76 Lagrange' s Memoirs. 



to this theorem and direct proofs of many other propositions, to 

 which Euler had arrived only by way of induction, and in which, 

 after having enriched the analysis of Diophantus and Fermat, the 

 author passes to the theory of equations with partial differences ex- 

 plains a striking paradox noticed by Euler, makes known an entire 

 class of equations of w^iich there were only some isolated exam- 

 ples, and puts out of sight the paradox by showing to what belong, 

 both the complete integral of these equations, and the singular 

 solution which is not comprised in this integral. 



A formula for the return of series, remarkable by its generality 

 and the simplicity of the law, of which he makes a happy applica- 

 tion to the problem of Kepler, and thence succeeds in rendering 

 sensible the convergency of the analytical expression of the equa- 

 tion of the center, a convergency w4iich we had always supposed, 

 without being able to demonstrate. 



An important memoir on the solution of numerical equations, con- 

 taining also new remarks on that of algebraical equations. This 

 work served as the base to a treatise which he afterwards pubhshed, 

 under the same title, and of which he gave two editions. 



Another memoir, no less important, and still more original, where 

 he reduces to operations of pure algebra, every process of the differ- 

 ential and integral calculus, which he separates from every idea of 

 infinitely small, of fluxions, of limits and of vanishing, and demon- 

 strates the lawfulness of the abbreviations permitted in these two cal- 

 culs, which he also frees from all difficulties, and from all paradoxes 

 that had sprung up in an imperfect and suspected metaphysique. 



The demonstration of a curious theorem on primal numbers ; a 

 demonstration that no one had been able to find, and the more diffi- 

 cult, as we know how to express algebraically propositions of this 

 kind. 



The integration of partial differences of the first order, by a fruit- 

 ful principle, sufficient for the greater part of cases where this inte- 

 gration is possible. 



A purely analytical solution of the problem of the rotation of a 

 body of any figure, wherein he at last surmounts difficulties that had 

 long stopped him, and by which geometers seemed to expect, with 

 curiosity, some ulterior developments, that they hoped to find in the 

 second volume of his new Mecanigue Analytique. 



Many memoirs on the obscure and difficult theory of probabilities, 

 wherein we admire the integral that forms its base, the number and 



