554 GEOMETRY 



mum 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 true calling. 



He published in 1869 a first work on the interpretation of imagin- 

 aries in geometry, and from 1870 he was in possession of the directing 

 ideas of his whole career. I had at this time the pleasure of seeing 

 him often, of entertaining him at Paris, where he had come with his 

 friend F. Klein. 



A course by M. Sylow followed by Lie had revealed to him all the 

 importance of the theory of substitutions; the two friends studied 

 this theory in the great treatise of C. Jordan; they were fully con- 

 scious of the important r61e it was called on to play in so many 

 branches of mathematical science where it had not yet been applied. 



They have both had the good fortune to contribute by their works 

 to impress upon mathematical studies the direction which to them 

 appeared the best. 



In 1870, Sophus Lie presented to the Academy of Sciences of Paris 

 a discovery extremely interesting. Nothing bears less resemblance 

 to a sphere than a straight line, and yet Lie had imagined a singular 

 transformation which made a sphere correspond to a straight line, 

 and permitted, consequently, the connecting of every proposition 

 relative to straight lines with a proposition relating to spheres, and 

 vice versa. 



In this so curious method of transformation, each property relative 

 to the lines of curvature of a surface furnishes a proposition relative 

 to the asymptotic lines of the surface attained. 



The name of Lie will remain attached to these deep-lying relations 

 which join to one another the straight line and the sphere, those two 

 essential and fundamental elements of geometric research. He de- 

 veloped them in a memoir full of new ideas which appeared in 1872. 



The works which followed this brilliant de"but of Lie fully con- 

 firmed the hopes it had aroused. Pluecker's conception relative to 

 the generation of space by straight lines, by curves or surfaces 

 arbitrarily chosen, opens to the theory of algebraic forms a field 

 which has not yet been explored, which Clebsch scarcely began to 

 recognize and settle the boundaries of. But, from the side of infini- 



