RELATIONS OF APPLIED MATHEMATICS 601 



this lecture. The fundamental equations of mechanics do not alter 

 their form in the slightest way when the algebraic sign of the time is 

 changed. All pure mechanical events can therefore occur equally 

 well in one sense as in its opposite, that is, in the sense of increasing 

 time or of diminishing time. We remark, however, that in ordinary 

 life future and past do not coincide as completely as the directions 

 right and left, but that the two are distinctly different. 



This becomes still more definite by means of the second law of the 

 mechanical theory of heat, which asserts that when an arbitrary 

 system of bodies is left to itself, uninfluenced by other bodies, the 

 sense in which changes of condition occur can be assigned. A certain 

 function of the condition of all the bodies, the entropy, can be 

 determined, which is such that every change that occurs must be in 

 the sense of carrying with it an increase of this function; thus, 

 with increasing time the entropy increases. This law is indeed an 

 abstraction, just as the principle of Galileo; for it is impossible, in 

 strict rigor, to isolate a system of bodies from all others. But since 

 it has given correct results hitherto, in connection with all the other 

 laws, we assume it to be correct, just as in the case of the principle of 

 Galileo. 



It follows from this law that every closed system of bodies must 

 tend toward a definite final condition for which the entropy is a 

 maximum. The outcome of this law, that the universe must come 

 to a final state in which nothing more can occur, has caused aston- 

 ishment; but this outcome is only comprehensible on the assump- 

 tion that the universe is finite and subject to the second law of the 

 mechanical theory of heat. If one regards the universe as infinite, 

 the above-mentioned difficulties of thought arise again if one does 

 not consider the infinite as a mere limit of the finite. Since there is 

 nothing analogous to the second law in the differential equations 

 of mechanics, it follows that it can be represented mechanically only 

 by the initial conditions. In order to find the assumptions suit- 

 able for this purpose, we must reflect that, to explain the appar- 

 ent continuity of bodies, we had to assume that every family 

 of atoms, or more generally, of mechanical individuals, existed in 

 incredibly many different initial positions. In order to treat this 

 assumption mathematically, a new science was founded whose pro- 

 blem is, not the study of the motion of a single mechanical system, 

 but of the properties of complexes of very many mechanical systems 

 which begin with a great variety of initial conditions. The task of 

 systematizing this science, of compiling it into a large book, and of 

 giving it a characteristic name, was executed by one of the greatest 

 American scholars, and in regard to abstract thinking, purely theo- 

 retic investigation, perhaps the greatest, Willard Gibbs, the recently 

 deceased professor at Yale University. He called this science statis- 



