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 Göttingen 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 l'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, according to the judgment of Jacobi, Gabriel Lamé, 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 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 à propos of the geometry of finite quantities.
Some, among whom must be placed J. Bertrand and O. Bonnet, wish to constitute an independent method resting directly on the employment of infinitesimals. The grand 'Traité de Calcul différentiel,' 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 Lamé in introducing his theory of differential parameters. Thus did Beltrami in extending with great ingenuity the employment of these differential invariants to the case of two independent variables, that is to say, to the study of surfaces.
