the Invention of cardan’s Rules , 8tc. 229 
ftead of x, in the terms x 3 and qx, in hopes of fuch an 
extermination of equal terms, and confequential reduc- 
tion of the equation to one of a limpler and more ma- 
nageable form, as was found to be fo ufeful in the cafe 
of the former equation x 3 +qx=r. We will therefore 
try both thefe fubftitutions ; and, as that of the difference 
y-z has in the former cafe proved fo fuccefsful, we will 
begin by that. 
Art. 6. Now, by fubflituting the difference y-z in- 
ftead of x in the equation x 3 -qx-r, we fhall transform it 
into the following equation, to wit, y 3 - ^yyz + ^yzz- z 3 - q 
x fy—z]=r, or y 3 -^yzx y-^z-z 3 -qxy-z-r', in which the 
terms 3 yzxy-z and qxy-z have both of them the fame 
lign — prefixed to them, and confequently can never ex- 
terminate each other, whether q,yz be equal or unequal 
to q. This fubftitution therefore is in this cafe of no 
ufe. 
Art. 7. We will now therefore try the fubftitution of 
the fum of y and z, inftead of their difference, in the 
equation x 3 —qx-r. 
Now, if x be fuppofed to be equal toy + z, and y+z be 
fubftituted inftead of it in the equation x 3 -qx~r , that 
equation will be thereby transformed into the following 
one, to wit, 
