224 M. Arago's Historical Eloge of Joseph Fourier, 



loss of heat in a body, is proportional to the excess of its tem- 

 perature above that of the medium in which it is immersed ; 

 but I hasten to shew you the geometrician penetrating, timidly 

 at first, into the questions of the propagation of heat, and in- 

 troducing the first germs of his prolific modes of investigation. 



It is to Lambert of Miihlhausen that we are indebted for this 

 first step. This ingenious geometrician had undertaken the 

 solution of a very simple problem, of which every body can 

 understand the meaning. 



A slender metallic bar is exposed, at one of its extremities, to 

 a steady and continued heat. The parts next the source of heat 

 are the first to become heated. By degrees the heat is commu- 

 nicated to the distant portions, and in a short time, each point 

 is found to have acquired the maximum of temperature which 

 it can ever attain. Although the experiment should be con- 

 tinued for a hundred years, the thermometrical state of the bar 

 would not be altered. 



As might be expected, this maximum of heat is much less, 

 the farther it is removed from the source. Is there any con- 

 nexion between the final temperatures and the distances of dif- 

 ferent parts from the extremity directly heated ? This con- 

 nexion exists ; it is very simple ; Lambert reduced it to calcu- 

 lation, and experiment confirmed the theoretical results. 



Along with the comparatively elementary question of the lon- 

 gitudinal propagation of heat, treated of by Lambert, there had 

 arisen the more general, but much more difficult problem of 

 this same propagation, in a body of three dimensions, termi- 

 nated by any sort of surface. This problem required the ap- 

 plication of the highest kind of analysis. Fourier was the first 

 to put it into a mathematical form. It is to Fourier also that 

 we are indebted for certain theorems by means of which we 

 may ascend from differential to integral equations, and carry 

 out the solutions, in the greater number of cases, to the ultimate 

 numerical apphcations. 



The first memoir of Fourier on the theory of heat, is dated 

 so far back as 1807. The Academy, to which it had been sub- 

 mitted, wishing to induce the author to extend and complete 

 it, made the question of the propagation of heat the subject of 

 ^he great mathematical prize which it was to give in 1812. 



