PRELIMINARY DISCOURSE. 7 



founded on new theorems, whose object we could not in this 

 place make known. The method which is derived from them 

 leaves nothing vague and indeterminate in the solutions, it leads 

 them up to the final numerical applications, a necessary condition 

 of every investigation, without which we should only arrive at 

 useless transformations. 



The same theorems which have made known to us the 

 equations of the movement of heat, apply directly to certain pro 

 blems of general analysis a.nd dynamics whose solution has for a 

 long time been desired. 



Profound study of nature is the most fertile source of mathe 

 matical discoveries. Not only has this study, in offering a de 

 terminate object to investigation, the advantage of excluding 

 vague questions and calculations without issue ; it is besides a 

 sure method of forming analysis itself, and of discovering the 

 elements which it concerns us to know, and which natural science 

 ought always to preserve : these are the fundamental elements 

 which are reproduced in all natural effects. 



We see, for example, that the same expression whose abstract 

 properties geometers had considered, and which in this respect 

 belongs to general analysis, represents as well the motion of light 

 in the atmosphere, as it determines the laws of diffusion of heat 

 in solid matter, and enters into all the chief problems of the 

 theory of probability. 



The analytical equations, unknown to the ancient geometers, 

 which Descartes was the first to introduce into the study of curves 

 and surfaces, are not restricted to the properties of figures, and to 

 those properties which are the object of rational mechanics ; they 

 extend to all general phenomena. There cannot be a language 

 more universal and more simple, more free from errors and from 

 obscurities, that is to say more worthy to express the invariable 

 relations of natural things. 



Considered from this point of view, mathematical analysis is as 

 extensive as nature itself; it defines all perceptible relations, 

 measures times, spaces, forces, temperatures ; this difficult science 

 is formed slowly, but it preserves every principle which it has once 

 acquired ; it grows and strengthens itself incessantly in the midst 

 of the many variations and errors of the human mind. 



Its chief attribute is clearness ; it has no marks to express con- 



