DEVELOPMENT OF GEOMETRIC METHODS. 427 



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, usually 

 disappear if one employs the notion on the invariants and the covari- 

 ants of quadratic powers of differentials which we owe to the researches 

 of Lipschitz and Christoffel, inspired by Eiemann'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 with- 

 out 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 Eibau- 

 cour, Halphen, S. Lie and others, with a multitude of propositions, 

 some concerning these surfaces envisaged in a general manner; 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 nega- 

 tive. 



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 imperfection 

 of the theory of imaginaries, the great geometer could not get from 

 its formulas any mode of generation of these surfaces, nor even any 

 particular surface. We will not here retrace the detailed history which 

 we have presented in our ' Legons sur la theorie des surfaces ' ; but it 

 is proper to recall the fundamental researches of Bonnet which have 

 given us, in particular, the notion of surfaces associated with a given 

 surface, the formulas of Weierstrass which establish a close bond be- 

 tween the minimal surfaces and the functions of a complex variable, 

 the researches of Lie by which it was established that just the formulas 

 of Monge can to-day serve as foundation for a fruitful study of minimal 

 surfaces. 



In seeking to determine the minimal surfaces of smallest classes 

 or degrees, we were led to the notion of double minimal surfaces which 

 is dependent on Analysis situs. 



