AND OTHER SURD EQUATIONS. 



213 



Let (a) be multiplied by a: -4- n V2a x -\- b; then, bear- 



1 *^ 1 111 



ing in mind that 1 -|- n z= 0, we shall have 



x^ -\- 2a nx -\- b 71 ziz } 

 1 111 



.'. x^ -j- 2anx in 6 n^ ; 



11 1 I 



or, X z:z a n^ -\- n a/o' n^ 4- b, (1). 



II II 



The second and third steps of the above solution may 

 need, perhaps, a little explanation. In transposing the 

 quantity 6 n to the right-hand side of the equation, it will 



be observed, that instead of aflfecting it (according to the 

 usual method) with the minus sign, we have multiplied it by 

 n. That these operations are equivalent, is too evident to 

 need demonstration; and it is easy to see also, that the 

 introduction of the symbol n for the negative sign is indis- 

 pensable, to prevent the ambiguity that would otherwise 

 result from the obliteration of that sign by involution. But 

 it may be asked, would not the same ends be answered 

 equally well, were we (instead of multiplying) to divide by 

 n? In replying to this question, it is important to observe, 

 that in order to enable us to retrace the several steps of the 

 solution with unerring certainty, the symbol n must always 

 be employed in conformity with some invariable principle of 

 operation ; so that, by adopting an inverse principle, we may 

 return with confident correctness, firom any part of the in- 

 vestigation, through the successive steps, to the original 

 equation. Unless the operation be thus conducted, it is 

 obvious that ambiguity and error will attach to our results. 

 In fact, we assume as the great principle that should guide us 

 in the solution of surd equations, that every successive trans- 

 formation should be made to bear with it an unmistakeable 

 index of its immediate origin ; for it is only by this means, 

 we conceive, that those rejective roots (Art. 1) which enter 

 into the ordinary solution may be excluded. Now, if we 



