686 



SCIENTIFIC THOUGHT. 



42. 



Principle of 

 substitu- 

 tion. 



counterpart in the transformation of algebraical forms 

 by the processes of substitution, these latter had 

 already been extensively studied for their own sakes in 

 the theory of algebraical equations, which in the first 

 quarter of the century had undergone a great develop- 

 ment under the hands of two brilliant mathematical 

 talents both lost to science at an early age the 

 Norwegian Abel and the Frenchman fivariste Galois. 1 

 Like all algebraical expressions, those termed equa- 

 tions were originally invented and commanded attention 



1 Evariste Galois is held to have 

 been one of the greatest mathema- 

 tical geniuses of modern times, who, 

 if he had lived, might have been a 

 rival of Abel : he was born in 1811, 

 and died before he was twenty -one, 

 in consequence of a duel. For a 

 long time his writings remained un- 

 published and unknown, till Liou- 

 ville published them in the llth vol. 

 of his 'Journal' (1846). Liouville 

 was also the first to recognise the 

 importance and absolute correctness 

 of Galois's method, which, when sub- 

 mitted to the Academy in the year 

 1831, and reported on by Lacroix 

 and Poisson, had appeared almost 

 unintelligible. On the eve of his 

 death Galois addressed a letter to 

 his friend Auguste Chevalier, which 

 is a unique document in mathema- 

 tical literature, forming a kind of 

 mathematical testament. He de- 

 sires this letter to be published 

 in the ' Revue Encyclopedique,' 

 referring publicly the "import- 

 ance," not the "correctness," of his 

 discoveries to the judgment of 

 Jacobi and Gauss, and expressing 

 the hope that some persons would 

 be found who would take the 

 trouble to unravel his hieroglyphics. 

 The first attempt to make Galois's 

 ideas generally accessible is to be 

 found in Serret's ' Algebre Superi- 

 eure' (3rd ed., 1866), but it was 



not till after the publication of 

 Camilla Jordan's ' Theorie des 

 Substitutions' (1870) that the 

 short papers of Galois were recog- 

 nised as containing the germs and 

 beginnings of an entirely novel and 

 comprehensive mathematical theory 

 viz., the "Theory of Groups." 

 The relation between the writings 

 of Abel and Galois is exhaustively 

 treated in Prof. Sylow's Paper on 

 Abel's work, contained in the ' Me- 

 morial Volume,' 1892, p. 24. He 

 there says : " Le merite de Galois 

 ne consiste pas essentiellemeut dans 

 ses propositions, mais dans la gene"r- 

 alite" de la me"thode qu'il appliqua. 

 C'est son admirable theoreme fonda- 

 mental qui a donne a la theorie 

 des Equations sa forme definitive, 

 et d'ou est sortie, en outre, la theorie 

 des groupes ge'neralise'e, qui est 

 d'une si grande importance, on peut 

 le dire, pour toutes les branches des 

 mathematiques, et qui deja, entre 

 les mains de Jordan, de Klein, de 

 Lie, de Poincar4 et d'autres, a en- 

 richi la science d'une longue suite 

 de de'couvertes irnportantes:" The 

 memoirs of Abel and Galois re- 

 ferring to the Theory of Equations 

 have been conveniently edited, in a 

 German translation, by H. Maser, 

 1889. See also Cayley's article on 

 "Equation" in the 'Ency. Brit.,' 

 32. 



