The origins of Lagrange’s theory of analytic functions are be found in Newton’s theory of series and fluxions. During his long stay in London in 1766, Langrage was most likely in touch with the more recent developments of Newtonian calculus. Among these, an important role was played by the residual analysis of John Landen. The method of fluxions is founded on the fundamental principle that every quantity may be supposed to be generated by a continuous increment, in the same way that a line is described by the motion of a point. John Landen (1719-1790) did not believe it to be the most correct approach: since it was a question of algebraic quantities it had to be considered a branch or a development of algebra, which has its own foundations and does not have to resort to those of the science of motion.

Preceded in 1758 by an announcement (A Discourse concerning the Residual Analysis), Landen’s treatise appeared in 1764: The Residual Analysis, a New Branch of the Algebraic Art, in which only finite increments are considered, and the term of “special value” is introduced for the basic definition of the value of the residual ratio when the denominator is null. Then the rules of the residual calculus are constructed and applied to the main problems of analysis: maxima and minima of functions, curvature, quadrature and rectification of curves.

In his introduction to Théorie des fonctions analytiques, Lagrange presented the calculus of derived functions as the true foundation of infinitesimal analysis, and described the incongruences and inadequacies of previous attempts to found infinitesimal calculus on infinitesimals of different orders (Leibniz, Bernoulli, L’Hôpital), or on the limit of ratios of finite differences (Euler, d’Alembert). He reproached Newton for introducing motion to his calculus of fluxions, and objected, moreover, to the formulation based on “the ultimate ratios of evanescent quantities” since the quantities were considered when they cease to exist. According to Lagrange, it was to avoid all these defects that a skillful mathematician, (“habile géomètre”) John Landen, had proposed a purely analytical method, in which the finite differences of the variables substitute the infinitesimal differences, although Lagrange adds: "on doit convenir que cette manière de rendre le Calcul différentiel plus rigoureux dans ses principes lui fait perdre ses principaux avantages, la simplicité de la méthode et la facilité des opérations".