688 



SCIENTIFIC THOUGHT. 



and what were the conditions i.e., the special proper- 

 ties of an equation which rendered it soluble ? These 

 were some of the questions which the great mathe- 

 maticians, such as Gauss, Abel, and Galois, placed before 

 themselves during the earlier part of the century. There 

 are other unsolved problems which the nineteenth cen- 

 tury inherited from preceding ones, where the same line 

 of reasoning was adopted i.e., where the question was 

 similarly reversed. Instead of trying to solve problems 

 as yet unsolved, it was proposed to prove their general 

 insolubility, and to show the reason of this ; also to 

 define the conditions which make a solution possible. 



earthed and placed in their cor- 

 rect historical perspective. Prof. 

 Burkhardt of Gottingen, to whom 

 we also owe the chapter on this 

 subject in the first volume of the 

 ' Encyklopadie,' &c., contributed in 

 the year 1892 a most interesting 

 historical paper, " Die Anfange der 

 Gruppentheorie und Paolo Ruffini " 

 (' Abhandl. zur Gesch. der Math.,' 

 6 Heft). In this paper he also 

 goes back to other earlier analysts, 

 among them Prof. Waring of Cam- 

 bridge, who during his lifetime used 

 to complain that he knew of no one 

 who read his mathematical tracts. 

 It appears that during nearly the 

 last thirty years of the eighteenth 

 century nothing had been added re- 

 garding the general theory of equa- 

 tions, and that Ruffini was the first 

 to begin a new epoch in the year 

 1799, with the distinct assertion that 

 a general solution of algebraic equa- 

 tions beyond the fourth degree, by 

 means of radicals, was impossible, 

 and with an attempt to prove this. 

 His researches were therefore con- 

 temporaneous with those of Gauss, 

 who published his ' Dissertation ' 

 (see note p. 644) in the same year, 

 and his great arithmetical work 



in 1801. Although Gauss seems 

 to have arrived at the same con- 

 clusion, and perhaps even to have 

 anticipated much later attempts to 

 solve the general equation of the fifth 

 degree by other than algebraical 

 operations (see Sylow, loc. tit., p. 

 16), his published researches rather 

 took the line of the study of a 

 definite class of soluble equations 

 which were connected with the 

 celebrated problem of the division 

 of the circle ; a satisfactory proof of 

 Ruffini's statement being withheld 

 till Abel published his celebrated 

 memoir in the year 1825 in the first 

 volume of Crelle's ' Journal.' With 

 this memoir the theory of equations 

 entered a new phase, towards 

 which the labours of Ruffini were 

 preparatory. As in so many other 

 cases, so also in this, the solu- 

 tion of the problem depended upon 

 stricter definitions of what was 

 meant by the solution of an equa- 

 tion, and by " algebraical " and 

 other ("transcendental") functions 

 and operations. We know that 

 both Abel and Galois began their 

 research by futile attempts to find 

 a solution of the general equation 

 of the fifth degree. 



