430 POPULAR SCIENCE MONTHLY. 



geometry, of projective geometry and of infinitesimal geometry. Many 

 other orthogonal systems have been made known. Among these it is 

 proper to signalize the cyclic systems of Kibaucour, for which one 

 of the three families admits circles as orthogonal trajectories, and the 

 more general systems for which these orthogonal trajectories are simply 

 plane curves. 



The systematic employment of imaginaries, which we must be 

 careful not to exclude from geometry, has permitted the connection 

 of all these determinations with the study of the finite deformation 

 of a particular surface. 



Among the methods which have permitted the establishment of 

 all these results it is proper to note the systematic employment of 

 linear partial differential equations of the second order and of systems 

 formed of such equations. The most recent researches show that this 

 employment is destined to renovate most of the theories. 



Infinitesimal geometry could not neglect the study of the two 

 fundamental problems set it by the calculus of variations. 



The problem of the shortest path on a surface was the object of 

 masterly studies by Jacobi and by Ossian Bonnet. The study of 

 geodesic lines has been followed up; we have learned to determine 

 them for new surfaces. The theory of ensembles has come to permit 

 the following of these lines in their course on a given surface. 



The solution of a problem relative to the representation of two 

 surfaces one on the other has greatly increased the interest of dis- 

 coveries of Jacobi and of Liouville relative to a particular class of 

 surfaces of which the geodesic lines could be determined. The results 

 concerning this particular case led to the examination of a new ques- 

 tion: to investigate all the problems of the calculus of variations of 

 which the solution is given by curves satisfying a given differential 

 equation. 



Finally, the methods of Jacobi have been extended to space of 

 three dimensions and applied to the solution of a question which 

 presented the greatest difficulties: the study of properties of minimum 

 appertaining to the minimal surface passing through a given contour. 



XIII. 



Among the inventors who have contributed to the development of 

 infinitesimal geometry, Sophus Lie distinguishes himself by many 

 capital discoveries which place him in the first rank. 



He was not one of those who show from infancy the most char- 

 acteristic aptitudes, and at the moment of quitting the University of 

 Christiania in 1865, he still hesitated between philology and mathe- 

 matics. 



It was the works of Pluecker which gave him for the first time 

 full consciousness of his veritable calling. 



