72 Lagrange^ s Memoirs, 



stratlons. These were not found at his death, being perhaps suppress- 

 ed as insufficient, or for some other cause hard to be conjectured. 

 These theorems, in other respects may appear more curious than 

 useful. We know however, that difficulty is an attraction, for all 

 men and especially for geometers. Without such attraction, can we 

 believe that they would have placed so much importance on the 

 problems of the brachystochrone, of isoperimeters, and of orthogonal 

 trajectories ? In truth, they wished to create the science of the cal- 

 culus, and to invent or bring to perfection methods which could not fail 

 of finding one day useful applications. Under this light, they would 

 devote themselves to the first question which required the employ- 

 ment of new resources. 



Such was for them as fine a fortune as the system of the world dis- 

 covered by Newton. Never had transcendant analysis been able to find 

 a more worthy or grand subject. Whatever progress is made therein 

 the first inventor will preserve his rank. Lagrange who often called 

 him the greatest genius which had ever existed, added himself also 

 et le plus heureux ; on ne trouve gu'une fois un systeme du monde 

 a etahlir. It required a hundred years of labors and of discoveries to 

 raise the edifice of which Newton had laid the foundations. Yet he 

 has received the praise of all, and has been supposed to have finish- 

 ed entirely the career which he simply began ; began, however, with 

 an eclat which should encourage his successors. 



Many geometers, undoubtedly, practised upon the theorems of 

 Fermat, but not one ever succeeded. Euler alone had made some 

 progress in this difficult path wherein have since distinguished them- 

 selves M. Legendre and M. Gauss. Lagrange, upon demonstrating 

 or correcting some attentive glimpses of Euler resolved a problem 

 which appeared to be the hnot of all the rest, and from which he 

 made flow a useful result, that is to say, the complete resolutions of 

 equations of the second degree, with two unknown quantities that 

 must be entire numbers. The memoir, printed like the preceding 

 among those of the Academy of Turin, is nevertheless dated at Ber- 

 lin, the 20th September, 1768. This date points out to us one of 

 the events, (few indeed,) which show that the life of Lagrange is not 

 all in his works. 



The residence at Turin pleased him little. He saw there no one 

 that cultivated mathematics with success : he was impatient to see 

 the savans of Paris with whom he corresponded. M. de Caraccioli, 

 with whom he lived in the greatest intimacy, had just been nominated 



