168 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



the genius of Descartes, Newton, Leibnitz. The employment of 

 rectangular coordinates and of the elements of differential and integral 

 calculus has become so familiar to us that we might be tempted at 

 times to forget that these admirable instruments date only from 

 the seventeenth century. And in the same way the theory of partial 

 differential equations dates only from the eighteenth century. In 

 1747 d'Alembert obtained the general integral of the equation of 

 vibrating cords. It was the study of physical phenomena that 

 suggested the notions of continuity, derivative, integral, differ- 

 ential equation, vector, and the calculus of vectors; and these notions, 

 by a just return, form part of the necessary scientific equipment of 

 every physicist; it is through these that he interprets the results of 

 his experiments. There is evidently nothing mysterious in the fact 

 that mathematical theories constructed on the model of certain 

 phenomena should h«ave been capable of being developed and of 

 furnishing the model for other phenomena. This fact is neverthe- 

 less worthy of holding our attention, for it permits an important 

 practical result. If new physical phenomena suggested new math- 

 ematical models, mathematicians will be in duty bound to devote 

 themselves to the study of these new models and their generaliza- 

 tions, with the legitimate hope that the new mathematical theories 

 thus erected will be found fruitful in furnishing in their turn to the 

 physicists forms of useful thought. In other words, to the evolution 

 of physics there should correspond an evolution of mathematics 

 which, without abandoning the study of the classical and tested 

 theories, should be developed in takuig into account the results of 

 experiment. It is in this order of ideas that I would examine to-day 

 the influence that molecular theories may exercise on the develop- 

 ment of mathematics. 



II. 



At the end of the eighteenth century and in the first half of the 

 nineteenth there was created on the hypothesis of continuity what 

 we may call classical mathematical physics. As types of the theories 

 thus constructed we may take hydrodynamics and elasticity. In 

 hydrodynamics, every liquid was by definition considered to be 

 homogeneous and isotropic. It was not quite the same in the study 

 of the elasticity of solid bodies. The theory of crystalline forms had 

 led one to admit the existence of a periodic network, that is to say, a 

 discontinuous structure; but the period of the network was supposed 

 to bo extremely small witli reference to the elements of matter 

 physically regarded as the differential elements. The crystalline 

 structure therefore led only to anisotrophy, but not to discontinuity. 

 The partial differential equations of elasticity, as well as those of 

 hydrodynamics, imply continuity of the medium studied 



