. The population of England has increased by 

 60 percent in the last half-century, but we should not be justified in 

 concluding that England is only a few half-centuries old. The 

 brightness of the star Mira Ceti has changed by about 60 percent in 

 the last month, but we must not conclude that Mira Ceti cannot be 

 more than a few months old. Thus general considerations of prob- 

 ability can at best give a presumption, and not a very strong one at 

 that. 



THE MOTIONS OF THE STARS 



Another line of argument, which seems to me far more convincing 

 than the foregoing, leads to very different conclusions. If I set a 



