the Invention of cardan’s Rules, See. 227 
evidently of the form of a quadratic equation, and may 
therefore be refolved, fo far as to find the value of the 
cube of y, in the fame manner as a quadratic equation ; 
after which it will be poffible to find the value of y itfelf 
by the mere extraction of the cube root; and then at 
laft, from the relation of y to a? (derived from the fore- 
going fuppofitions thatjy-sr was equal to x, and that 3 yz 
was equal to q , and confequently z equal to — ) we fhall 
be able to determine the value of x. 
Art. 4. It would therefore remain for the inveftigator 
of this method to inquire, whether or no the fuppofition, 
“ that 3 yz was equal to q was a poffible fuppofition ; 
that is, whether it was poffible (whatever might be the 
magnitudes of q and r) for two quantities,^ and z, to exift* 
whofe nature fhould be fuch that their difference y—z 
fhould be equal to the unknown quantity x in the equa- 
tion x* + qx=r, and that three times their produCl fhould 
at the fame time be equal to q, or their fimple produCl to 
the third part of q. And this fuppofition he would foon 
find to be always poffible, whatever may be the magni- 
tudes of q and r; becaufe, if the leffer quantity £ is fup- 
pofed to increafe from 0 ad infinitum , and the greater 
quantity y is likewife fuppofed to increafe with equal 
fwiftnefs, or to receive equal increments in the fame 
G g 2 times, 
