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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 

 Klein and his " program " that the 

 space problem belonged to the 

 theory of groups. . . . Ever since 

 1880 he had been pondering over 

 these questions ; he published his 

 views first in 1886 on the occasion 

 of the Berlin meeting of natural 

 philosophers. Helmholtz's concep- 

 tion was itself unconsciously (but 

 remarkably so, inasmuch as it 

 dates from 1868) one belonging to 

 the theory of groups, trying, as it 

 did, to characterise the groups of 

 the sixfold infinite motions in 

 space, which led to the three 

 geometries, in comparison with all 

 other groups. He did this by 

 fixing on the free mobility of rigid 

 bodies i.e., on the existence of an 

 invariant between two points as 



the only essential invariant. When 

 Lie took up this problem in prin- 

 ciple, as one belonging to the theory 

 of groups, he recognised that for 

 our space that part of the axiom of 

 monodromy was unnecessary which 

 added periodicity to the free mo- 

 bility round a fixed axis. . . . 

 The value of these investigations 

 lies mainly in this, that they permit 

 of our fixing for every kind of geo- 

 metry the most appropriate system 

 of axioms. . . . And they justly 

 received in the year 1897 the first 

 Lobatchevski prize awarded by the 

 Society of Kasan" (M. Nbther, 

 'Math. Ann.,' vol. liii. p. 38). A 

 lucid exposition of Lie's work will 

 be found in Mr B. Russell's ' Essay/ 

 &c., p. 47 sqq. 



