The mathematical methods employed by Fourier in his treatment of special problems repay a careful study. As they stand they would often fail to satisfy even a lenient standard of mathematical rigour, and indeed they appear to have raised doubts in the minds of Laplace, [Pg 12] Lagrange, and Legendre, who formed the distinguished commission charged to examine one of his memoirs. But they are models of what may be called mathematical experiment; and at any rate they are successful in the end, and the results are easily verified. The form, again, in which these results are presented is I think quite unlike anything that had gone before, especially in the occurrence of definite integrals, but a slight examination shews that it would be difficult to imagine anything more adapted to the particular circumstances, or really more lucid. One special question examined by Fourier may be noticed for its connection with more recent speculations. It had been debated whether the earth has an intrinsic store of heat, or whether it was altogether dependent on the sun. Fourier’s conclusion is that the internal temperatures are independent of the solar influence, but that the latter is mainly responsible for the superficial temperatures. Among Fourier’s anticipations of modern practice, we may cite his recourse to graphical methods for the solution of equations, and especially his insistence [Pg 13] on the necessity that results should be capable of reduction, when needed, to numerical form. The general equations of Hydrodynamics date from Euler (1755), but a long period elapsed before any but the simplest applications were made of them. The theory of waves on water was propounded by the French Academy as the subject of a prize essay for the year 1815. The problem proposed was to trace the effect of a given initial disturbance of the surface. The memoir of Cauchy, to whom the prize was awarded, is remarkable as containing the first satisfactory proof of the persistence of the irrotational quality in a portion of fluid which possesses it at any one instant. The analytical difficulties of the special problem are considerable, owing mainly to the fact that there is no definite wave-velocity, but the genius of the author supplied what was wanting, and the notes afterwards appended to his memoir contain a store of important analytical results, relating chiefly to definite integrals. In particular we meet here for the first time with [Pg 14] the integrals known afterwards by the name of Fresnel, who encountered them in his work on Physical Optics. A parallel and independent memoir by Poisson, who was himself debarred from competing for the prize, confines itself more closely to the terms of the problem, but agrees in the main results. It is remarkable that neither writer pauses to consider the simpler and more fundamental properties of a simple-harmonic train of waves. This was left for Green and Airy, and extended in various ways by Stokes.