DEVELOPMENT OF MATHEMATICAL THOUGHT. 683 



dealt with as if they were special things having special 

 properties, though the latter depend only on the pro- 

 perties of the numbers they are made up of and their 

 mode of connection ; as powers and surds are separately 

 examined ; so the arrangements called determinants can 

 be subjected to a special treatment, their properties 

 ascertained, and themselves subjected to the ordinary 

 operations of arithmetic. This doctrine, which con- 

 stitutes the beginning and centre of the theory of 

 algebraical forms or " quantics " and of algebraical 

 operations or " tactics," was pretty fully worked out 

 and first introduced into the course of teaching by 

 Cauchy in France ; then largely adopted by Jacobi in 

 Germany, where Otto Hesse, trained in the ideas of 

 Pliicker, first showed its usefulness in his elegant 

 applications to geometry. In France it was further 

 developed by Hermite, who, together with Cayley and 

 Sylvester in England, proclaimed the great importance 

 of it as an instrument and as a line of mathematical 

 thought. 1 In the latter country the idea of abbrevi- 

 ating and summarising algebraical operations had become 

 quite familiar through another device which has not 

 found equal favour abroad namely, the Calculus of 



1 "For what is the theory of , p. 301) refers to Otto Hesse's 



determinants ? It is an algebra ! " problem of reducing a cubic 



upon algebra ; a calculus which , function of three letters to another 



enables us to combine and foretell j consisting only of four terms by 



the results of algebraical opera- , linear substitutions a problem 



tions, in the same way as algebra j which appears to set at defiance all 



enables us to dispense with the 

 performance of the special opera- 

 tions of arithmetic. All analysis 

 must ultimately clothe itself under 

 this form." In this connection 

 Sylvester ('Phil. Mag.,' 1851, Apl., 



the processes and artifices of com- 

 mon algebra," as " perhaps the 

 most remarkable indirect question 

 to which the method of determin- 

 ants has been hitherto applied." 



