It is to Lambert of Mulhouse, that we owe this first step.  This ingenious geometer had proposed a very simple problem which any person may comprehend.  A slender metallic bar is exposed at one of its extremities to the constant action of a certain focus of heat.  The parts nearest the focus are heated first.  Gradually the heat communicates itself to the more distant parts, and, after a short time, each point acquires the maximum temperature which it can ever attain.  Although the experiment were to last a hundred years, the thermometric state of the bar would 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 particles of the bar from the extremity directly heated?  Such a relation exists.  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 longitudinal propagation of heat, there offered itself the more general but much more difficult problem of the propagation of heat in a body of three dimensions terminated by any surface whatever.  This problem demanded the aid of the higher analysis.  It was Fourier who first assigned the equations.  It is to Fourier, also, that we owe certain theorems, by means of which we may ascend from the differential 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 mathematical 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 favourable judgment.  The illustrious commissioners of the prize, Laplace, 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 rigour.  They did not, however, support their opinion by any arguments.