SopJius Lie. r,l 



The publication* of his first paper, " Representation der Imagi- 

 naren der Flange ometrie," in 1869, led to a travelling scholarship 

 which enabled him to visit foreign universities. The succeeding winter 

 was spent at Berlin, where Lie and Klein met and contracted their 

 long friendship, which was based, in the first instance, upon their 

 common interest in the range of ideas associated with Pliicker's 

 geometry. Thence in the early summer of 1870 they went to Paris, 

 coming into personal relations with Darboux and Jordan ; and their 

 growing interest in the theory of groups, first awakened in Lie by 

 Sylow's lectures, which he had attended as a student, was stimulated 

 for both of the two friends, as they recognised different modes in 

 which that theory could be newly applied to branches of mathematics. 

 In particular, Lie then discovered his now famous transformation, 

 which makes a sphere correspond to a straight line, and by which, 

 therefore, theorems on aggregates of lines can be translated into 

 theorems on aggregates of spheres. The paperf containing this result 

 was communicated to the Academy of Sciences by Chasles. 



The visit to Paris was brought to an abrupt end by the outbreak of 

 the war between France and Prussia. Klein was of course, as a German,, 

 bound to leave Paris; but the same obligation did not rest upon a 

 Norwegian, and Lie remained until August, when he conceived a plan 

 of walking through France to Italy. He had gone only as far as 

 Fontainebleau, when a misadventure suspended his journey for a time. 

 Let it be told in the words of Darboux J ' 



" Occupe sans cesse des idees qui fermentaient dans sa t6te, il allait 

 chaque jour dans la forest, s'arretant dans les sites les plus eloignes 

 des sentiers battus, prenant des notes, dessinant des figures au 

 crayon. II n'en fallait pas taut a cette epoque pour eveiller les 

 sotip9ons. Arre'te' et incarcere' a Fontainebleau, dans des condi- 

 tions d'ailleurs fort douces, il se reclamait de M. Chasles, de M. 

 Bertrand, d'autres encore ; je fis le voyage de Fontainebleau et 

 n'eus aucune peine a convaincre le procureur imperial ; toutes les 

 notes que 1'on avait saisies et oil figuraient des complexes, des 

 systemes orthogonaux, des noms de geometres, ne se rapportaient. 

 en aucune fac^on a la defense nationale. . . . 



After a delay of four weeks, he continued his journey to Italy ; then, 

 travelling homewards through Switzerland and Germany, he returned 

 to Christiania in December, after a joint note by Klein and himself had 

 been communicated in that month to the Berlin Academy by Kummer. 



* Reprinted, under a different title, in ' Crelle's Journal,' vol. 70 (1869), pp. 

 346353. 



t ' Comptes Rendus,' vol. 71 (1870), pp. 579583. 



+ See the ' Notice sur M. Sophus Lie,' spoken on the occasion when Lie's death 

 was announced to the Academy of Sciences (' Comptes Kendus,' vol. 128 (Feb. 27,. 

 1899), pp. 525529). 



