Resolution of Algebraic Equations. 27 9 
skilful analyst, is sufficient for the reduction, if not the solu- 
tion, of any particular numeral equation whatsoever, and the 
more so the larger its dimension : for, from the endless variety 
of relations numbers bear to each other, hardly any set of them 
can occur, as the coefficients of an equation, or perhaps exist, 
that, upon being compared, do not exhibit some peculiarity (of 
greater or less extent) sufficient to afford a clue to the corre- 
spondent relation in their roots. And, if no such clue is imme- 
diately given by the equation itself, taking the equation of the 
differences or sums in pairs, or of the squares, &c. of the roots, 
will soon find one. But, as peculiarities of that sort (though 
never so frequent) may be deemed always accidental, and evi- 
dently, no general method can be founded upon them, even 
where the coefficients are given, it may be asked, How any use can 
be made of them in cases of indeterminate equations ? 
14. To this I answer, that there are some properties of quan- 
tities that depend only on the index of the equation, without 
any regard to the value of its coefficients ; or, in other words, 
there are some peculiar properties which merely depend upon 
the number of any set of quantities, abstracted from all conside- 
ration of their nature and values. For example, two quantities 
(a)and(6) have their differences the same quantity [a — b), only 
taken both affirmatively and negatively, ( a — 6) and ( b — a): 
when squared, these differences become equal; (# a — 2 ab -f- b z ) 
is the square of both : therefore, let the quantities themselves 
be chosen as they may, the equation of the squares of their 
if 2, 3, or more pairs of roots are equal, the reduction can only be carried down to 
a quadratic, cubic, &c. for, every pair of equal roots being equally to be found by 
the method, of course the final or resulting equation must be of a dimension as great 
as their number 
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