862 



SCIENCE. 



[N. S. Vol. XX. No. 521. 



Theory never had a more beautiful tri- 

 umph; perhaps one might add that it was 

 too complete, because it was at this mo- 

 ment above all that were conceived for 

 natural philosophy the hopes at least pre- 

 mature of which I spoke above. 



In all this period, especially in the sec- 

 ond half of the eighteenth century, what 

 strikes us with admiration and is also some- 

 what 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 insufficien- 

 cies when he published his theory of an- 

 alytic functions where he strove to give a 

 precise foundation to analysis. 



One can not too much admire the mar- 

 velous presentiment he had of the role 

 which were to play the functions we call, 

 with him, analytic; but we may confess 

 that we stand astonished before the dem- 

 onstration he believed to have given of the 

 possibilty of the development of a func- 

 tion in Taylor's series. 



The exigencies in questions of pure an- 

 alysis were less at this epoch. Confiding 

 in intuition, one was content with certain 

 probabilities and agreed implicitly about 

 certain hypotheses that it seemed useless 

 to formulate in an explicit way ; in reality, 

 one had confidence in the ideas which so 

 many times had shown themselves fecund, 

 which is very nearly the mot of d'Alem- 

 bert. 



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. 



III. 



We have now reached the first years of 

 the nineteenth century. As we have ex- 

 plained, 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 es- 

 thetic preoccupations which are to give a 

 physiognomy so different to so many 

 mathematical works, above all in the latter 

 two thirds of the nineteenth century. 



Not to anticipate, however, after so 

 many examples of the influences of physics 

 on the developments of analysis, we meet 

 still a new one, and one of the most mem- 

 orable, in Fourier's theory of heat. He 

 commences by forming the partial differ- 

 ential equations which govern temperature. 



What are for a partial differential equa- 

 tion the conditions- at the limits permitting 

 the determination of a solution? 



For Fourier, the conditions are suggested 

 by the physical problem and the methods 

 that he followed have served as models to 

 the physicist-geometers of the first half of 

 the last century. 



One of these consists in forming a series 

 with certain simple solutions. Fourier 

 thus obtained the first types of develop- 

 ments more general than the trigonometric 

 developments, as in the problem of the 

 cooling of a sphere, where he applies his 

 theory to the terrestrial globe, and investi- 

 gates the law which governs the variations 

 of temperature in the ground, trying to go 

 even as far as numerical applications. 



In the face of so many beautiful results, 

 we understand the enthusiasm of Fourier 

 which scintillates from every line of his 

 preliminary discourse. Speaking of mathe- 

 matical analysis, he says, "there could not 

 be a language more universal, more simple, 

 more exempt from errors and from obscuri- 

 ties, that is to say, more worthy to express 



