the Invention of cardan's Rules , Sic. 235 
as it is fuppofed to be. Confequently, XX X X muft be 
greater than qx; and, a fortiori , ^x 2 x + ^xx 2 +x 3 (which 
is more than triple of x 2 x) muft be greater than qx ; that 
is, the increment of x l will be greater than the contem- 
porary increment of qx. Therefore, the excefs of x 5 
above qx , or the compound quantity x 3 -qx, will increafe 
continually, without decreafing, while x increafes from 
V q ad infinitum, q . e . d . 
Art. 13. It follows, therefore, upon the whole of 
thefe inquiries, that if the compound quantity x 3 -q x, or, 
its equal, the abfolute term r, is lefs than VLll or r S is 
lefs than ~ , it will be impoflible for yz to be equal to 
- , and confequently the foregoing method of refolving 
3 
the equation x 3 -qx-r will be impracticable; but, if 
x 3 - qx or r is greater than » or ~ is greater than 
it will be poffible for yz to be equal to and confe- 
quently, the foregoing method of refolving the equation 
x 3 -qx=r will be practicable. And thus we fee in what 
manner it is pr.obable that cardan’s rule for refolving 
the cubic equation x 3 ~qx~r in the firft cafe of it, or 
when r is greater than ~~ , or ~ is greater than |y, to- 
gether with the reftriCtion of it to that firft cafe, may 
have been difcovered. 
H h 2 
Of 
