DEVELOPMENT OF GEOMETRIC METHODS 549 



tangents, of asymptotic lines which have taken so important a place 

 in recent researches. Nor should we forget the determination of the 

 surface of which all the lines of curvature are circles, nor above all 

 the memoir on triple systems of orthogonal surfaces where is found, 

 together with the discovery of the triple system formed by surfaces 

 of the second degree, the celebrated theorem to which the name of 

 Dupin will remain attached. 



Under the influence of these works and of the renaissance of syn- 

 thetic methods, the geometry of infinitesimals retook in all researches 

 the place Lagrange had wished to take away from it forever. 



Singular thing, the geometric methods thus restored were to receive 

 the most vivid impulse in consequence of the publication of a memoir 

 which, at least at first blush, would appear connected with the purest 

 analysis; we mean the celebrated paper of Gauss, Disquisitiones 

 generates circa superficies curvas, which was presented in 1827 to the 

 Gottingen Society, and whose appearance marked, one may say. 

 a decisive date in the history of infinitesimal geometry. 



From this moment, the infinitesimal method took in France a free 

 scope before unknown. 



Frenet, Bertrand, Molins, J. A. Serret, Bouquet, Puiseux, Ossian 

 Bonnet, Paul Serret, develop the theory of skew curves. Liouville, 

 Chasles, Minding, join them to pursue the methodic study of the 

 memoir of Gauss. 



The integration made by Jacobi of the differential equation of the 

 geodesic lines of the ellipsoid started a great number of researches. 

 At the same time the problems studied in the Application de I' Analyse 

 of Monge were greatly developed. 



The determination of all the surfaces having their lines of curvature 

 plane or spheric completed in the happiest manner certain partial 

 results already obtained by Monge. 



At this moment, one of the most penetrating of geometers, ac- 

 cording to the judgment of Jacobi, Gabriel Lam4, who, like Charles 

 Sturm, had commenced with pure geometry and had already made to 

 this science contributions the most interesting by a little book pub- 

 lished in 1817 and by memoirs inserted in the Annales of Gergonne, 

 utilized the results obtained by Dupin and Binet on the system of 

 confocal surfaces of the second degree, and, rising to the idea of 

 curvilinear coordinates in space, became the creator of a wholly new 

 theory destined to receive in mathematical physics the most varied 

 applications. 



XI 



Here again, in this infinitesimal branch of geometry are found the 

 two tendencies we have pointed out a propos of the geometry of finite 

 quantities. 



