AND OTHER SURD EQUATIONS. 



223 



this incongruous result, is not strictly receivable. In like 

 manner the root marked (3), viz.. 



I 111 



though it appears to satisfy the proposed equation, is re- 

 jective, because it involves the acceptance of the impossible 

 equality, 



Jlax 4- h rz w (a + V«* + hn-*\ 

 111 1 1 1 



an equality which can have no more existence than the 

 relation 1 zr — 1. On the same grounds we reject n* 

 as being strictly a root of 



] 4- s/x — 0. 

 There can be no doubt, I think, that algebraically this 

 value satisfies the equation, but arithmetically it does not ; 

 and to accept it, seems to me to be nothing less than an 

 evasion of the authority of the sign (-{-) prefixed to the 

 radical. 



An eminent analyst, to whose researches we have already 

 had occasion to refer, in an interesting paper * published in 

 the Philosophical Magazine for October 1850, seems to 

 take a very similar view of this subject to that which we 

 have just been expounding. After giving an elegant dis- 

 cussion of the equation 



1 4" Va; — 4 — Vic — - 1 n 0, 

 Mr. Cockle remarks, *' if, in the above instances, the diffi- 

 culty is to be evaded, it is only by greatly refining our 

 solution, and, as it has occurred to me, by using expressions 

 of the form m (-}- \y -\-n ( — 1)', and by following certain 

 rules respecting our reductions, and the signs to be affixed 

 to the radicals. To those who would attempt such a com- 

 plex and artificial system of solution, rather than admit the 



* " On impossible equations, on impossible quantities, and on tessarines. 

 By James Cockle, Esq., M. A. of Trinity College, Cambridge ; Barrister- 

 at-law of the Middle Temple." Phil. Mag., third series, vol 37, pp. 281-3. 



