2 66 Mr. Wilson’s Essay on the 
and its breaking off entirely after the fourth, have been very 
puzzling and mortifying circumstances to the cultivators of 
algebra. Having in the first degrees proceeded upon apparently 
very general principles, and made a seeming progress towards 
a general resolution of equations, it is provoking to find it sud- 
denly interrupted, not to be resumed by any contrivance. Va- 
rious 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 Medita- 
tiones Algebraicce, (p. 182.) maybe seen several concurrent rea- 
sons assigned, why the methods there shewn, and Dr. Waring’s 
own, (undoubtedly the most general of any of them, since it 
proceeds upon one principle to the fifth degree,) cannot apply 
further : but, all reasons drawn from the data of any particular 
method, (like that coipmonly given for the imperfection in 
Cardan’s Rule, which I shall examine hereafter, ) though very 
just in themselves, cannot be conclusive: they indisputably 
shew, why the precise method to which they respectively apply 
must fail ; but that does not exclude the expectation that some 
other, founded upon different principles, may succeed. The 
question therefore recurs : Is there not some paramount funda- 
