250 Biography of M. Le Comte Lagrange, 



in part of persons whose language he did not understand. 

 He was then right in dividing the office in order that it 

 might be completely filled. 



We must not he expected to follow Lagrange step by 

 step, in the learned researches with which he has filled the 

 Memoirs of Berlin, after his arrival in that city, on the 6th 

 November, 1766, and even some volumes of the Academy 

 of Turin, that owed to him, in all respects, its existence. 

 But we cannot omit pointing out, at least in a few words, 

 the most remarkable which they contain. He wrote the 

 following Memoirs in the Transactions of the Berlin 

 Academy : — 



1 . A great memoir wherein are found the demonstration 

 of a curious proposition that Euler could not demonstrate, 

 a new extension given 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 explains 

 a striking paradox noticed by Euler, makes known an 

 entire class of equations of which there were only some 

 isolated examples, 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. 



2. A formula for the return of series, remarkable by its 

 generality and the simplicity of the law, of which he makes 

 a happy application to the problem of Kepler, and thence 

 succeeds in rendering sensible the convergency of the 

 analytical expression of the equation of the center, a con- 

 vergency which we had always supposed, without being 

 able to demonstrate. 



3. An important memoir on the solution of numerical 

 equations, containing also new remarks on that of alge- 

 braical equations. This work served as the basis of a 

 treatise which he afterwards published, under the same 

 title, and of which he gave two editions. 



4. Another memoir, no less important, and still more 

 original, where he reduces to operations of pure algebra, 

 every process of the differential and integral calculus, which 

 he separates from every idea of infinitely small, of fluxions, 



