374 POPULAR SCIENCE MONTHLY. 



While these examples exhibit a very close relation between con- 

 tinuous and discontinuous groups of infinite order, yet the methods 

 employed to investigate problems belonging to these groups are gen- 

 erally quite different. The theory of the former is mainly due to 

 Sophus Lie and has been developed principally with a view to the solu- 

 tion of differential equations. The theory of the latter has been 

 developed largely in connection with questions in function theory and 

 owes its rapid growth to the influence of Klein. A large part of Lie's 

 results are contained in his ' Transf ormationsgruppen, ' consisting of 

 three large volumes, while the ' Modulf unctionen ' and 'Automorphe 

 Functionen' of Klein and Fricke are the best works on the discon- 

 tinuous groups of infinite order. 



Although the notion of group is one of the most fundamental ones 

 in mathematics, yet it is one which is more useful to arrive at reasons 

 for certain results and at connections between apparently widely 

 separated developments than to furnish methods for attaining these 

 results or developments. Its greatest service so far has been its uni- 

 fying influence and its usefulness in proving the possibility or the 

 impossibility of certain operations. In fact, it is generally conceded 

 that group theory had its origin in the use which Euffini and Abel made 

 of it to prove that the general equation of the fifth degree can not be 

 solved by radicals. 



While it may be said to have 'shown its dominating influence in 

 nearly all parts of mathematics, not only in recent theories, but also 

 far towards the foundation of the subject, so that this theory can no 

 longer be omitted in the elementary text-books,'* yet this influence 

 is largely a guiding influence. The bulk of mathematics is not group 

 theory and the main part of the work must always be accomplished by 

 methods to which this notion is foreign. On the other hand, it seems 

 safe to say that this theory is not a fad which will pass into oblivion 

 as rapidly as it rose into prominence. Its applications are so ex- 

 tensive and useful that it must always receive considerable attention. 

 Moreover, it presents so many difficulties that it will doubtless offer 

 rich results to the investigator for a long time. 



* Pund, ' Algebra mit Einschluss der elementaren Zahlentheorie,' 1899, 

 preface. 



