674 PHILOSOPHICAL TRANSACTIONS, [aNNO I78O. 



been made. He would therefore suppose x to be equal io y — z; and by sub- 

 stituting this quantity instead of x in the original equation a^ + qx = r, he 

 would transform that equation into the following one, to wit, y^ — 3yz X 

 {jj — z) — 7? -\- q X (y — z) = r. Now in this equation it is evident, that the 

 terms 3yz X {y — z) and q X {y — z) have contrary signs; and therefore, if 

 their co-efficients 3yz and q can be supposed to be equal to each other, those 

 terms will mutually destroy each other, and the equation will be reduced to the fol- 

 lowing short one, y^ — z^ = r. And if in this equation we substitute, instead of z, 

 its value -^, derived from the same supposition of the equality of 9 and 3yz, 



the equation will be^^ — ~—, = r; and, by multiplying both sides by y, it will be 



y^ — ^ ■=. ry^; which equation, though it rises to the 6th power of the un- 

 known quantity y, is evidently of the form of a quadratic equation, and may 

 therefore be resolved, so far as to find the value of the cube of y, in the same 

 manner as a quadratic equation; after which it will be possible to find the value 

 of y itself by the mere extraction of the cube root; and then at last, from the 

 relation of 3/ to x (derived from the foregoing suppositions that 3/ — z was equal 



to X, and that 3yz was equal to q, and consequently z equal to -|-) we shall be 

 able to determine the value of x. 



Art. 4. It would therefore remain for the investigator of this method to 

 inquire, whether the supposition of 3yz being equal to q, was a possible supposi- 

 tion; that is, whether it was possible (whatever might be the magnitude of q 

 and r) for 2 quantities, y and z, to exist, whose nature would be such that their 

 difference y — 2 should be equal to the unknown quantity x in the equation 

 x^ -^ qx =■ r, and that 3 times their product should at the same time be equal 

 to q. And this supposition he would soon find to be always possible, whatever 

 may be the magnitudes of q and r; because, if the lesser quantity z be supposed 

 to increase from O ad infinitum, and the greater quantity y be likewise supposed 

 to increase with equal swiftness, or to receive equal increments in the same times, 

 and thus to preserve their difference y — z always of the same magnitude, or 

 equal to x, it is evident that the product or rectangle yz will increase continually 

 at the same times from O ad infinitum, and consequently will pass successively 

 through all degrees of magnitude, and therefore must at one point of time 

 during its increase become equal to A^q. 



And having thus found this supposition of the equality of yz and 49, or of 

 33/z and q, to be always possible, whatever might be the magnitudes of q and r, 

 our investigator would justly consider his solution of the equation x^ -\- qx = r, 

 which was founded on that supposition, as legitimate and complete. And thus 

 we see in what manner it seems probable, that Cardan's rule for resolving the 

 cubic equation x^ -\- qx =: r may have been discovered. 



