MATHEMATICS 29 



(according to the 2nd axiom) be the same in each case. 

 But if we take - cy from ^ax + cy - cy, the result, of 

 course, is $ax + cy. Therefore, if we take - cy from 

 $ax, the result must also be $ax + cy. 



From ax 3 - bx z + x 

 Take px 3 - qx* + 2x 



The result or difference = (a - p) x s (b - q)x z + (i - 2) x. 



Here we have the quantity X B twice repeated, each 

 time with a different coefficient, and the coefficient +j9, 

 has to be subtracted from + a, the result necessarily 

 being ax 3 -px 3 , which may be written (a -p) x 3 . 



Of the two squares of x, the negative coefficient q, 

 has to be taken from - b ; we must then, as before, change 

 the sign of q for subtraction, and so we have - bx* + qx z , 

 and this may be expressed in a bracket (b q *) x*. 

 Finally the simple quantity x and the coefficient 2 have 

 to be subtracted from the quantity ix (since x standing 

 alone is one x) and so we have (i - 2) x. 



In algebraic multiplication the explicit sign of that 

 process ( x ) is often omitted, and any two letters written 

 with only a point between them (a. b), or merely side by 

 side (or ab) mean (as in arithmetical multiplication) 

 that a has to be taken b times, or that b has to be taken 

 a times. 



If the quantity which is to be multiplied (or the 

 multiplicand), and the quantity by which it has to be 

 multiplied (or the multiplier) have both the same sign 

 (both + or both - ), then the result must have the 



* The portion b remaining after q has been subtracted from 

 it, or (b-q), being of course equal to that produced by the 

 subtraction of the whole of b, followed by the addition of 

 q, or - b + q. 



