426 POPULAR SCIENCE MONTHLY. 



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 

 generales 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, 

 €hasles, 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 

 T* Analyse ' of Monge were greatly developed. 



The determination of all the surfaces having their lines of curva- 

 ture 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 Lame, who, like Charles 

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

 this science contributions the most interesting by a little book published 

 in 1817 and by memoirs inserted in the Annates 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. 



XL 



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. 



Some, among whom must be placed J. Bertrand and 0. Bonnet, 

 wish to constitute an independent method resting directly on the em- 

 ployment of infinitesimals. The grand ' Traite de Calcul differentiel,' 

 of Bertrand, contains many chapters on the theory of curves and of 

 surfaces, which are, in some sort, the illustration of this conception. 



Others follow the usual analytic ways, being only intent to clearly 

 recognize and put in evidence the elements which figure in the first 

 plan. Thus did Lame in introducing his theory of differential param- 

 eters. Thus did Beltrami in extending with great ingenuity the em- 

 ployment of these differential invariants to the case of two inde- 

 pendent variables, that is to say, to the study of surfaces. 



