720 SCIENTIFIC THOUGHT. 



variance. He also expressed some doubt regarding the 

 logical consistency of the assumptions of Helmholtz. 

 Sophus Lie undertook this investigation, and thus 

 brought the logical side of the labours of Eiemann 

 and Helmholtz to a final conclusion. 1 This is one of 

 the celebrated instances where the rigorous algebraical 

 methods have detected flaws in the more intuitional or 

 purely geometrical process, and extended our knowledge 

 of hidden possibilities. 



But there is yet another branch of the great science 

 of number, form, and interdependence, the principles 

 and foundations of which had been handed down from 

 earlier ages, where the critical and sifting process of the 

 nineteenth century has led to an expansion and revolu- 

 tion of our fundamental ideas. Here also, as in so 

 many other directions, the movement begins with Gauss. 

 Hitherto I have spoken mainly of algebra or general 

 arithmetic, of geometry, of the connections of both in the 



1 " Lie was early made aware by the only essential invariant. When 



Klein and his "program" that the Lie took up this problem in prin- 



space problem belonged to the ciple, as one belonging to the theory 



theory of groups. . . . Ever since of groups, he recognised that for 



1880 he had been pondering over our space that part of the axiom of 



these questions ; he published his monodromy was unnecessary which 



views first in 1886 on the occasion , added periodicity to the free mo- 



of the Berlin meeting of natural | bility round a fixed axis. . . . 



philosophers. Helmholtz's concep- | The value of these investigations 



tion was itself unconsciously (but I lies mainly in this, that they permit 



remarkably so, inasmuch as it of our fixing for every kind of geo- 



dates from 1868) one belonging to metry the most appropriate system 



the theory of groups, trying, as it of axioms. . . . And they justly 



did, to characterise the groups of received in the year 1897 the first 



the sixfold infinite motions in Lobatchevski prize awarded by the 



space, which led to the three Society of Kasan " (M. Neither, 



geometries, in comparison with all ' Math. Ann.,' vol. liii. p. 38). A 



other groups. He did this by i lucid exposition of Lie's work will 



fixing on the free mobility of rigid ' be found in Mr B. Russell's ' Essay," 



bodies i.e., on the existence of an &c., p. 47 tqq. 

 invariant between two points as 



