660 



SCIENCE 



[N. S. Vol. XXXIX. No. 1007 



In fact, when the kinetic theory of gases 

 first defines its swarms of molecules, with 

 their countless paths and collisions, it ap- 

 pears to be viewing a gas simply as a com- 

 plex mechanism; and in certain respects 

 this seeming is well founded. But the logic 

 of the theory of probabilities, which the 

 kinetic theory uses in deducing the physical 

 properties of gases from the statistical aver- 

 ages of collisions and free paths of the 

 hypothetical molecules, is no longer re- 

 ducible to the logic of mechanics. For the 

 velocity, the path, and the collision of each 

 individual molecule are all indifferent facts 

 for this kinetic theory of gases; which de- 

 votes itself to the study of probabilities and 

 of tendencies. And its methods are in part 

 those which the procedure of the insurance 

 actuaries exemplifies. The logic in ques- 

 tion is one which in some respects still needs 

 further elucidation. For even up to the 

 present time the logic of the theory of prob- 

 abilities is a controverted topic. But there 

 are a few features of the situation about 

 which nobody who looks carefully into the 

 subject can retain, I think, any serious 

 doubt. 



First, then, the average behavior of a 

 very large collection of irregularly moving 

 objects has characters which are decidedly 

 lawful, even although the laws in question 

 are what may be called laws of chance. 



The recent familiar use of statistical dia- 

 grams for illustrative purposes has made 

 this law of chance more familiar to many 

 classes of students than it was in the day 

 when Maxwell wrote certain words, which 

 you will fiind in his ' ' Theory of Heat. ' ' ^ 

 These words give you the very heart of the 

 statistical aspect of nature. 



The distribution of the molecules according to 

 their velocities is found to be of exactly the same 

 mathematical form as the distribution of observa- 

 tions according to the magnitude of their errors, 



1 Page 309 of the Appleton edition of 1875. 



as described in the theory of errors of observation. 

 . . . Whenever in physical phenomena some cause 

 exists over which we have no control, and which 

 produces a scattering of the particles of matter, 

 a deviation of observations from the truth, or a 

 diffusion of velocity or of heat, mathematical ex- 

 pressions of this exponential form are sure to make 

 their appearance. 



This, then, is in concrete form the law of 

 random distribution, the form of iron neces- 

 sity which one finds in the realm of chance. 



All this law of chance variation was, of 

 course, at that time no novelty, although 

 the popular use of statistics has since made 

 it more familiar. What was new, however, 

 was the fact that when Maxwell computed 

 the consequences which followed from sup- 

 posing the existence of his swarm of col- 

 liding molecules with their chance distri- 

 bution of velocities, he was able to deduce 

 not only the principal physical properties 

 of gases, but in particular those properties 

 which, like all the phenomena which illus- 

 trate the second law of the theory of 

 energy, are not expressible in terms of 

 merely mechanical laws, unless these laws 

 are applied to the case of a system complex 

 enough to ens.ure that the velocities of its 

 molecules shall approximate closely to this 

 chance distribution. 



Since Maxwell 's time, the same theoretical 

 methods have been applied to a vast range 

 of physical phenomena, with the general 

 result that the second law of the theory of 

 energy is now generally regarded, by all 

 except the extreme Energetiker, as essen- 

 tially a statistical law. So viewed, the sec- 

 ond law of energy becomes a principle 

 stated wholly in terms of the theory of 

 probability. It is the law that the physical 

 world tends, in each of its parts, to pass 

 from certain less probable to certain more 

 probable configurations of its moving par- 

 ticles. As thus stated the second prin- 

 ciple not only becomes a law of evolution, 

 an historical principle, but also ceases to 



