654 SCIENCE PROGRESS 



we adopt; obviously we cannot consider it in detail now. But 

 clearly this restoration of available energy may occur in the 

 universe, and the probability that it does so occur is of the same 

 order as that in Boltzmann's estimate. 



The probability is, as we have seen, unimaginably small. 

 But the universe is as unimaginably great. What do we mean 

 by saying that the universe has infinite extension and duration ? 

 We do not mean that it has, literally, no boundaries ; nor that 

 it had no beginning and will have no end. To say as much is 

 to play with pseudo-ideas. When we say that the universe has 

 infinite extension we mean that no matter how great, in finite 

 numbers, it may be conceived to be, it can still be conceived 

 as greater; and so also with its duration. That is to say, we 

 can make the universe as big as we like, or as old as we like, 

 while still regarding the dimensions we ascribe to it as such 

 as are capable of mathematical treatment. Its extension and 

 duration are "infinite" in the sense that we speak of infini- 

 tesimally small magnitudes in the theory of the differential 

 calculus. The radius of the earth, the distance of the earth 

 from the nearest fixed star, and its distance to stars of no 

 parallax are to be regarded as infinitesimals of the {n— i)st, 

 (n— 2)nd, and (n — 3)rd orders. The duration of a man's life, the 

 age of the habitable earth, the age of the solar system, may 

 also be so regarded. It does not matter now that the probability 

 of a restorative of available energy is incredibly small. We 

 can suppose the duration of the universe to be as much greater 

 as we wish. 



The universe that we know is the material universe in which 

 there are encrgy-transformdtions. We may regard it as an 

 infinitesimally small part of all that exists — the entire universe, 

 let us say, in short. Its duration we may also regard as an 

 infinitesimally small part of the duration of the entire universe. 

 The latter is physically dead : the sum of its entropy has 

 attained its maximum value. But here and there in it are 

 regions of the magnitude of our known stellar universe — 

 individual universes, Boltzmann calls them — but infinitesimal 

 in their extension when compared with the entire universe. 

 In these individual universes, for moments when compared 

 with the duration of the entire universe, but for eternal eras 

 (Aenonen) when compared with a human life-time, the second 

 law of thermodynamics becomes reversed, just as it does in 



