DEVELOPMENT OF MATHEMATICAL THOUGHT. 689 



In following this altered course of investigation, an 

 enormous amount of mathematical knowledge was gained, 

 and problems were solved which had previously never 

 been thought of. Especially through the theory of equa- 

 tions the abstract doctrine of algebraical forms was 

 created and greatly advanced long before it was gener- 

 ally recognised that it had peculiar importance through 

 the correspondence or parallelism which existed be- 

 tween algebraical expressions and geometrical con- 

 figurations. 



Out of these earlier algebraical and later combined al- 44. 



Theory of 



gebraical and geometrical investigations, a novel and very groups. 

 useful point of view has been gradually gained which 

 represents the most general conception of mathematical 

 tactics. This centres in the notion of a group of ele- 

 ments. These elements may be quantities or opera- 

 tions, so that the theory of Groups embraces not only 

 the doctrines which deal with quantities but also those 

 which deal with arrangements and their possible changes. 

 The older combinatorial analysis dealt mainly with 

 assemblages of a quantity of separate elements, their 

 number, their variety : the modern theory of groups 

 deals rather with the processes and operations by which 

 different arrangements can be transformed one into the 

 other. It is an algebra of operations. The methods 

 of transformation which presented themselves first of 

 all were the methods known in algebra as substitution. 

 Accordingly the first comprehensive treatise on the 

 theory was the ' Treatise on Substitutions,' published in 

 1870 by M. Camille Jordan. This book forms a land- 

 mark in modern mathematics ; it brought into a system 

 VOL. II. 2 x 



