682 



SCIENTIFIC THOUGHT. 



40. 



Determin- 

 ants. 



contained the same number of factors. Such forms 

 could be written down on the pattern or model of 

 one of their terms by simple methods of exchange or 

 permutation of the elements. It would then not be 

 necessary to write down all the terms but only to indicate 

 them by their elements, these also being abbreviated by 

 the use of indices. Rows and columns or arrangements 

 in squares suggested themselves as easy and otherwise 

 well-known artifices by which great masses of statistics 

 and figures are marshalled and controlled. Out of these 

 manifold but simple devices there grew an algebra of 

 algebra, a symbol for denoting in a very general way 

 symmetrical and homogeneous algebraical expressions. 1 

 Gauss termed such expressions Determinants : they 

 turned up in his ' Disquisitiones Arithmetics ' as they had 

 done half a century before in Cramer's ' Analyse des lignes 

 courbes algebriques.' Just as common fractions can be 



garb, which soon led to a more 

 general conception. The Bary- 

 centric co-ordinates were the first 

 instance of homogeneous co-ordin- 

 ates, . . . and already with Mobius 

 the advantages become evident 

 through the symmetry and ele- 

 gance of his formulae" (Hankel, 

 'Project. Geom.,' p. 22). 



1 Determinants were first used 

 by Leibniz for the purpose of 

 elimination, and described by him 

 in a letter to the Marquis de 

 1'Hospital (1693). The importance 

 of his remarks was not recognised 

 and the matter was forgotten, to 

 be rediscovered by Cramer in the 

 above - named work (1750, p. 

 657). It is interesting to note 

 that the same difficulty of the 

 process of elimination induced 

 Pliicker to resort to geometrical 



interpretation of analytical ex- 

 pressions, and that whilst he "saw 

 the main advantage of his method 

 in avoiding algebraical elimination 

 through a geometrical considera- 

 tion, Hesse showed how, through 

 the use of Determinants, algebraical 

 operations could receive that pliabil- 

 ity the absence of which was the 

 reason for Pliicker to discard it." 

 (See the account of Clebsch's work 

 in 'Math. Ann.,' vol. vii. p. 13.) 

 Through this invention the com- 

 binatorial analysis, which, in the 

 hands of the school in Germany, 

 had led into a desert, was raised 

 again into importance. It has be- 

 come still more important since the 

 general theory of forms and of 

 groups began to play an increasing 

 part in modern analysis. 



