DEVELOPMENT OF MATHEMATICAL ANALYSIS 503 



able. Soon appear partial differential equations, with vibrating cords, 

 the mechanics of fluids and the infinitesimal geometry of surfaces. 



This was a wholly new analytic world; the origin itself of the 

 problems treated was an aid which from the first steps permits no 

 wandering, and in the hands of Monge geometry rendered useful 

 services to the new-born theories. 



But of all the applications of analysis, none had then more renown 

 than the problems of celestial mechanics set by the knowledge of the 

 law of gravitation and to which the greatest geometers gave their 

 names. 



Theory never had a more beautiful triumph; perhaps one might 

 add that it was too complete, because it was at this moment above 

 all that were conceived for natural philosophy the hopes at least 

 premature of which I spoke above. 



In all this period, especially in the second half of the eighteenth 

 century, what strikes us with admiration and is also somewhat 

 confusing, is the extreme importance of the applications realized, 

 while the pure theory appeared still so ill assured. One perceives it 

 when certain questions are raised like the degree of arbitrariness in 

 the integral of vibrating cords, which gives place to an interminable 

 and inconclusive discussion. 



Lagrange appreciated these insufficiencies when he published his 

 theory of analytic functions, where he strove to give a precise foun- 

 dation to analysis. 



One cannot too much admire the marvelous presentiment he had 

 of the role which the functions, which with him we call analytic, 

 were to play; but we may confess that we stand astonished before 

 the demonstration he believed to have given of the possibility of the 

 development of a function in Taylor's series. 



The exigencies in questions of pure analysis were less at this 

 epoch. Confiding in intuition, one was content with certain probabil- 

 ities, and agreed implicitly about certain hypotheses that it seemed 

 useless to formulate in an explicit way; in reality, one had con- 

 fidence in the ideas which so many times had shown themselves 

 fecund, which is very nearly the mot of d'Alembert. 



The demand for rigor in mathematics has had its successive 

 approximations, and in this regard our sciences have not the absolute 

 character so many people attribute to them. 



Ill 



We have now reached the first years of the nineteenth century. 

 As we have explained, the great majority of the analytic researches 

 had, in the eighteenth century, for occasion a problem of geometry, 

 and especially of mechanics and of physics, and we have scarcely 

 found the logical and aesthetic preoccupations which are to give a 



