278 Mr. Wilson’s Essay on the 
and consequently may be found, without resolving either of them, 
by continual division or subtraction, according to the ordinary 
rule for finding the common measure.* 
12. Any other relation frojn the knowledge of which one 
character may be made to represent two or more roots , evi- 
dently answers the same end. Indeed all relations of that kind 
may be converted into equality itself, by taking, instead of the 
given equation, some other properly derived from it. Thus if, 
instead of [a and b) being the same, f b ) had been supposed the 
negative of [a) or ( — a), and then, instead of the former equa- 
tion, that of the squares of the roots were taken, the relation 
would be made equality; for [a] and ( — a) have the same square. 
If arithmetical proportion was known to be the relation of any 
number of the roots, by taking the equation of their differences, 
it would also be converted into equality. 
13. If three or more roots, or any number of parcels of roots, 
are known to be related, and their common relation be used 
to represent them, of course the number of distinct characters 
to be determined will proportionably be diminished : and, as the 
number of subordinate equations furnished by the coefficients 
remains always the same, while the dimension of the proposed 
equation is unaltered, more of them may be used together to 
discover the related roots, and their investigation be propor- 
tionably facilitated. This single observation, in the hands of a 
* Vide Sanderson’s Algebra, (quarto ed. Vol. i. p. 86, 87, 88,) where the rule 
is well given; and Maclaurin’s Algebra, (P. II. cap. iv. p. 162.) ; or Mr. 
Hellins’s Essay upon the Reduction of Equations having equal Roots. But of the 
last it should be observed, that some qualification must be made to the assertion, 
that the reduction may be carried on till a simple equation is obtained. In cases 
where there is only one pair of roots equal, that proposition is undoubtedly true ; but. 
