the Invention of cardan’s Rules, &cc. 231 
3 yz is equal to q , and confequently that z is equal to 
A j we may determine the value of x. 
Art. 8. The only thing, therefore, that would remain 
for the inveftigator of thefe rules to do, in order to know 
w'hether the foregoing method of refolving the equation 
x*+qx=r was practicable or not, would be to inquire, 
whether it was poflible in all cafes, that is, in all magni- 
tudes of the known quantities q and r, for 3 yz to be 
equal to q , or for yz (or the product or redtangle of the 
two quantities y and z, whofe fum is equal to x) to be 
equal to | , and, if it was not poflible in all cafes, but 
only in fome, to determine in what cafes it was poflible, 
or what mu ft be the relation between q and r to make it 
poflible. 
Art. 9. Now, in order to determine this queftion, it 
would be proper and natural to obferve, that the quan- 
tity yz, or the product of the two quantities y and z, 
whofe fum is fuppofed to be equal to x, can never be 
greater than the fquare of half that fum, that is, than 
the fquare of -, or than — , by El. 2, 5, but may be of 
2 , 4 
any magnitude that does not exceed that fquare. There- 
fore, if -is greater than—, it will be impoflible for yz to 
be equal to it ; but, if q - is either equal to, or lefs than, 
3 
XX 
