684 



SCIEiNTIFlC THOUGHT. 



41. Operations, the idea of treating algebraical operations 



Calculus of "^ ' o o r 



Operations. ai^(j their symbols as quantities, and of subjecting them 

 to arithmetical treatment separately from the mate- 

 rial operated on. The genius of Arthur Cayley was 

 specially fertile in this direction, as was that of 

 Sylvester in the nomenclature or language of the 

 doctrine of forms.^ The merit, however, of having 

 brought together the new ideas which emanated from 

 the schools of Poncelet and Chasles in France, of Cayley 

 and Sylvester in England, into a connected doctrine, and 

 of having given the impetus to the fundamental re- 



^ The theory of invariants was 

 gradually evolved from many inde- 

 pendent beginnings. In 1864 Syl- 

 vester wrote ('Phil. Trans.,' p. 

 579), "As all roads are said to lead 

 to Rome, so I find, in my own case 

 at least, that all algebraical in- 

 quiries, sooner or later, end at the 

 Capitol of Modern Algebra, over 

 whose shining portal is inscribed 

 the Theory of Invariants." About 

 the same time (1863) Aronhold de- 

 veloped the principal ideas which 

 lay at the foundation of the theory 

 in organic connection and in com- 

 plete generality, hereby domiciling 

 in Germany the doctrine which had 

 previous!}' owed its development 

 mainly to English, French, and 

 Italian mathematicians (see Meyer, 

 'Bericht,' &c., p. 95). The differ- 

 ent roads which Sylvester refers to 

 can be traced, first, in the love of 

 symbolic reasoning of Boole, who 

 was " one of the most eminent of 

 those who perceived that the sj'm- 

 bols of operation could be separated 

 from those of quantity and treated 

 as distinct objects of calculation, 

 his principal characteristic being 

 perfect confidence in any result 

 obtained by the treatment of sym- 

 bols in accordance with their 



primary laws and conditions, and 

 an almost unrivalled skill and 

 power in tracing out these results " 

 (Stanley Jevons in article " Boole," 

 'Ency. Brit.'); secondly, in the 

 independent geometrical labours of 

 Hesse in Germany (whose mathe- 

 matical training combined Pliicker's 

 and Jacobi's teaching) and Dr 

 Salmon in Dublin (who, after 

 having transplanted Poncelet and 

 Chasles to British soil, recog- 

 nised the importance of Cayley 's 

 and Sylvester's work, and in- 

 troduced in the later editions of 

 his text - book modern algebraical 

 methods) ; thirdly, in the independ- 

 ent investigations belonging to the 

 theory of numbers of Eisenstein in 

 Germany and Hermite in France. 

 In full generality the subject was 

 taken up and worked out by Syl- 

 vester in the ' Cambridge and 

 Dublin Mathematical Journal ' 

 (1851 - 54), and by Cayley in the 

 first seven memoirs upon Quantics 

 (1854-61), which "in their many- 

 sidedness, together with the ex- 

 haustive treatment of single cases, 

 remain to the present day, for the 

 algebraist as well as for the geo- 

 metrician, a rich source of dis- 

 covery" (Meyer, loc, cit., p. 90). 



I 



