282 Mr. J. Cockle on Impossible Equations. 



As by linear equations, taken in their utmost generality, 

 we are led to contemplate negative quantity; and as by qua- 

 dratics, cubics, and the higher equations, we are in like manner 

 led to form the notion of unreal quantity; so by surd or irra- 

 tional equations we may be conducted to the idea of impos- 

 sible quantity. 



Let 



1+ \/x—k — Vx— 1=W, 



1— Vx—^ + Vx— 1 = X, 

 1+ */x— 4 + Vx— 1=Y, 

 1_ s/x—k — Vx—\ = Z. 

 Then, by actual multiplication, we obtain 



6-2^ + 2 \/x*— 507 + 4 = W.X, 



6 — 2a?— 2 Vx* — 5^ + 4 = Y.Z; 

 and hence, on multiplication and reduction, 



4(5-a?) = W.X.Y.Z. 



Let WXYZ = V, then 5 is the only value of x which satis- 

 fies V = 0. Hence, no value of x other than 5 can make 

 either of the factors W, X, Y, or Z equal to zero; for, if so, 

 such other value must make one at least of the remaining 

 factors infinite, and we should have to subject x to incompa- 

 tible conditions. 



Now W = may, by a transposition, be rendered identical 

 with the equation (I) given by Gamier at p. 335 of the second 

 edition of his Analyse, and is satisfied by the value x = 5. But 

 X = (which may, by transposition, be rendered identical 

 with the equation numbered (2) by Garnier at the page just 

 cited) is not satisfied by that value; at least, not if we con- 

 sider the symbol V to be such that the quantities included 

 under it are necessarily affected with ( + l) 2 . This appears to 

 me to be the true meaning of that symbol of radicality, provided 

 that, in the development of algebra from arithmetic, we adhere 

 as closely as we can to the analogies afforded by the latter 

 science, in the most general form of which (universal arith- 

 metic or arithmetical algebra) all the quantities (i. e. symbo- 

 lized numbers) employed are, implicitly at least, considered as 

 affected with ( + 1) 2 . Accordingly I adhered to this view in 

 forming the equation which led me to my Theory of Tessa- 

 rines, and by which I sought to connect that theory with 

 ordinary algebra. There is no real loss of generality by thus 

 restricting the symbol \/ . The root corresponding to the 



