DEVELOPMENT OF GEOMETRIC METHODS. 43 * 



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 epoch 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 conscious 

 of the important role 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 resem- 

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

 singular transformation which made a sphere correspond to a straight, 

 and permitted, consequently, the connecting of every proposition rela- 

 tive to straights 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 debut 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, that Clebsch scarcely began to recognize 

 and settle the boundaries of. But, from the side of infinitesimal 

 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 afterward that of con- 

 tact 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 in- 

 tegration which Jacobi had instituted for partial differential equa- 

 tions 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 char- 

 acteristics of Monge is fully applicable to equations of the second order 

 with two independent variables. 



