DEVELOPMENT OF MATHEMATICAL THOUGHT. 703 



The cases of Cauchy, Abel, and Jacob! are the best- 

 known instances. Through their labours an entirely 

 new field had been prospected and partially cultivated. 

 It was to this that Weierstrass, the other great leader 

 in modern theory, was attracted. He made the clear 

 definition and logical coherence of the novel concep- 

 tions which it involved his principal aim. Gauss 

 had laboured without assistance at similar problems, 

 making many beginnings which even his colossal intellect 

 could not adequately develop. Weierstrass early gathered 

 around him a circle of ardent and receptive pupils and 

 admirers, 1 to whose care and detailed elaboration he 



1 The researches of Weierstrass 

 (1815 to 1897) began somewhat 

 earlier than those of Riemann, but 

 only became generally known and 

 appreciated in their fundamental 

 originality through his pupils his 

 academic influence dating from 

 the year 1861. Some account of 

 Weierstrass's activity is given by 

 Emil Lampe in the 6th volume 

 (1899) of the ' Bericht der Math. 

 Verein.,' p. 27, &c. The genesis 

 of his ideas is traced by Brill 

 and Nbther in the Report quoted 

 in the last note, and by M. 

 Poincar^ in ' Acta Math.,' vol. xxii. 

 The former divides his Researches 

 roughly into two periods, during 

 the first of which (1848-56) he 

 dealt with what Cayley would 

 call " known " functions ; progress 

 during this period depending not 

 so much upon fundamentally new 

 ideas as upon an investigation of 

 special problems and great analyti- 

 cal skill. The second period begins 

 in the year 1869, and is devoted to 

 nothing less than the building up 

 of the entire structure of mathe- 

 matical thought from the very 

 beginning upon altered definitions, 

 through which the dilemmas and 



paradoxes would be obviated that 

 had shown themselves ever since 

 the middle of the eighteenth cen- 

 tury in consequence of a too 

 confident application and extension 

 of conventional ideas suggested 

 mainly by practical problems. The 

 elements of this grand edifice are 

 now largely accepted, not only in 

 Germany, but also in France, Italy, 

 and England. In Germany Prof. 

 0. Stolz, through his works on 

 General Arithmetic, 2 vols. (1885 

 and 1886), and the Calculus, 3 vols. 

 (1893 to 1899), has probably done 

 more than any other academic 

 teacher to utilise the new system 

 of mathematical thought for the 

 elementary course of teaching. It 

 seems of importance to state, how- 

 ever, that outside of the circle of 

 Weierstrass's influence, and quite 

 within the precincts of Riemann's 

 school, the necessity was felt of 

 strengthening the foundations on 

 which research in higher mathe- 

 matics was carried on, by going 

 back to the fundamental ideas of 

 arithmetic. The principal repre- 

 sentative of this line of research 

 was Hermann Hankel (1839-73), a 

 pupil of Riemann's, who, in the 



