532 PHILOSOPHICAL TRANSACTIONS. [ANNO \7QQ. 



unity is one of the roots of that equation, there is no discernible connection be- 

 tween the simple equation expressing x = 1 , and the original equation ; no trans- 

 formation of one will produce the other. This latter equation r=± !, though 

 truly expressing a numerical root of the former, is no more a resolution of it than 

 of the equations x x — 6x + 5 = O, x* — Jx -f- 6 = O, or any other of the infinite 

 number of equations of which unity is a root; whereas the algebraic resolution of 

 a? 2 — 5x + 4 = O, viz. # = — — — ij— — '-, which equally expresses ], and 4 the 

 other root, needs only to be cleared of its radical, to show itself but another form 

 of the same equation; and gives x 1 — 5x -\- 4 = 0, as at first. 



7. This view of the algebraic resolution of an equation shows, that it does not 

 so much aim at giving us the roots themselves, as the basis or common principle 

 of their artificial combination in the equation to which it applies; pointing out 

 some form of a perfect power, of which they may be conceived to be the cor- 

 respondent natural roots. From which it follows, that if the transformation 

 required to be made in the given equation be possible, or such as can really be 

 effected, the resolution will be real; for every real power has some real root: but 

 that if, on the contrary, the power into which the equation is conceived to be 

 transformed be merely imaginary, the resolution must be so too; for all the roots 

 of an imaginary power are themselves imaginary. It doth not therefore depend on 

 the nature of the roots of the equation themselves, but on the form which the 

 equation must assume to become a perfect power, to determine whether the reso- 

 lution be real or imaginary : so that the nature of the resolution, and that of the 

 roots of an equation may be very different, as we know is frequently the case; 

 particularly in the resolution of cubic equations by Cardan's Rule, where, when the 

 roots are real, the resolution is almost always imaginary. This has seemed to sur- 

 prize and perplex some writers very much, who have treated it as at best a paradox, 

 if not a contradiction,* but surely without cause; for, as the formula affects only 

 to be an ideal representation of the mechanism or structure of a perfect power 

 answering to the given affected equation, it may be expected to be clear or com- 

 plicated, real or imaginary, not as the roots themselves are simple and real, but as 

 the principle of their union, of which only it is truly the index, is near or remote: 

 it merely shows the central point of their combination, which, like the centre of 

 gravity, suspension, or any other power, may not actually exist in any of the bodies 

 whose motions it governs, but in some imaginary point without, and remote from 

 them all. Had the nature of the algebraic resolution of an equation been con- 

 sidered in this light, and the forms to which they are proposed to be reduced, been 

 compared with the original forms of the roots in the given equation, no surprize 

 or appearance of paradox could have arisen in the matter; but it must have been 

 clearly perceivable what cases would admit of real, and what only of imaginary 

 resolutions, as will be shown hereafter. I have dwelt the longer on the nature of 



* Vide Playfair on the Arith. of Impossibles, Phil. Trans. 1778, p. 318 ; Dr. Hutton on Cubic 

 Equations, ditto, 1780, p. 387 j and Mr. Baron Maseres, Script. Logarith. vol. 2, p. 246\ — Orig. 



