678 ' PHILOSOPHICAL TRANSACTIONS. [aNNO 1780. 



thod of resolving the equation x^ — qx = ?■ will be practicable. And thus we 

 see in what manner it is probable that Cardan's rule for resolving the cubic equa- 

 tion x^ — qx = r, in the first case of it, or when r is greater than 

 ^^ -, or ^- is greater than |- , together with the restriction of it to that first case, 



may have been discovered. 



Art. 14. In the 3d equation qx — x^ = r, the terms x^ and qx have different 

 signs, as well as in the 2d equation x^ — qx z=r; and therefore it seems to have 

 been natural for the inventor of Cardan's rules to try both the substitutions of 

 y — z and y -\- z instead of x in this equation, as well as in that 2d equation, in 

 hopes of an extermination of equal terms that are marked with contrary signs, 

 and a consequent reduction of the equation to another which, though of double 

 the dimensions of the equation qx — x^ =i r, should have been of a simpler 

 form and more easy to be resolved. But it will be found on trial, that neither of 

 these substitutions will answer the end proposed. 



Art. 15. For, in the first place, let us suppose x ioht ■= y — 2. Then we 

 shall have x^ z=.y^ — 3yz X {y — z) — z?, and qx = q X (y — z), and conse- 

 quently qx — x^ = q X {y — z) — y^ + 3yz X {y —z) + z^ Tiierefore, q X 

 (3/ — z) — z/^ -f 3yz X iy — z + z^ will be = r. Now in this equation it is 

 evident that the terms q x (2/ — z) and 3yz X (y — z) have the same signs, 

 and therefore can never destroy each other. Therefore no such method of re- 

 solving this equation qx — x^ = r, as was found above for the 2 former equa- 

 tions .r^ -\-qx = r and x^ — qx =z r, can be obtained by substituting the differ- 

 ence y — z \n it instead of x. 



Art. id. We may now try the substitution oi y -\- z instead of x in the terms 

 of this equation. Now if x be supposed to be =: ^ + z, we shall have x^ =3/* 

 -|- 3yz X (y + z) -j- 2^, and qx z= q X {y -\- z), and consequently, qx — ,r' = 

 q X {y + z) — y'' — 3yz X {y + z) — z^ Therefore, q X {y + z) — j/' — 

 3yz X (j/ + z) — z^ will be = r. In this equation it is true indeed that the 

 terms q X {y + z) and 3yz X {y — z) have different signs. But they cannot 

 be equal to each other : for since the 3 terms y^ and 3yz X {y — z) and z^ are 

 all marked with the sign — , or are to be subtracted from the first term q X 

 {y -\- z), and the remainder is = r, it is evident, that q X (?/ -f- z) must be 

 greater than the sum of all the 3 terms y^, 3yz X {y -\- z), and z^, taken toge- 

 ther, and therefore, a fortiori, greater than 3yz X {y -\- z) alone. Therefore 

 no such extermination of equal terms marked with contrary signs, as took place in 

 the transformed equations derived from the 2 former equations .r' -^ qx =■ r, and 

 x^ — qx=i r, can take place in this tranL-formed equation, derived from the equation 

 qx — x^= r, by substituting j/ -f z in its terms instead of a.' ; and consequently no 

 such method of resolving tlie equation qx — a' = r, as has been found for the 



