VOL. XIV.]| PHILOSOPHICAL TRANSACTIONS. 3Q 



dioristic limits rational, have the coefficient of the roots to be the triple of a 

 square number, such is a^ — 48 a = N. 



Assume a rank of roots in Arithmetical progression, and raise resolvends to 

 them. Thus : 



Draw a base line, and a perpendicular to it; then from in the base line 

 set off the negative resolvends downwards, and the affirmative ones upwards, 

 and raise their roots upon them as ordinates ; then a curve passing through the 

 same, is one half of the curve or locus on the right hand, for affirmative roots; 

 and the other half on the left hand is described in the same manner, by 

 assuming a rank of negative roots, and raising resolvends to them. See the 

 curve fig. 1, pi. 2. 



And 16, the third part of the coefficient of the roots, cubed, is equal to the 

 square of 64, half the resolvend, or dioristic limit. Which, in composing 

 Cardan's canon, is always subtracted from the square of half the absolute, as 

 in the example following: 



If I were to find the root belonging to the resolvend 297. 



The square of its half is 22052-^- 



The square of 64 half the dioristic limit 4096 



The difference is I7936:i- 

 And the rule is 148^ + /1 79564-, 

 I48i — -/l7956i. 

 That is, in a quadratic equation, if 297 were the sum of the two roots, and 

 64 the root of the rectangle : then if from the square of half the sum, the 

 rectangle be subducted, there remains the square of half the difference of the 

 roots; and giving them an universal cube root, it is v'148^ + v/17956j- + 

 v'148i — / 1798564- = 9 the root sought. 



In the scheme, QB and QP may denote the roots of Cardan's binomials, 

 that run infinitely upward, and terminate at Q. 



