Resolution of Algebraic Equations. 267 
mental reason for this general failure ? If there can be shewn 
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 distinctions 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 de- 
gree of equations, than why our success is so limited, the true 
principle upon 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 equa- 
tions in general; pointing out the common difficulty, 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 upon 
a subject already so beaten, I persuade myself, such an inves- 
tigation will lead to some conclusions which have not been 
remarked, and which are both curious and important. 
CHAP. I. 
Of 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, 
