550 GEOMETRY 



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 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 con- 

 ception. 



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 para- 

 meters. Thus did Beltrami in extending with great ingenuity the 

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

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



It seems that to-day is accepted a mixed method whose origin is 

 found in the works of Ribaucour, under the name perimorphie. The 

 rectangular axes of analytic geometry are retained, but made mobile 

 and attached as seems best to the system to be studied. Thus dis- 

 appear most of the objections which have been made to the method 

 of coordinates. The advantages of what is sometimes called intrinsic 

 geometry are united to those resulting from the use of the regular 

 analysis. Besides, this analysis is by no means abandoned; the com- 

 plications of calculation which it almost always carries with it, in its 

 applications to the study of surfaces and rectilinear coordinates, usu- 

 ally disappear if one employs the notion on the invariants and the 

 covariants of quadratic powers of differentials which we owe to the 

 researches of Lipschitz and Christoffel, inspired by Riemann's studies 

 on the non-Euclidean geometry. 



XII 



The results of so many labors were not long in coming. The notion 

 of geodesic curvature which Gauss already possessed, but without 

 having published it, was given by Bonnet and Liouville; the theory 

 of surfaces of which the radii of curvature are functions one of the 

 other, inaugurated in Germany by two propositions which would 

 figure without disadvantage in the memoir of Gauss, was enriched 

 by Ribaucour, Halphen, S. Lie, and others, with a multitude of propo- 

 sitions, some concerning these surfaces envisaged in a general man- 

 ner; others applying to particular cases where the relation between 

 the radii of curvature takes a form particularly simple; to minimal 

 surfaces for example, and also to surfaces of constant curvature, 

 positive or negative. 



The minimal surfaces were the object of works which make of 

 their study the most attractive chapter of infinitesimal geometry. 

 The integration of their partial differential equation constitutes one 

 of the most beautiful discoveries of Monge; but because of the im- 

 perfection of the theory of imaginaries, the great geometer could not 



