xxiv JOSIAH WILLARD GIBBS. 



triumphant use in the last three chapters shows that such purely 

 mechanical systems as he has been considering will exhibit, to human 

 perception, properties in all respects analogous to those which we 

 actually meet with in thermodynamics. No one can understandingly 

 read the thirteenth chapter without the keenest delight, as one after 

 another of the familar formulae of thermodynamics appear almost 

 spontaneously, as it seems, from the consideration of purely mechanical 

 systems. But it is characteristic of the author that he should be more 

 impressed with the limitations and imperfections of his work than 

 with its successes ; and he is careful to say (p. 166) : " But it should be 

 distinctly stated, that if the results obtained when the numbers of 

 degrees of freedom are enormous coincide sensibly with the general 

 laws of thermodynamics, however interesting and significant this 

 coincidence may be, we are still far from having explained the 

 phenomena of nature with respect to these laws. For, as compared 

 with the case of nature, the systems which we have considered are of 

 an ideal simplicity. Although our only assumption is that we are 

 considering conservative systems of a finite number of degrees of 

 freedom, it would seem that this is assuming far too much, so far as 

 the bodies of nature are concerned. The phenomena of radiant heat, 

 which certainly should not be neglected in any complete system of 

 thermodynamics, and the electrical phenomena associated with the 

 combination of atoms, seem to show that the hypothesis of a finite 

 number of degrees of freedom is inadequate for the explanation of the 

 properties of bodies." While this is undoubtedly true, it should, also 

 be remembered that, in no department of physics have the phe- 

 nomena of nature been explained with the completeness that is here 

 indicated as desirable. In the theories of electricity, of light, even in 

 mechanics itself, only certain phenomena are considered which really 

 never occur alone. In the present state of knowledge, such partial 

 explanations are the best that can be got, and, in addition, the 

 problem of rational thermodynamics has, historically, always been 

 regarded in this way. In a matter of such difficulty no positive 

 statement should be made, but it is the belief of the present 

 writer that the problem, as it has always been understood, has been 

 successfully solved in this work ; and if this belief is correct, one of 

 the great deficiencies in the scientific record of the nineteenth century 

 has been supplied in the first year of the twentieth. 



In methods and results, this part of the work is more general than 

 any preceding treatment of the subject ; it is in no sense a treatise on 

 the kinetic theory of gases, and the results obtained are not the 

 properties of any one form of matter, but the general equations of 

 thermodynamics which belong to all forms alike. This corresponds to 

 the generality of the hypothesis in which nothing is assumed as to 



