MATHEMATICS IN THE NINETEENTH CENTURY 481 



spheres, or other configurations at pleasure, the geometry of birational 

 transformation, the analysis situs, etc., etc. Klein finds this answer: 

 In each geometry we have a system of objects and a group which 

 transforms these objects one into another. We seek the invariants 

 of this group. In each case it is the abstract group and not the con- 

 crete objects which is essential. The fundamental role of a group in 

 geometrical research is thus made obvious. Its importance is the 

 solution of algebraic equation, in the theory of differential equations 

 in the automorphic functions we have already seen. The immense 

 theory of algebraic invariants developed by Cayley and Sylvester, 

 Aronhold, Clebsch, Gordan, Hermite, Brioschi, and a host of zealous 

 workers in the middle of the century, also finds its place in the far 

 more general invariant theory of Lie's theory of groups. The same is 

 true of the theory of surfaces, so far as it rests on the theory of differ- 

 ential forms. In the theory of numbers, groups have many important 

 applications, for example, in the composition of quadratic forms and 

 the cyclotomic bodies. Finally, let us note the relation between hyper- 

 complex numbers and continuous groups discovered by Poincare. 



In resume, we may thus say that the group concept, hardly not- 

 iceable at the beginning of the century, has at its close become one 

 of the fundamental and most fruitful notions in the whole range of 

 our science. 



Infinite Aggregates 



Leaving the subject of groups, we consider now briefly another 

 fundamental concept, namely, infinite aggregates. In the most 

 diverse mathematical investigations we are confronted with such 

 aggregates. In geometry the conceptions of curves, surface, region, 

 frontier, etc., when examined carefully, lead us to a rich variety of 

 aggregates. In analysis they also appear, for example, the domain 

 of definition of an analytic function, the points where a function of 

 a real variable ceases to be continuous or to have a differential coeffi- 

 cient, the points where a series of functions ceases to be uniformly 

 convergent, etc. 



To say an aggregate (not necessarily a point aggregate) is infinite 

 is often an important step; but often again only the first step. To 

 penetrate farther into the problem may require us to state how 

 infinite. This requires us to make distinctions in infinite aggregates, 

 to discover fruitful principles of classification, and to investigate the 

 properties of such classes. 



The honor of having done this belongs to George Cantor. The 

 theory of aggregates is for the most part his creation; it has en- 

 riched mathematical science with fundamental and far-reaching 

 notions and results. 



The theory falls into two parts; a theory of aggregates in general, 



