530 PHILOSOPHICAL TRANSACTIONS. [ANNO 1799. 



to cubics; but no formula exhibiting to the eye the actual resolution of a biqua- 

 dratic has yet appeared; and for the 5th degree, and all upwards, not even a clue 

 which promises a general resolution has been struck out, by the continued labour 

 and ingenuity of mathematicians for several centuries. 



3. This failure in the chain, beginning at the 3d degree, and its breaking off 

 entirely after the 4th, have been very puzzling and mortifying circumstances to 

 the cultivators of algebra. Having in the first degrees proceeded on apparently 

 very general principles, and made a seeming progress towards a general resolution 

 of equations, it is provoking to find it suddenly interrupted, not to be resumed by 

 any contrivance. Various causes have been assigned for so remarkable a difficulty: 

 but the generality of those causes, as commonly given, do not reach the principle. 

 It has been usual for operators, when they found their methods fail, to look back 

 till they could detect some inconsistence or impossibility in their work, and to sup- 

 pose the difficulty explained, by pointing out the period at which such an error is 

 made. The power and richness of the algebraic calculus affords numerous ways of 

 compassing the same thing, and, as all of them fail when applied to this object, 

 there is necessarily a point in every one of them, at which some inconsistence or 

 impossibility is introduced: thence, a number of different causes may be imagined. 

 In Dr. Waring's Meditationes Algebraicae, p. 1 82, may be seen several concurrent 

 reasons assigned, why the methods there shown, and Dr. Waring's own, un- 

 doubtedly the most general of any of them, since it proceeds on one principle to 

 the 5th degree, cannot apply further: but all reasons drawn from the data of any 

 particular method, like that commonly given for the imperfection in Cardan's Rule, 

 which I shall examine hereafter, though very just in themselves, cannot be con- 

 clusive: they indisputably show why the precise method to which they respectively 

 apply must fail; but that does not exclude the expectation that some other, founded 

 on different principles, may succeed. The question therefore recurs: Is there not 

 some paramount fundamental reason for this general failure ? If there can be shown 

 to be any thing in the nature of abstract quantity, which governs the several orders 

 of quantities from which equations are framed, and leads directly to the distinc- 

 tions and limitations practice discovers, that will reach the difficulty at its source, 

 and afford the satisfaction desired. 



4. I think, that by turning the course of our inquiry rather to examine how we 

 come to succeed at all, in resolving any degree of equations, than why our success 

 is so limited, the true principle on which their resolution must depend will appear; 

 and with what probability, and by what means, if possible, we may expect to 

 render our methods more perfect. With this idea, I shall take a concise view of 

 the nature and resolution of equation in general; pointing out the common diffi- 

 culty, and by what circumstances that difficulty is, in certain cases, lessened or 

 removed; confining myself always to the principle of each step, and a strict 

 analysis of the result, avoiding all detail of mere operation ; and, without pretending 

 to much novelty on a subject already so beaten, I persuade myself, such an inves- 



