Page:Popular Science Monthly Volume 64.djvu/378

374 While these examples exhibit a very close relation between continuous and discontinuous groups of infinite order, yet the methods employed to investigate problems belonging to these groups are generally quite different. The theory of the former is mainly due to Sophus Lie and has been developed principally with a view to the solution 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 'Transformationsgruppen,' consisting of three large volumes, while the 'Modulfunctionen' and 'Automorphe Functionen' of Klein and Fricke are the best works on the discontinuous 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 unifying 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 Ruffini 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 extensive 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.