504 ALGEBRA AND ANALYSIS 



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 in- 

 fluences of physics on the developments of analysis, we meet still a 

 new one, and one of the most memorable, in Fourier's theory of heat. 

 He commences by forming the partial differential equations which 

 govern temperature. 



What are for a partial differential equation 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 solu- 

 tions. Fourier thus obtained the first types of developments 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 investigates 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 enthu- 

 siasm of Fourier which scintillates from every line of his preliminary 

 discourse. Speaking of mathematical analysis, he says, " There could 

 not be a language more universal, more simple, more exempt from 

 errors and from obscurities, that is to say, more worthy to express 

 the invariable relations of natural things. Considered under this 

 point of view, it is as extended as nature herself; it defines all sen- 

 sible relations, measures times, spaces; forces, temperatures. This 

 difficult science forms slowly, but it retains all the principles once 

 acquired. It grows and strengthens without cease in the midst of 

 so many errors of the human mind." 



The eulogy is magnificent, but permeating it we see the tendency 

 which makes all analysis uniquely an auxiliary, however incom- 

 parable, of the natural sciences, a tendency, in conformity, as we 

 have seen, with the development of science during the preceding two 

 centuries; but we reach just here an epoch where new tendencies 

 appear. 



Poisson having in a report on the Fundamenta recalled the re- 

 proach made by Fourier to Abel and Jacobi of not having occupied 

 themselves preferably with the movement of heat, Jacobi wrote to 

 Legendre: "It is true that Monsieur Fourier held the view that 

 the principal aim of mathematics was public utility, and the ex- 

 planation of natural phenomena; but a philosopher such as he 

 should have known that the unique aim of science is the honor of 

 the human spirit, and that from this point of view a question about 

 numbers is as important as a question about the system of the 



