DEVELOPMENT OF MATHEMATICAL THOUGHT. 691 



of groups investigates. Its immediate application, and 

 the purpose for which it was elaborated, was the theory 

 of Equations. Every equation constitutes an arrange- 

 ment in which a finite number of independent elements, 

 called constants or coefficients, is presented under a 

 certain algebraical form. The solution of the equation 

 means the finding of such an arrangement as when 

 substituted in the equation for the unknown quantity, 

 will satisfy the equation. 



The conception of a group of operations standing in 

 the defined relations is, however, capable of a great 

 and fundamental extension into that region of mathe- 

 matics which deals, not with fixed or constant, but with 

 variable or flowing; quantities ; not with elements which 

 are disconnected or discontinuous, but with such as are 

 continuous. To understand the development of modern 

 mathematical thought, it is accordingly necessary to go 

 back somewhat and review the progress which the 



. 



Continuous 



and dis- 

 continuous 



studying any manifold (e.g., such as 

 projective geometry, line geometry, 

 geometry of reciprocal radii, Lie's 

 sphere geometry, analysis situs, 

 &c.) as there are continuous groups 

 of transformations that can be 

 established ; and there are as many 

 invariant theories (see ' Ency. Math. 

 Wiss.,' vol.ii. p. 402 ; Nother, loc. cit., 

 p. 22). From that date onward the 

 different kinds of groups have been 

 defined and systematically studied, 

 notably by Klein and Lie and their 

 pupils. In this country, although 

 many of the relevant ideas were 

 contained in the writings notably 

 of Cayley and of Sylvester, the 

 systematic treatment of the subject 

 was little attended to before the 

 publication (1897) of Prof. Burn- 



j side's ' Theory of Groups of Finite 

 Order,' and latterly of his article 

 on the whole Theory of Groups in 

 the 29th volume of the ' Eucy. Brit.' 

 It has been remarked by those who 

 have studied most profoundly the 

 development of the two great 

 branches of mathematical tactics 

 viz., " The Theory of Invariants " 

 and the " Theory of Groups " that 

 the progress of science would have 

 been more rapid if the English 

 school had taken more notice of the 

 general comprehensive treatment 

 by Lie, and if Lie himself had not 

 refrained from entering more fully 

 into the special theories of that 

 school (see Dr F. Meyer, ' Bericht,' 

 &c., p. 231). 



