DEVELOPMENT OF GEOMETRIC METHODS 555 



tesimal geometry, this conception has been given its full value by 

 Sophus Lie. The great Norwegian geometer was able to find in it 

 first the notion of congruences and complexes of curves, and after- 

 ward that of contact transformations of which he had found, for the 

 case of the plane, the first germ in Pluecker. The study of these 

 transformations led him to perfect, at the same time with M. Mayer, 

 the methods of integration which Jacobi had instituted for partial 

 differential equations of the first order; but above all it threw the 

 most brilliant light on the most difficult and the most obscure parts 

 of the theories relative to partial differential equations of higher 

 order. It permitted Lie, in particular, to indicate all the cases in 

 which the method of characteristics of Monge is fully applicable to 

 equations of the second order with two independent variables. 



In continuing the study of these special transformations, Lie was 

 led to construct progressively his masterly theory of continuous 

 groups of transformations and to put in evidence the very important 

 role that the notion of group plays in geometry. Among the essential 

 elements of his researches, it is proper to signalize the infinitesimal 

 transformations, of which the idea belongs exclusively to him. 



Three great books published under his direction by able and de- 

 voted collaborators contain the essential part of his works and their 

 applications to the theory of integration, to that of complex units and 

 to the non-Euclidean geometry. 



XIV 



By an indirect way I have arrived at that non-Euclidean geometry 

 the study of which takes in the researches of geometers a place which 

 grows greater each day. 



If I were the only one to talk with you about geometry, I should 

 take pleasure in recalling to you* all that has been done on this sub- 

 ject since Euclid or at least from Legendre to our days. 



Envisaged successively by the greatest geometers of the last cen- 

 tury, the question has progressively enlarged. 



It commenced with the celebrated postulatum relative to parallels; 

 it ends with the totality of geometric axioms. 



The Elements of Euclid, which have withstood the action of so 

 many centuries, will have at least the honor before ending of arous- 

 ing a long series of works admirably enchained which will contrib- 

 ute, in the most effective way, to the progress of mathematics, at the 

 same time that they furnish to the philosophers the most precise and 

 the most solid points of departure for the study of the origin and of 

 the formation of our cognitions. 



I am assured in advance that my distinguished collaborator will 

 not forget, among the problems of the present time, this one, which is 

 perhaps the most important, and with which he has occupied himself 



