VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 675 



Art. 5. In the 2d equation x^ — qx ■= r, in which the 2d term qx is sub- 

 tracted from the first, or marked with the sign — , it seems to have been natural 

 for the person who invented these rules to substitute the sum as well as the dif- 

 ference of 2 other quantities, y and z, instead of x, in the terms x^ and qx, in 

 hopes of such an extermination of equal terms, and consequential reduction of 

 the equation to one of a simpler and more manageable form, as was found to be so 

 useful in the case of the former equation x^ + qx = r. We will therefore try 

 both these substitutions, and, as that of the difference y — z has in the former 

 case proved so successful, we will begin by that. 



Art. 6. Now, by substituting the difference y — z instead of x in the equa- 

 tion x^ — qx ^= r, we shall transform it into the following equation, to wit, 

 y^ — 3yz X iy — z) — 7? — q X {y — z) =■ r; in which the terms 3yz X 

 {y — z) and q x {y — z) have both of them the same sign — prefixed to them, 

 and consequently can never exterminate each other, whether 3j/z be equal or 

 unequal to q. This substitution therefore is in this case of no use. 



Art. 7. We will now therefore try the substitution of the sum of y and z, instead 

 of their difference, in the equation x^ — qx = r. Now, if x be supposed to be 

 equal to !/ -|- z, and?/ -|- z be substituted instead of it in the equation x^ — qx = r, 

 that equation will be thus transformed into the following one, to wit, 

 y^ -{- 3yz X (y + z) -{- z^ — q X (y + z) =: r. Now in this equation, 3yz X 

 (y + z) and q X (y -{- z) have contrary signs. Consequently, if they can be 

 supposed to be equal to each other, they will destroy each other, and the equa- 

 tion will be thus reduced to the following short one, y^ -\- z^ = r; that is, if 

 3yz and q can be supposed to be equal to each other, or if yz can be supposed to 

 be equal to ^q, the equation will be reduced to the short equation y^ -{- z^ = r. 



And if in this short equation we substitute, instead of z its value ^, derived from the 



same supposition of the equality of 3yz and q, the equation thence resulting 



will be 3^^ -|- -^ = r; and by multiplying both sides by y\ it will be ^^ -}- |-, 



= ry'; and, by subtracting y^ from both sides, it will be ry' — ^^ = 5^; which, 

 though it rises to the 6th power of y, is evidently of the form of a quadratic 

 equation, and consequently may be resolved in the same manner as a quadratic 

 equation, so far as to find the value of y% or the cube of the rooty; after 

 which it will be possible to find the value of y itself by the mere extraction ot 

 the cube root; and lastly, from the relation of y to x (contained in the 2 sup- 

 positions, that y -{- z IS equal to x, and that 3yz is equal to q, and consequently 

 that z is equal to |-) we may determine the value of x. 



Art. 8. The only thing therefore, that would remain for the investigator of 

 these rules to do, in order to know whether the foregoing method of resolving 



4 R 2 



