A Conjecture concerning 
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Of the Equation qx-x l -r. 
Art. 14. In the third equation qx-x*~r the terms x l 
and qx have different figns, as well as in the fecond 
equation x 3 -qx-r; and therefore it feems to have been 
natural for the inventor of cardan’s rules to try both 
the fubftitutions of y-z and y+z inftead of x in this 
equation, as well as in that fecond equation, in hopes of 
an extermination of equal terms that are marked with 
contrary figns, and a confequent reduction of the equa- 
tion to another which, though of double the dimenfions 
of the equation qx-x^-r, fhould have been of a fimpler 
form and more eafy to be refolved. But it will be found 
upon trial, that neither of thefe fubftitutions will anfwer 
the end propofed. 
Art. 15. For, in the firft place, let us fuppofe x to be 
—y-z. Then we (hall have x’ i —y % —? ) yyz+'2 > yyz-z i =- 
y 1 — syzxy-z-z*, and qx=qxy-z, and confequently 
qx-x' s =qxy-z-y 3 + zyzx ly—z+z*. Therefore, qxy-z 
-y l + 3 yz x ly- z+z 3 will be = r. Now in this equation it 
is evident, the terms qxy-z and 3 yzxy-z have the 
fame figns, and therefore can never deftroy each other. 
Therefore, no fuch method of refolving this equation 
qx 
