690 



SCIENTIFIC THOUGHT. 



the beginnings of the new and comprehensive calculus 

 of operations which were contained in the writings of 

 Lagrange, Abel, Cauchy, and Galois, and established the 

 terminology and the algorithm. A group of substitu- 

 tions is defined as having the property that each two or 

 more operations belonging to it and successively applied 

 can be replaced by another single operation contained in 

 the same group. Succeeding operations are symbolically 

 represented by the product of two or more letters. This 

 product has certain algebraical properties, and in analogy 

 with common products it has factors, a degree, an index ; 

 the substitution may be cyclical and symmetric, and may 

 have many other remarkable properties which the theory ^ 



1 The "Theory of Groups" has 

 now grown into a very extensive 

 doctrine which, according to the late 

 Prof. Marius Sophus Lie (1842-99), 

 is destined to occupy a leading and 

 central position in the mathemati- 

 cal science of the future. " The 

 concej^tion of Group and Invariant 

 was for him not only a methodical 

 aspect from which he intended to 

 review the entire older region of 

 mathematics, but also the element 

 which was destined to permeate 

 and unify the whole of mathemati- 

 cal science " (M. Nother, ' Math. 

 Ann.,' voh liii. p. 39). But though it 

 is an undoubted fact that the largest 

 systematic works on the subject 

 emanate from that great Norwegian 

 mathematician, and that his ideas 

 have won gradual recognition, 

 especially on the part of prominent 

 French mathematicians, notably 

 M. Picard ('Traite d' Analyse,' 

 1896, vol. iii.) and M. Poincare, 

 the epoch - making tract which 

 pushed the novel conception into 

 the foreground was Prof. F. 

 Klein's ' Erlangen Programme ' 

 (1872), entitled " Vergleichende I 



Betrachtungen iiber neuere geo- 

 metrische Forschungen. " To those 

 who read and re-read this short 

 but weighty treatise, it must in- 

 deed have been like a revelation, 

 opening out entirely new avenues 

 of thought into which mathematical 

 research has been more and more 

 guided during the last generation. 

 The tract, which has now been 

 translated into all the important 

 modern languages, remained for a 

 long time comparatively unnoticed, 

 and, twenty years after its publica- 

 tion, was reprinted by the author 

 in the 4'3rd volume of the ' Math. 

 Annalen,' with some introductory 

 remarks which indicate the changes 

 that had taken place in the in- 

 terval as regards the scope of the 

 idea. The main result of the dis- 

 sertation is this : That, primarily, 

 for all geometrical investigations, 

 the characteristic properties of any 

 manifold (or arrangement) is not 

 the element out of which it is com- 

 posed, but the group, the transfor- 

 mations of which reveal its invarian- 

 tive properties. There are, accord- 

 ingly, as many different ways of 



