The memoir presented by Lagrange, which this paper examines, is usually considered as an elegant, but scarcely practicable, contribution to numerical analysis. The purpose of this study is to show the significance of the novel mathematical ideas it contains, and in particular to look at this essay from the perspective of generating function theory, for which the theoretical foundations would be laid some little time later by Laplace. This excursus of Lagrange's does indeed proffer an abundance of procedures that were to become standard in this latter theory. Further, Lagrange's memoir introduces some quite extraordinary elements, e.g. an algorithm for the approximation of an integral series by means of rational fractions — quite analogous in some cases to the determination of Padé approximants ; or the introduction of polynomials formally akin to Chebyshev polynomials, to cater for tasks that would only devolve to the latter in the 20th century.