We owe to Leibnitz the first suggestion of Differentiation, with fractional and negative indices, but no definite notion of the theory was attained until Euler expounded it in the Petersburgh Commentaries for 1731. Still Euler wrote only a few pages on the subject, so that the theory could scarcely be said to have come into existence, until Laplace, in his Théorie des Probabilités, and Fourier, in his Théorie de la Propagation de la Chaleur, shewed how general differential coefficients might be deduced by means of definite integrals, provided we assume or prove, by means of some elementary definition, that the differential coefficient of a circular or of an exponential function has a certain form. The formula given by M. Fourier is a very simple one; and our astonishment is great, when we reflect on the time which elapsed from its announcement to the first application that was made of it. This took place in 1832, in a memoir by M. Liouville, entitled Questions of Geometry and Mechanics resolved by a new analysis, which memóir is followed by two others on the more immediate theory of the analysis itself. Although M. Liouville regards his analysis as a new invention, we have no doubt that the idea is due to Fourier; but still to M. Liouville belongs the honour of moulding it into a shape capable of being made use of in the solution of problems.