218 MR. R. HARLEY ON IMrOSSIBLE 



deservedly celebrated. The general conclusion to which 

 the Professor's reasoning tends, is thus elegantly expressed 

 in the closing sentence of the article referred to : — " If, by 

 any management or contrivance, we force, in a particular 

 case, a violation of a general law, I need scarcely say that 

 our result will be inadmissible." This remark is peculiarly 

 applicable to the case now under consideration. For, by 

 substituting for n its numerical value in (1), the law of 

 generation is lost sight of, and consequently, (5) and (6) 

 being severally substituted in the irrational equation (a), 



we violate, in a certain case already specified, the general 

 law controlling the square root of expressions affected by 

 the symbol n^. To illustrate this clearly, let us consider 

 the simple surd equation 



1 + V^=r0 (a), 



Transposing, ^/xzz — Izzn: 



.'.azzn^ (a'). 



Now (a}) evidently satisfies (a) ; for 1 -|- V^^ zz I -\-n 

 zz ; and yet, though n^ zz 1, /v zz 1 is not the root of (a), 

 but of its congener, 



I — V^=0 (h). 



The reason is, that Vl and A,/n^ are not equal, the latter 

 being n times the former.* So, in like manner, by substi- 

 tuting from (3) and (6), we get respectively (4) and (8) ; 



* Possibly it may be objected that Vl is eitJier + 1 or — 1, and that, 

 therefore, unity is the root of both (a) and (6). In answer to this, it 

 might seem sufficient simply to refer to Art. 5, in which it is shown, we 

 think, that such a conclusion is not consistent with rigorous reasoning ; 

 that it involves, in fact, a virtual violation of the law of signs. As, how- 

 ever, the entire theory of impossible equations depends for its existence 

 on the non-identity of such equations as (a) and (b), we may further re- 

 mark, that if these be treated as simultaneous equations, (a) -f- (b) will 

 give 2 = 0, an arithmetical absurdity. Whether or not n* is philosophi- 

 cally admissible as a root of (a), will be hereafter considered. 



