VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 335 



The three roots are: 



or = jb 4- -/ r + v/ r'i — ^a 4- V r — V r^ — q^ 



x = p-'^^-^ X ^r+/r^-^^- l±-^X ^r^\/?'::rf 



2 " . . . ^ 2 



3 



Or, to make the arithmetical calculation easier and readier, if we put m -j- \/ ?* 

 for the cube root of the irrational binomial r + v^r^ + ^^ then will the same 

 three roots of the equation he x = p -\- 2m, and x = p — m ± V — 3n. 



Therefore when there is given any cubic equation, we must compare its terms 

 with the several terms of the general equation, by which the values of />, q, r 

 will be easily found : then these being known, all the roots of the given equa- 

 tion will thence be known. And of this solution here follow some examples in 

 numbers. 



1 . Let it be proposed to find the root x of this equation, x"^ = 2x^ + 3* + 4. 

 First, by comparing or equating the like terms, there will he3p =. 2, or /)= ^; 

 secondly, 3q — 3p'^ = 3, that is, 3^ — 4- = 3, or ^ = '^^ ; thirdly, 2r 4-jb^ -- 

 3pq = 4, that is, 2r — -^-^ =4, or r = fA; also r^ — q^ = Vr 5 therefore 

 :r = -I- + ^if + i/Vt ^~ ^ir— /Vr- The two other roots are impos- 

 sible. 



2. In the equation a^ — 12x^ — 41a: + 42, it will be first Sjb = 12, or jb = 4; 

 next, 3q — 3/>" = — 41, that is 3^ — 48 = — 41, or^ = 4-; thirdly 2r -\- p^ 

 — 3pq = 42, that is, 2r + 36 = 42, or r = 3 ; hence r^ — ^' = — y./ : then 

 the cube root of r + ^^ r^ — q^, that is, of the binomial surd 3 + v' — Vt» 

 extracted by the methods of the arithmetic of surds, is — 1 -j- »/ — 4. = m-{- 

 -v/ n: therefore x := p -{- im = 4 — 2 = 2, also x = p — m + V — 372 = 4-}- 



work, consisting of annotations, illustrations, and supplements. At the date of this publication 

 Mr, Colson was in the situation of master of Sir Joseph Williamson's free mathematical school at 

 Rochester, and a F. R. S. Three years after this we find he succeeded Mr, Sanderson as Lucasian 

 Professor of Mathematics at Cambridge, a situation which he held for 20 years, after which he was 

 succeeded, in 1759> by Dr. Waring. Though Mr. Colson died only Dec. 20, 176o, The character of 

 our author seems to have been chiefly that of minute precision, and patient laborious industry. The 

 present paper, on the solution of cubic and biquadratic equations, is founded nearly on the idea of the 

 solution by Descartes, by assuming the biquadratic as equal to the product of two quadratics with 

 indeterminate coefficients. The construction of the equations also, by means of the circle and para- 

 bola, is in imitation of the like constructions of Descartes in his geometry, and of Mr. Baker in his 

 Gteometrical Key. -Mr. Colson, it seems, was so well pleased with the Analytical Institutions of the 

 Signora Agnesi, that he made an entire translation of that lady's work, from the Italian, the copy 

 of which has lately been found among his papers, and published at the exoence of Baron Maseres, 

 in two volumes, 4to. 



