Chapter 5 (Roots as sums of radicals, pp. 104-120) of Stedall's book (bibliographic information below) discusses:
- The John Colson paper you asked about (mainly on pp. 104-106).
- A paper by de Moivre published in the same volume of the Philosophical Transactions that follows immediately after Colson's paper.
- The impact of Colson's and de Moivre's work on some of Euler's investigations in the 1730s.
- Related investigations by Euler and Bezout that was published in 1764. (Euler's paper was originally communicated to the Berlin Academy in 1753, but the associated publication did not appear in print until 1764.)
Jacqueline Anne Stedall, From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra, Heritage of European Mathematics, European Mathematical Society, 2011, xii + 224 pages. MR 2012a:01012; Zbl 1231.01006
Excerpt from pp. 104-106 of Stedall's book
In 1545 Cardano had written at some length about the number of positive or negative roots one could expect to find for a given cubic or quartic equation (see pages 14-16). He wrote very much more briefly about the form those roots could take (page 16). In his view a solution to a quadratic equation was the sum of a rational and a square root while a solution to a cubic equation was the sum of a rational and two cube roots. He did not explicitly discuss the structure of the roots of quartic equations, which from experience he knew to be rather more complicated (for an example see page 14). His only comment on equations of higher degree was that a fifth root, for example, could satisfy only an equation of the simplest kind, a fifth power equal to a number; conversely, such equations could not be satisfied by a sum of two or more such roots.
From now on, to avoid confusion between roots of numbers and roots of equations, we will use the term 'radicals' to describe square, cube, and all higher roots of integers or rational numbers. These are central to the content of this chapter.
Until the early years of the eighteenth century, no other author explicitly considered the form that roots of equations might take. When Dulaurens in 1667 solved some special equations of degrees $5,$ $7,$ and $11$ they turned out to have roots compossed of sums of pairs of radicals of degrees $5,$ $7,$ and $11,$ respectively, but Dulaurens did not comment on it. In Paris in 1675 Leibniz and Tschirnhaus briefly pursued the idea of roots as sums (or other expressions) composed of radicals, but Leibniz complained of the labour involved in trying to eliminate the radical signs, so the idea came to nothing and was never published (see pages 64-65). In the spring of 1707, however, two papers on equations were published in the Philosophical Transactions of the Royal Society, the first by John Colson, the second by Abraham de Moivre. Both introduced new conjectures about the structure of roots of equations. De Moivre's paper in particular was the mathematical starting point for the developments outlined in this chapter, and was quoted frequently by later writers.
In this chapter we will first discuss the papers of Colson and de Moivre. We will then examine the way the ideas presented in them were taken up first by Euler, who was quick to spot their potential, and later also by Étienne Bezout. The consequence was that Euler and Bezout were independently but almost simultaneously able to develop an important new technique of equation-solving, which will be described in the final part of this chapter.
John Colson, born in 1680, entered Christ Church, Oxford, in 1699 but never took his degree. Ten years later he took up a teaching post at the new mathematical school at Rochester in Kent. In 1739 he became a lecturer at Cambridge, and later that year became the fifth Lucasian Professor. Despite ending up in such a prestigious post, Colson's mathematical output over his lifetime was of little significance. He was better known as a publisher and translator of mathematical texts than as an innovator, and his paper on equations of 1707 was one of only three original pieces of work that he published. Nevertheless, it contained one important new idea.
The first part of the paper is devoted to the rules for solving cubic and quartic equations. For cubic equations Colson first stated the solution formula, then gave several worked examples to show its use. Only after that did he offer a derivation of it. His method, for solving the general equation $z^3 = 3qz + 2r,$ was to suppose that $z = a + b,$ so $z^3 = 3abz + a^3 + b^3.$ Comparing this identity with the proposed equation we have $q = ab$ (or $q^3 = a^{3}b^{3})$ and $2r = a^3 + b^3.$ These equations are easily combined to give $2ra^3 = a^6 + q^3,$ which is a quadratic equation in $a^3$ with solutions (1) $a^3 = r + \sqrt{r^2 – q^3}$ $\;\;$ (2) $b^3 = r - \sqrt{r^2 – q^3}.$
There was nothing new or remarkable in this (see, for example, similar derivations by Hudde and Dulaurens, pages 54-55 and 57-58). At this point, however, Colson observed that any quantity has three cube roots, and that the cube roots of unity are $1,$ $-\frac{1}{2} + \frac{1}{2}\sqrt{-3},$ $-\frac{1}{2} - \frac{1}{2}\sqrt{-3}.$ Equations (1) and (2) therefore yield three possible values for $a$ and three for $b.$ Thus there are potentially nine possible values of $z = a+b.$ Colson tested out the various combinations, and found that the juxtapositions that satisfy the original equation are [lengthy displayed equations omitted]. Thus he had found not just one root, as most of his predecessors had been satisfied to do, but all three roots of the original cubic. footnote: Leibniz had asserted privately to Huygens that Cardano's rule could produce all the roots of a cubic, but had not explained how. See Chapter 1, note 19.
[some comments about Colson and quartic equations omitted] The fact that a cubic equation has three roots and a quartic equation has four had been recognized for at least a century but Colson was the first to give explicit formulae for each root. His paper ends with geometric constructions which need not concern use here.
(ADDED NEXT DAY) Below are some books that you might also want to know about. I do not think Colson is mentioned in any of these books, but given what you've written in your question and in your follow-up comments, I would encourage you to consider getting a copy of some or all of these books.
[1] Isabella G. Bashmakova and Galina S. Smirnova, The Beginnings and Evolution of Algebra, translated from the Russian by Abe Shenitzer, The Dolciani Mathematical Expositions #23, Mathematical Association of America, 2000, xvi + 179 pages. MR 2000h:01002; Zbl 942.01001 [See also here.]
[2] Roger Lee Cooke, Classical Algebra. Its Nature, Origins, and Uses, John Wiley and Sons (Wiley-Interscience), 2008, xii + 206 pages. MR 2009b:00001; Zbl 1139.00001 [Review at MAA website.]
[3] Jacques Sesiano, An Introduction to the History of Algebra. Solving Equations from Mesopotamian Times to the Renaissance, Mathematical World #27, American Mathematical Society, 2009, viii + 174 pages. MR2514537; Zbl 1182.01002 [See also here.]
[4] Jacqueline Anne Stedall, A Discourse Concerning Algebra: English Algebra to 1685, Oxford University Press, 2002, xii + 294 pages. MR 2005c:01015; Zbl 1035.01006 [Review at MAA website.]
[5] Veeravalli Seshadri Varadarajan, Algebra in Ancient and Modern Times, Mathematical World #12, American Mathematical Society, 1998, xvi + 142 pages. MR 99d:01007; Zbl 917.01002 [See also here.]
[6] Girolamo Cardano, The Great Art or the Rules of Algebra, translated and edited by T. Richard Witmer, MIT Press, 1968, xxiv + 267 pages. MR 40 #4074 and 50 #12562; Zbl 191.27704 [Reprinted by Dover Publications in 1993 (MR 94k:01038; Zbl 862.01034) and in 2007.]
[7] François Viète, The Analytic Art, translated by T. Richard Witmer, The Kent State University Press, 1983, i + 450 pages. MR 86b:01012; Zbl 558.01041 [Reprinted by Dover Publications in 2006 (Zbl 1115.01017).]
[8] Edward Waring, Meditationes Algebraicae. An English Translation of the Work of Edward Waring, edited and translated from the Latin by Dennis Weeks, American Mathematical Society, 1991, lx + 459 pages. MR 93a:01026
In modern terminology, Khayyam solves the cubic $x^3+cx=d$ (your case (b)) as the $x$-coordinate of the point of intersection in the first quadrant between the parabola $x^2=\sqrt{c}y$ and the circle $x\big({d\over c}-x\big)=y^2$. (The center of the circle is at $\bigl({c\over 2d},0\bigr)$, and it passes through the origin $(0,0)$.)
He comments that this gives a unique positive $x$-value, and he shows that this $x$-value solves the cubic equation since
$${c\over x^2}={x^2\over y^2}={x^2\over x\bigl({d\over c}-x\bigr)}={x\over {d\over c}-x}$$
which implies that $d-cx=x^3$, or $x^3+cx=d$.
Edit: For your case (a), I haven't seen exactly how Khayyam did this. But looking at the above, with the equation $x^3=cx+d$, it would be reasonable to try the same parabola $x^2=\sqrt{c}y$ and the equation
$${c\over x^2}={x^2\over y^2}={x^2\over x\bigl(x+{d\over c}\bigr)}={x\over x+{d\over c}}$$
which, by comparison of the outermost terms, implies that $x^3=cx+d$. For the middle equality to hold, you need the equation $y^2=x^2+{d\over c}x$, which is the equation of a hyperbola with center at $\bigl(-{c\over 2d},0\bigr)$, and which passes through the point $(0,0)$.
Also note that Khayyam of course did not have our algebraic notation at his disposal, so he would have to carry out his arguments using only geometric properties of the conic sections, as developed by Apollonius of Perga.
Best Answer
There's a brief note in this book on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in the first quadrant.
Another note should be made that the curves of the time were constructed with geometric tools (straightedge, compass, and a bunch of other contraptions), and $y=x^3$ isn't really a sort of curve that easily lends itself to such a construction (but is now easily constructed thanks to our current knowledge of coordinate geometry).
Here is a more explicit mention of the hyperbola-circle intersection problem Khayyam studied and was mentioned in the OP.
Here is a (more or less) complete table of all the intersection cases Khayyam studied. (The book has an appendix containing a (translated) section of Khayyam's work.)
Here is yet another reference.
(I'll keep updating this answer as I comb through more books; watch this space! As an aside, it's funny that my attempts to look for answers to this question are leading me to references for this question!)