VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 533 



the algebraic resolution of an equation, because it is a very curious subject, about 

 which many errors and inconsistencies have been fallen into, though hardly any 

 direct examination of it is to be found in any of our books. It is the sole method 

 of obtaining a complete general answer to any problem. It makes algebra con- 

 sistent with itself and sufficient to solve its own difficulties, without foreign aid, 

 from series or other branches; and, in all cases where any general ulterior use is to 

 be made of the resolution of an equation, is the only method that avails at all. 



8. In order to obtain this general resolution, the common methods have been, 

 without considering the nature of the roots, to attempt some universal reduction in 

 the forms of equations; as, 1st. The destroying their intermediate terms, and con- 

 verting them into pure powers. Or, 2dly. The discovering some constant com- 

 plement which will always raise them to the nearest perfect power. In both which 

 cases, the resolution will afterwards be nothing more than simple extraction of the 

 proper root. Or, 3dly. The assuming some convenient formula with indeterminate 

 co-efficients; and, by assigning their values properly, adapting it to every case. 



It would be going to too great a length, to give distinct examples here, of the 

 application of these methods. Numerous instances of each of them are given in 

 the common books of algebra, which usually treat them as separate and distinct 

 from each other; but the fact is, they are all in -truth the same. Whoever tries 

 them separately, will find, however variously they seem to set out, they lead pre- 

 cisely to the same conclusion, and fail precisely in the same points. A quadratic, 

 whether resolved by completing the square, or by expunging the 2d term: a cubic, 

 whether resolved by Cardan's rule, or by completing the cube, or by assuming a 

 resolution, as suggested in Dr. Waring's Meditationes Algebraicae, p. 179, 180, 

 present the same formula of resolution, and the same limitations and irreducible 

 cases. And the reason is easily found. To complete the requisite power, accord- 

 ing to the index of the equation, or to destroy the intermediate terms, occasions 

 an alteration in just the same number of terms; it is only the particular relation 

 they are required to bear to each other that is varied. In the one case, they are all 

 to be equal, or equal to nothing; in the other, to correspond respectively with the 

 known law of the binomial theorem, which gives the uncise of a regular power. 

 Both depend on the practicability of a more general problem, of which they are 

 but specific cases; viz. the problem " to give the co-efficients of an equation any 

 general determinate relation." If that were practicable, and it were possible to 

 mould them so as to establish a general relation between them, or any required 

 number of them, it is easy to perceive, that the particular relation must be a 

 secondary consideration; and that, wherever the same number of terms are to be 

 acted on, the same means that might make them equal, might give them any other 

 proportion at pleasure. 



9. However, of all these methods, and any other of the kind, it is to be ob- 

 served, that the principle is demonstrably a false assumption. For, if it be once 

 admitted that the construction of equations, and the laws of the successive co-efrl- 



