34 DISSERTATION SECOND. (.fAiu i 



Another advantage resulting from the use of the same nota- 

 tion, consists in the reduction of all the different relations 

 among quantities to the simplest of those relations, that of 

 equality, and the expression of it by equations. This 

 gives a great facility of generalization, and of comparing 

 quantities with one another. A third arises from the sub- 

 stitution of the arithmetical operations of multiplication and 

 division, for the geometrical method of the composition 

 and resolution of ratios. Of the first of these, the idea is 

 so clear, and the work so simple ; of the second, the idea is 

 comparatively so obscure, and the process so complex, that 

 the substitution of the former for the latter could not but 

 be accompanied with great advantage. This is, indeed, 

 what constitutes the great difference in practice between 

 the algebraick and the geometrick method of treating quan- 

 tity. When the quantities are of a complex nature, so as 

 to go beyond what in algebra is called the third power, the 

 geometrical expression is so circuitous and involved, that 

 it renders the reasoning most laborious and intricate. The 

 great facility of generalization in algebra, of deducing one 

 thing from another, and of adapting the analysis to every 

 kind of research, whether the quantities be constant orvari 

 able, finite or infinite, depends on this principle more than 

 any other. Few of the early algebraists seem to have been 

 aware of these advantages. , 



The use of the signs plus and minus has given rise to 

 some dispute. These signs were at first used, the one to 

 denote addition, the other subtraction, and for a long time 

 were applied to no other purpose. But as, in the multipli- 

 cation of a quanlity, consisting of parts connected by those 

 signs, into another quantity similarly composed, it was al- 

 ways found, and could be universally demonstrated, that, 

 in uniting the particular products of which the total was 

 made up, those of which both the factors had the sign mi 



