gx i Extenfion of Cardan’s Rule to the 
to two unequal parts a and b, of which a was fuppofed 
to be the greater ; and we firft found the value of the 
greater part a, and then determined that of the leffer 
part b from its relation to a, which is expreffed by the 
equation 3 ab = q. But we may with the fame eafe firft 
determine the value of the leffer part b, and then derive 
from it that of the greater part a ; which would produce 
the firft of the two expreflions of the value of x men- 
tioned in the laft article. This may be done as follows. 
Since it has been fliewn in Art. 5. that r is — cP-vP, 
and yib is = q, and confequently a is =^, and <3 3 to — t 
it follows that r will be -■~ J rb % . Therefore rR is = - +R. 
27P 27 7 
and (fubtradting b 6 from both fides )rb*—b f> is=~. There- 
fore (fubtradling both fides from than which they are 
evidently lefs), we fhall have r b- rb z +b (, — r -^ _ L. There- 
fore the fquare-root of - — rb z + Z> 6 will be = » / rr - — • 
But the fquare-root of --rR+f is the difference of the 
quantities ~ and R, that is (becaufe b z is the lefler part 
of a l +b\ or r, and confequently is lefs than the half of 
it, or — ), it is = - — b\ Therefore — - R is = . /! --i. 
Therefore (adding b 3 to both fides) — will be = R + 
v/ 
