the Invention of cardan’s Rules , See. 233 
X is greater than — the compound quantity x % -qx will 
be greater than and, e converfo , if the compound 
quantity x % -qx is lefs than X will be lefs than 
and, if the compound quantity a 3 - qx is greater than 
2 qVq 
3V 7 ? 
gagi 
2 \/ Q 
, at will be greater than — -. Confequently, if the 
v 3 
compound quantity x z -qx, or, its equal, the abfolute term 
2 q \/ q 
r in the equation x z -qx=r, is lefs than or — is lefs 
3 
than — , it will be impollible for yz to be equal to - ; but, 
2 7 3 
if x^-qXy or r, is greater than 0 r — is greater than 
3^3 4 
— , it will be poffible for yz to be equal to - . Therefore, 
2 7 3 
rr 
if x^-aXi or r, is lefs than iddhd or — is lefs than the 
foregoing method of refolving the cubic equation 
x^-qx-r will be impracticable; but, if x 3 -qx=r, or r, is 
greater than or ~ is greater than ~ , it wn^be prac- 
rr 
ticable. 
Art. 1 o. It now only remains to be proved, that while 
x increafes, from being equal to V q, ad infinitum , the 
compound quantity x l -qx will likewife increafe from 0 
ad infinitum , without ever decreafing. Now this may be 
demonftrated as follows. 
Vol. LXX. 
Hh 
Art. 
