EVOLUTION OF THE UNIVERSE. 585 
be either positive, negative, or zero, and which is interpreted as the 
reciprocal of the square of the radius of curvature. But both this 
interpretation, and the assumption tacitly made that it is positive (thus 
making the three-dimensional universe closed) are entirely gratuitous, and 
not demanded by the theory. I will, however, go on using this convenient 
metaphorical speech. : 
Lemaitre’s theory not only gives a complete solution of the difficulties 
it was intended to solve, a solution of such simplicity as to make it appear 
self-evident, once you know it (like Columbus’s famous solution of the 
problem how to stand an egg on its small end)—it also incidentally 
contains the answer to some questions of long standing, such as the 
question : What becomes of the energy which is continually poured out into 
space by the stars? It is, in fact, used up by the work done in the adiabatic 
expansion of the universe. There can be not the slightest doubt that 
Lemaitre’s theory is essentially true, and must be accepted as a very real 
and important step towards a better understanding of nature. 
Now if we adopt this theory of the expanding universe, it is very 
tempting to seek a connection between this expansion and the evolution 
of the material bodies constituting the universe, and to identify the 
beginning of the expansion with the beginning of that evolution. But 
the time elapsed since the beginning of the expansion is only a few 
thousand million years—an interval that we have learned to consider as 
very short from the evolutionary point of view. There is no escape from 
this. We can, in fact, make the interval logarithmically infinite, but that 
is only a mathematical trick: we call zero minus infinity, but that does 
not make the interval during which anything really happens any longer. 
It is a consequence of Lemaitre’s equations that the time taken by the 
* universe to increase its radius from anywhere near its minimum to its 
present value is of the order of magnitude of the radius itself, if we adopt 
corresponding units of space and time, e.g. years for time and lightyears 
for space. The real origin of the difficulty is that the ratio of the natural 
units of space and time is determined by the velocity of light, whilst the 
velocities which determine the rate of progress of the evolutionary process, 
say in the case of a stellar system, are those of material bodies (stars) 
which are of the order of one ten-thousandth of that of light. Con- 
sequently, if we wish to construct a causal connection between the com- 
mencement of the expansion and events which are supposed to have 
happened at a very early stage of the evolution of the stellar systems— 
such as the first formation of condensations, or the imprisoning of free 
energy inside matter, called ‘ stagnation’ by Lemaitre—we unavoidably 
meet with the difficulty that the time elapsed since these two beginnings 
is some thousands of times longer in the one chain of events than in the 
other. I do not think it will ever be found possible to reconcile the two 
time scales. 
‘We thus, however reluctantly, come to the conclusion that the 
expansion of the universe on the one hand, and the evolution of stellar 
systems and stars on the other hand, are two different processes, taking 
place side by side, but without any apparent connection between them. 
The expansion has only been going on during an interval of time which is 
as nothing compared with the duration of the evolution. Leaving the 
