THE SCIENTIFIC SPIRIT IN GERMANY. 



187 



Jacobi, and still more through his contemporary Lejeune 

 Dirichlet (born 1804 at Diiren, of French extraction, 

 and trained in Paris under Laplace, Legendre, Fourier, 



the field of research which they 

 cultivated by fundamentally new 

 ideas of such breadth that fully 

 half a century was required be- 

 fore they were thoroughly appreci- 

 ated by mathematicians. Abel 

 (a Norwegian by birth) died in 

 1829 when only twenty - seven 

 years old, having during the four 

 years which embrace his published 

 memoirs extended the limits of 

 algebra and laid the foundations 

 for a more comprehensive treat- 

 ment of the higher or transcendent 

 functions, or forms of mathematical 

 dependence. Mathematicians be- 

 fore him had tried to solve algebra- 

 ically equations beyond the fourth 

 degree, but had failed. Abel proved 

 that the problem as then conceived 

 could not be generally solved. Le- 

 gendre had through his unaided 

 labours, extending over thirty 

 years, established the theory of 

 elliptic integrals as far as was 

 possible on the lines then adopted. 

 Abel and simultaneously Jacobi 

 treated the subject from an entirely 

 novel point of view, and by doing 

 so opened out quite a new field of 

 research, the extent and importance 

 of which Abel fully recognised when 

 he presented to the French Acad- 

 emy his memoir of 1826, in which 

 he dealt with functions of which 

 those studied by Legendre and 

 Jacobi were only special cases. 

 This memoir, containing Abel's 

 celebrated theorem, which he had 

 already discovered in 1825, and 

 which was published in a brief ar- 

 ticle in Crelle's Journal in 1829, re- 

 mained unnoticed, being, as Legen- 

 dre explained to Jacobi, almost un- 

 readable. See Enneper, ' Elliptische 

 Functionen,' 2d ed., p. 192 ; Jacobi's 

 Werke, vol. i. p. 439, &c. Abel 



has been called the greatest mathe- 

 matical genius that has yet existed 

 (Oltramare in ' La grande Encyclo- 

 pe"die,' art. "Abel"); his fellow- 

 worker, Jacob Steiner (1796-1863, 

 a Swiss by birth), has been termed 

 the greatest geometrician of modern 

 times. The progress of analysis 

 had thrown into the background 

 purely geometrical researches, al- 

 though a revival of these had com- 

 menced in France with Monge and 

 his followers, and had been further 

 promoted by Poncelet, as well 

 as simultaneously by Mobius and 

 Pliicker in Germany. The labours 

 of the two latter remained for a 

 long time unknown and unrecog- 

 nised. Steiner, who was self- 

 taught, who disliked the calculus, 

 and considered it a disgrace that 

 geometry could not solve her prob- 

 lems by purely geometrical methods, 

 undertook to find the common root 

 and leading principle which con- 

 nected all the theorems and por- 

 isms bequeathed to us by ancient 

 and modern geometry ; he brings 

 order into the chaos, and shows 

 how nature with a few elements 

 and the greatest economy succeeds 

 in giving to figures in space their 

 numberless properties. He not 

 only completed that part of geome- 

 try which had been treated by the 

 ancients the geometry of the line, 

 the conic sections or curves of the 

 second order, and the surfaces in 

 space corresponding to them but 

 he also attacked problems which 

 before him had been solved only by 

 the calculus, and even succeeded in 

 carrying his methods beyond the 

 reach of the calculus of varia- 

 tions, specially invented to deal 

 with geometrical questions. Like 

 Fermat in the theory of numbers, 



