432 POPULAR SCIENCE MONTHLY. 



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 

 of which the study 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 would 

 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 poskdatum 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 arousing 

 a long series of works admirably enchained which will contribute, in 

 the most effective way, to the progress of mathematics, at the same 

 time that they furnish to the philosophers the points of departure the 

 most precise and the most solid 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 

 with so much success; and I leave to him the task of developing it 

 with all the amplitude which it assuredly merits. 



I have just spoken of the elements of geometry. They have re- 

 ceived in the last hundred years extensions which must not be for- 

 gotten. The theory of polyhedrons has been enriched by the beautiful 

 discoveries of Poinsot on the star polyhedrons and those of Moebius 

 on polyhedrons with a single face. The methods of transformation 

 have enlarged the exposition. We may say to-day that the first book 

 contains the theory of translation and of symmetry, that the second 

 amounts to the theory of rotation and of displacement, that the 

 third rests on homothety and inversion. But it must be recognized 

 that it is thanks to analysis that the ' Elements ' have been enriched 

 by their most beautiful propositions. 



