40 PHILOSOPHICAL TRANSACTIONS. [aNNO 1684. 



As to Cardans Rules. — ^The limits are of two kinds ; viz. either the base 

 limits when the resolvend is O, and the equation falls a degree lower : or the 

 dioristic limits, by which a pair of roots gain or lose their possibility. 



Cardan's canons are but the sum of the roots of a solid quadratic equation, 

 arising out of half the dioristic limit as the root of the rectangle and the re- 

 solvend as the sum. 



If the roots of those binomials are separately set off, as ordinates on their 

 resolvends, they produce curves infinitely continued upward, and meeting in a 

 point bisecting the root that is equal to a pair of equal roots, when the equation 

 is just limited, or dioristic, in the figure at Q. 



If these binomials are laid off as ordinates to their resolvends, Mr. Newton 

 upon sudden thoughts, supposed they may describe both sides of a hyperbola. 



If these binomial curves be continued downward, and separately found, should 

 always added make the root of a cubic equation capable of 3 roots ; then Car- 

 dan's impossible or negative roots are proved possible, and we only in ignorance 

 how to extract them. 



Assume any root within the limits of 3 possible roots, and raise a resolvend 

 to it; and when you have done, by Cardan's rules improved, you may find that 

 root, and, with a little varying the same, both the other roots (for every num- 

 ber or magnitude capable of a cube root, is capable of two more) as follows : 



After having obtained the cube roots of Cardan's binomials, according to 

 Van Schooten, in Descartes, or Kersey, if you change the signs of the 

 rational parts of those roots, as also the signs of the radical parts, and mul- 

 tiply these last parts by 3, the results are also roots of the cubic equation 

 first proposed. 



Example. — Of the equation ggc — 21a — 20 = 0, the cube roots of the 

 binomials are + 2-|^ -f -/ — f, and -|- 2-1- — ■/— 4; and their sum is the 

 root sought = -j- 5. Also, the other two roots are — 2^ -f \/2\, and — 



2i - V^^- 



Also, in this equation a^ — 6o a— 32 = O, the binomial roots are -f 4 -f- 

 \/— A, and + 4 — \/ — 4. Hence the root sought is -j- 8j and the other 

 two roots are — 4 -|- y^-f 12, and — 4 — \/ -\- 12. 



If the roots in the former section be assumed in arithmetical progression, 

 and the equation with its several resolvends be depressed, there will come 

 out a regular series of quadratic equations ; whence an easy method will arise 

 of writing down such ranks as, multiplied by an arithmetical progression, shall 

 always produce the same cubic equation, the resolvend only varying. 



Let the roots of this series of quadratics be found as usual in binomials; 

 let these binomials be cubed; and then let it be observed, whether the results 



