VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 53 1 



tigation will lead to some conclusions which have not been remarked, and which 

 are both curious and important. 



On the Resolution of Equations in General. 



5. Equations, in that part of algebra which treats of their general resolution, are 

 usually considered to be reduced to one general form, for the greater convenience 

 of comparing them, i. e. to their lowest rational dimension, with unity always for 

 the co-efficient of the highest power of the unknown quantity; in which state, 

 every simple equation is already resolved. The resolution of all other degrees is 

 the finding the simple equations of which they are compounded: but to do this in 

 a general manner, it is evident we must seek, instead of the particular equations 

 themselves directly, a general expression representing them all; which general ex- 

 pression is called the formula of resolution, such as, the common quadratic resolu- 

 tion, or that given for cubics by Cardan's Rule. 



6. These formulae, properly speaking, are rather the reversion of an equation, 

 than the resolution of it: for though the unknown quantity be evolved or reduced 

 to a simple dimension, the known parts are necessarily involved or affected with a 

 surd at least as high as the dimension of the equation, in order to exhibit the 

 proper number of correspondent values belonging to the unknown quantity in an 

 equation of that degree. Thus, the equation, x 1 — px -f- q = 0, and its common 

 resolution x = — — -~— — , are both the same quadratic; only, under the first 

 form, the unknown quantity, being of the dimension of the 2d degree, has 2 

 values; whereas in the 2d form it has only 1, and the double value is transferred, 

 by the quadratic surd, to the known parts on the opposite side of the equation. 

 Thus also, the equation x 3 — qx + r = O, and the Cardanic formula belonging to 



iU = ^[-y + ^(J-f^)J + ^[--£ - </(£ -f^)] are, in the same man 

 ner, the same cubic merely reverted. But as equations are usually denominated 

 from the dimension of the unknown quantity, these resolutions are commonly 

 deemed simple equations: they may in this view be defined to be, the simple equa- 

 tions that the original quadratic, cubic, or other higher given equation, contained 

 in power, since they express the nature and form of a quantity which, by involu- 

 tion or reverting the operation, re-produces it; as the root of any power, being 

 re-involved, returns to the power from which it was extracted. This fixed and 

 visible connection between the equation and the general formula for its roots, 

 throws a beauty and elegance into the method of pure algebraic resolution, which 

 none of the others, such as the method of divisors, and all the contrivances for 

 approximation, can pretend to. For, when by any of those methods we have ob- 

 tained one or more separate roots, the relation to the original equation is no longer 

 perceivable; but here the chain is perfect. The equation leads to the resolution: 

 the resolution embraces at once all the correspondent roots ; and, when re-involved, 

 proves the operation, by reproducing the original equation. Thus, for example, if 

 a? 2 — 5x + 4 = O, and it be perceived, or found by any conjectural method, that 



3 Y 2 



