284 Method of finding the impossible Roots of an Equation. 



braic principles; that it is allowable to put the general algebraic 

 expression z + y^ —\ for the unknown quantity of an equation, 

 z and y being possible quantities, positive or negative. When 

 this is done, the equation is resolved into two others, containing 

 the two quantities z and y^, from which, by elimination, a result- 

 ing equation containing only y'^ may be obtained. Hence, ex- 

 cepting the case in which y is zero, I inferred that the resulting 

 equation must give a positive possible value of y". At this point 

 of the reasoning Mr. Airy submits " that at present we are not 

 entitled to assume that we can obtain a possible value of y'^ from 

 the resulting equation ;" and he goes on to argue that a positive 

 possible value of y'^ is proved to be obtainable only in case the 

 last term of the equation is shown to be negative. But the 

 general antecedent argument proved that the equation must be 

 satisfied by a positive possible value of ?/^, without reference to 

 the sign of the last term. Of course I am aware, from the way 

 in which Mr. Airy puts his objection, that he does not admit 

 the general argument ; but as he has adduced nothing against 

 it, I still maintain it to be good, and will add a few more consi- 

 derations in support of it. 



In arithmetic, the differences and the ratios of quantities are 

 visibly expressed by numbers, and by means of numbers the 

 ordinary rules of calculation, as addition, subtraction, multipli- 

 cation, &c., are established. In algebra these rules are adopted, 

 not proved; but whereas the letters a and b do not indicate 

 which of the quantities they represent is the greater, it is neces- 

 sary to operate at the same time by rules which shall make the 

 operations independent of relative magnitude. This is effected by 

 the use of the signs + and — ; and it is the use of these signs 

 for this purpose that constitutes the difference between algebra 

 and arithmetic. The rules of signs, and the various simple 

 forms of functions strictly algebraic, such as —x, a~y, ±^ — b, 

 aiVr^^ 3jc_^ all flow from this single principle. After establish- 

 ing the different kinds of algebraic operations, and deducing the 

 different forms of algebraic functions, it is found that z + y^^ — l 

 is a perfectly general algebraic expression, z and y being possible 

 quantities, positive or negative. From these principles of abs- 

 tract algebra we may proceed to use algebra as an instrument 

 of research, that is, for finding from given data an unknown 

 quantity. Being provided with rules of operation which are in- 

 dependent of relative magnitude, we can, if we call x the unknown 

 quantity, operate upon it algebraically, and bring it into sym- 

 bolic relation with the given quantities, without knowing its 

 magnitude relative to theirs. But the unknown quantity acquires 

 by these operations an algebraic expression, and must be included 



