iS€t.\.] DISSERTATION SECOND. 35 



nus before them, must be added into one sum with those of 

 which all the factors had the sign phis ; while those of 

 which one of the factors had the sign plus, and the other 

 the sign minus, must be subtracted from the same, — this 

 general rule came to be more simply expressed by saying, 

 that in multiplication like sign3 gave plus, and that unlike 

 signs gave minus. 



Hence the signs plus and minus were considered, not as 

 merely denoting the relation of one quantity (o another 

 placed before it, but, by a kind of fiction, they were con- 

 sidered as denoting qualities inherent in the quantities to 

 the names'of which they were prefixed. This fiction was 

 found extremely useful, and it was evident that no errour 

 could arise from it. It was necessary to have a rule for 

 determining the sign belonging to a product, from the signs 

 of ihe factors composing that product, independently of 

 every other consideration ; and this was precisely the pur- 

 pose for which the above fiction was introduced. So ne- 

 cessary is this rule in the generalizations of algebra, that 

 we meet with it in Diophanlus, notwithstanding the imper- 

 fection of the language he employed ; for he states, that 

 Atfpis into As<\J//s gives rvx^t?, &c. The reduction, there- 

 fore, of the operations on quantity to an arithmetical form, 

 necessarily involves this use of the signs plus or minus ; 

 that is, their application to denote something like absolute 

 qualities in the objects they collect together. The at- 

 tempts to free algebra from this use of the signs have of 

 course failed, and must ever do so, if we would preserve 

 to that science the extent and facility of its operations. 



Even the most scrupulous purist in mathematical lan- 

 guage must admit, that no real errour is ever introduced by 

 employing the signs in this most abstract sense. If the 

 equation x 3 + px 2 + qx -f r=o, be said to have one posi- 

 tive and two negative roots, this is certainly as exception- 



