the Invention of cardan’s Rules , &c. 225 
imagine, that he would fubftitute the fum or difference 
of two other quantities inftead of x, as being the moft 
limple and obvious fubftitutions that could be made. 
And by making thefe fubftitutions, the above mentioned 
rules would of courfe come to be difcovered, as well as the 
aforefaid limitation of them in the refolution of the equa- 
tion x 3 —qx=r, which reftrains the rule to thofe cafes only 
in which r is greater than or ~ is greater than 
and their utter inutility in all the cafes of the equation 
qx-x^—r. This will appear by examining each of thefe 
equations feparately in the following manner. 
Of the equation x*+qx=r. 
art . 3. In the equation x^ + qx-r the inveftigator of 
thefe rules would naturally be inclined to fubftitute the 
difference Gf two quantities (which we will here call y 
and z, and of which we will fuppofe y to be the greater) 
inftead of x, rather than their fum, of would fuppofe x 
to be equal to y—z , rather than to y + z; becaufe, if he 
Was to fuppofe x to be equal to the fum of the two quan- 
tities y and z, and was to fubftitute that fum, or the bi- 
nomial quantity y+z, inftead of x in the equation 
x^ + qx-r, it is evident, that (as the figns of x 3 and qx 
Vol, LXX. G g are, 
