G02 APPLIED MATHEMATICS 



tical mechanics, and it falls naturally into two parts. The first in- 

 vestigates the conditions under which the outwardly visible proper- 

 ties of a complex of very many mechanical individuals is not in any 

 wise altered; this first part I shall call statistical statics. The sec- 

 ond part investigates the gradual changes of these outwardly visible 

 properties when those conditions are not fulfilled; it may be called 

 statistical dynamics. At this point we may allude to the broad view 

 which is opened by applying this science to the statistics of ani- 

 mated beings, of human society, of sociology, etc., and not merely 

 upon mechanical particles. A development of the details of this 

 science would only be possible in a series of lectures and by means 

 of mathematical formulas. Apart from mathematical difficulties it is 

 not free from difficulties of principle. It is based upon the theory 

 of probabilities. The latter is as exact as any other branch of mathe- 

 matics if the concept of equal probabilities, which cannot be de- 

 duced from the other fundamental notions, is assumed. It is here 

 as in the method of least squares which is only free from objection 

 when certain definite assumptions are made concerning the equal 

 probability of elementary errors. The existence of this fundamental 

 difficulty explains why the simplest result of statistical statics, the 

 proof of Maxwell's speed law among the molecules of a gas, is still 

 being disputed. 



The theorems of statistical mechanics are rigorous consequences 

 of the assumptions and will always remain valid, just as all well- 

 founded mathematical theorems. But its application to the events 

 of nature is the prototype of a physical hypothesis. Starting from 

 the simplest fundamental assumption of the equal probabilities, we 

 find that aggregates of very many individuals behave quite ana- 

 logously as experience shows of the material world. Progressive or 

 visible rotary motion must always go over into invisible motion of 

 the minutest particles, into heat, as Helmholtz characteristically 

 says: ordered motion tends always to go over into not ordered 

 motion; the mixture of different substances as well as of different 

 temperatures, the points of greater or less intense molecular 

 motion, must always tend toward homogeneity. That this mixture 

 was not complete from the start, that the universe began in such 

 an improbable state, belongs to the fundamental hypotheses of the 

 entire theory; and it may be said that the reason for this is as little 

 known as the reason why the universe is just so and not otherwise. 

 But we may take a different point of view. Conditions of great mix- 

 ture and great differences in temperature are not absolutely impos- 

 sible according to the theory but are very highly improbable. If the 

 universe be considered as large enough there will be, according to the 

 laws of probability, here and there places of the size of fixed stars, 

 of altogether improbable distributions. The development of such 



