DEVELOPMENT OF MATHEMATICAL THOUGHT. 719 



commented on by Cayley) was later on supplied in con- 

 sequence of a suggestion of his. The researches of 

 Eiemann, and still more those of Helmholtz, had not 

 merely a mathematical, they had also a logical and a 

 psychological, meaning. Space was conceived to be a 

 threefold - extended manifold. There are other mani- 

 folds besides space such, for instance, as the threefold- 

 extended manifold of colours. Helmholtz came from the 

 study of this manifold to that of space. Now the 

 question arises as to the conditions or data which are 

 necessary and sufficient for the foundations of a science 

 like geometry. We have seen that the axiom of parallel 

 lines is not required ; we have also seen that the notion 

 of distance and number can be generalised. What other 

 data remain which cannot be dispensed with ? Helm- 

 holtz had attempted to answer this question. But 

 neither he nor Eiemann had considered the possibility 

 of a purely protective geometry. Now it is the merit of 

 Prof. Klein to have seen that there exists a purely alge- 

 braical method by which this problem can be attacked. 

 This is the method of groups referred to above, and 57. 



Sophus Lie. 



applied by Sophus Lie to assemblages of continuously 

 variable quantities. Klein was one of the first to recog- 

 nise the power of this new instrument. He saw that 

 the space problem was a problem of transformations, the 

 possible motions in space forming a group with definite 

 elements (the different freedoms of motion) which were 

 continuously variable i.e., in infinitesimal quantities 

 and which returned into themselves under certain well- 

 defined conditions. They possessed, moreover, in the 

 maintenance of distance the algebraic property of in- 



