JOSEPH FOURIER. 161 



heat of a body is proportional to the excess of its temperature above 

 that of the medium in which it is phinged ; but I have been desirous of 

 showing you geometry penetrating, timidly at lirst, into questions of 

 the propagation of heat, and depositing there the first germs of its fer- 

 tile methods. 



It is to Lambert, of Mulhouse, that we owe this first step. This inge- 

 nious geometer had proposed a very simple problem, which any person 

 may comprehend. A slender metallic bar is exposed at one of its ex- 

 tremities to the constant action of a certain focus of heat. Tlie parts 

 nearest the focus are heated first. Gradually the heat communicates 

 itself to the more distant parts, and, after a short time, each point ac- 

 quires the maximum temperature which it can ever attain. Although 

 the experiment were to last a hundred years, the thermometric state of 

 the bar vould not undergo any modification. 



As might be reasonably expected, this maximum of heat is so much 

 less considerable as we recede from the focus. Is there any relation 

 between the final temperatures and the distances of the different parti- 

 cles of the bar from the extremity directly heated ? Such a relation ex- 

 ists. It is very simple. Lambert investigated it by calculation, and 

 experience confirmed the results of theory. 



In addition to the somewhat elementary question of the lond'itudinal 

 propagation of heat, there ofiered itself the more general but much more 

 difficult problem of the propagation of heat in a body of three dimen- 

 sions terminated by any surface whatever. This problem demanded the 

 aid of the higher analysis. . It was Fourier who first assigned the equa- 

 tions. It is to Fourier, also, that we owe certain theorems, by means of 

 which we may ascend from the difierential equations to the integrals, 

 and push the solutions, in the majority of cases, to the final numerical 

 applications. 



The first memoir of Fourier on the theory of heat dates from the year 

 1807. The Academy, to which it was communicated, being desirous of 

 inducing the author to extend and improve his researches, made the 

 question of the propagation of heat the subject of the great mathemati- 

 cal prize which was to be awarded in the beginning of the year 1812. 

 Fourier did, in effect, compete, and his memoir was crowned. But, alas ! 

 as Fontenelle said, " in the country even of demonstrations, there are 

 to be found causes of dissension." Some restrictions mingled with the 

 favorable judgment. The illustrious commissioners of the prize, La- 

 place, Lagrange, and Legendre, while acknowledging the novelty and 

 importance of the subject, while declaring that the real differential 

 equations of the propagation of heat were finally found, asserted that 

 they perceived difficulties in the way in which the author arrived at 

 them. They added that his processes of integration left something to be 

 desired, even on the score of rigor. They did not, however, support 

 their opinion by any arguments, 



Fourier never admitted the validity of this decision. Even at the 

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