the Invention of cardan’s Rules , See. 237 
qx-x l -r as was found above for the two former equa- 
tions x l + qx~r and x 3 -qx-r, can be obtained by fubfti- 
tuting the difference in it inftead of x . 
Art. 16. We will now try the fubftitution of y+z in- 
ftead of x in the terms of this equation. 
Now, if x be fuppofed to be =y+z, we fhall have 
A 3 =y 3 + q,yyz+ zyzz+z 1 =y s + $yz xy+z+z*, and qx — 
qxy+z, and confequently, qx—x 3 =q xy + z—y 3 - ^yz xy+z 
-z 3 . Therefore, qxy+z-y 3 -q ) yzxy+z-z 3 will be — r. 
In this equation it is true indeed that the terms 
qxy+z and 3 yzxy-z have different figns. But they 
cannot be equal to each other : for, fince the three terms 
y 3 and %yzxy-z and z 3 are all marked with the flgn 
or are to be fubtradted from the firft term qxy+z , and 
the remainder is =r, it is evident, that qxy+z muft be 
greater than the fum of all the three terms y 3 , 3 yzxy+z>> 
and z 3 , taken together, and therefore, a fortiori , greater 
than zyzxy+z alone. Therefore, no fuch extermination 
of equal terms marked with contrary figns as took place 
in the transformed equations derived from the two for- 
mer equations x 3 + qx-r and x 3 -qx~r, can take place in. 
this transformed equation derived from the equation 
qx-x 3 —r by fubllituting y+z in its terms inftead of xy 
and confequently no fuch method of refolving the equa- 
tion 1 
