Resolution of Algebraic Equations. 
301 
ming them in pairs, is x — -J- — [2 q -}- \ fi\ x x 
£ T + 
— 4 s — IF+J r + TT^l x - r — 4] —W~ T+^l = °>* 
which, when (p) is supposed to vanish, becomes (a; 6 — 2 qx* HH 
q* — 4 fS\x* — r l =. o). 
30. These two methods, one applying to the biquadratic 
equations complete in their terms, and the other to those from 
which the second term has been expunged, are all that have yet 
been discovered ; and, notwithstanding the number of different 
methods attributed to different writers, which from their manner 
of setting out appear distinct, they will all be found to resolve 
themselves, in principle, into one of these. Dr. Hutton’s Ma- 
thematical Dictionary, under the article Biquadratic Equations, 
gives four methods ; viz. Ferrari’s, Des Cartes’s, Euler’s, 
and Simpson’s ; to which may be added another by Dr. Wa- 
ring,! and perhaps many more. They proceed upon a variety 
of different contrivances ; but, when analysed, and the real object 
gained is viewed apart from the process that led to it, Ferrari’s, 
which is the oldest, and does not require the extinction of the 
second term , will be seen to produce the cubic ( x z — qx* - f- 
pr — 45^ — p* s -f 4 qs — r*=o); and Des Cartes’s, .which 
supposes the second term to be first destroyed, terminates in 
the cubic-formed equation of the sixth degree ( x 6 — 2 qx* -\- 
q 1 — 4 s^x* — r* — o). The rest produce cubics, or cubic- 
formed sixth powers, whose roots are some parts or multiples 
of this last; except Waring’s method, which does not expunge 
the second term, and therefore produces a cubic whose roots 
are parts of the first. But, whether the resulting equation be 
* Waring’s Medit. Algeb . p. 133. 
+ Ibid. p. 138; and the Appendix to Dr. Hutton’s Dictionary. 
MDCCXCIX. R r 
