210 MR. R. HARLEY ON IMPOSSIBLE 



Now on trial we find that this value will not satisfy the 

 proposed equation, unless the sign (+) prefixed to the 

 radicals be taken to mean the algebraic addition of a root 

 of the quantities Qc — 3) and {x -{- 21) ; that is (to be 

 more explicit), unless we take the plus root of (x — 3) and 

 the mintis root of (x -f- 21). This hypothesis, however, 

 I regard as inadmissible, believing it to be at variance 

 with that definite and distinct signification of symbols, 

 which is so absolutely essential to the exact expression of 

 the conditions of a problem. It virtually transforms the 

 sign prefixed to the radical into a mere connecting link, 

 exercising no real control over that radical. It is true, 

 that, when the symbol V is introduced in the course of an 

 algebraic investigation, it is always capable of sustaining 

 both a plus and minus interpretation ; so that in every such 

 case the double sign -|- is understood as involved ia iL 

 But this does not seem by any means to justify an am' 

 biguouSf or rather I should say a variable, interpretation 

 of that symbol when it is necessarily employed with a par- 

 ticular sign of operation before it, that particular sign being 

 indispensable to the perfect symbolic representation of the 

 given conditions. In the one case, the symbol having 

 been introduced for the purpose of efPecling a transforma- 

 tion, may be taken both positively and negatively without 

 violating any stipulated conditions : in the other case, it is 

 given with a specific sign of operation prefixed to it, and 

 this sign cannot be altered or evaded consistently with the 

 conditions of the problem. For these reasons, therefore, 

 we cannot accept 4 as a root of the proposed equation ; but 

 this is the only value of x which will satisfy the rational 

 equation : hence (art, 3, cor. 3) 



4 -|- Va; — 3 -f Vx 4- 21 =: 



has no root whatever, in other words (art. 4), it is Art 

 impossible equation. The foreign root 4 was introduced by 



