MATHEMATICS 27 



using brackets, since y? is taken a times, 2 b times and 

 once ; and if is taken b times, and c times and once ; and 

 z is taken c times, and a times and minus once ; we may 

 evidently write it with the result as follows : 



(a + 2b + i) y? + (b + c + i) y 2 + (c + a - i) z. 



Thus the addition of algebraical quantities is per- 

 formed by connecting those that are unlike with their 

 proper signs, and collecting those that are similar into 

 one sum. 



In algebraical subtraction, on the other hand, we have 

 to change the sign, and then proceed as in addition. 



The reason of this change of sign is best seen by an 

 example ; and the reader must bear in mind the fifth 

 axiom before given. 



Let us suppose that from any quantity a, there has to 

 be subtracted the quantity b c. Now if we subtract b 

 from it (which would be expressed thus, a - b ), we shall 

 have subtracted too much, because the quantity to be sub- 

 tracted was not b, but only whatever might be left of b 

 after c had been taken away from it, It was not the whole 

 sum of 5, but only b diminished by c, or b - c, which had 

 to be taken from a : therefore evidently the operation 

 will be completed by adding c to the too much dimin- 

 ished sum, a - b. 



Thus we have a b + c, and so we have come to change 

 the sign before c from into + . It follows that, to 

 subtract b - c from a, we must change the signs and add. 



Therefore in order to subtract from 



+ a 

 the sum + b - c 



we must change the signs of the quantities to be sub- 

 tracted ; thus : 



