426 14. EQUATIONS AND ALGORITHMSFIGURE 4. Omar Khayyam's solution of x^3 + ax^2 + b^2 x = b^2 c.we would phrase it, x^3 + ax^2 + bx = c. In keeping with his geometric interpretation
of magnitudes as line segments, Omar Khayyam had to regard the coefficient b
as a square, so that we shall write b^2 rather than b. Similarly, he regarded the
constant term as a solid, which without any loss of generality he considered to be a
rectangular prism whose base was an area equal to the coefficient of the unknown.
In keeping with this reduction we shall write 6^2 c instead of c. Thus Omar Khayyam
actually considered the equation x^3 + ax^2 + b^2 x = 6^2 c, where a, b, and c are data
for the problem, to be represented as lines. His solution is illustrated in Fig. 4.
He drew a pair of perpendicular lines intersecting at a point Ï and marked off
OA — a and OC = c in opposite directions on one of the lines and OB = b on
the other line. He then drew a semicircle having AC as diameter, the line DB
through  perpendicular to OB (parallel to AC), and the rectangular hyperbola
through C having DB and the extension of OB as asymptotes. This hyperbola
intersects the semicircle in the point C and in a second point Z. From Æ he drew
ZP perpendicular to the extension of OB, and Æ Ñ represented the solution of the
cubic.
When it comes to actually producing a root by numerical procedures, Omar
Khayyam's solution is circular, a mere restatement of the problem. He has broken
the cubic equation into two quadratic equations in two unknowns, but any attempt
to eliminate one of the two unknowns merely leads back to the original problem.
In fact, no method of solution exists or can exist that reduces the solution of every
cubic equation with real roots to the extraction of real square and cube roots of real
numbers. What Omar Khayyam had created was an analysis of cubic equations
using conic sections. He said that no matter how hard you look, you will never
find a numerical solution "because whatever is obtained by conic sections cannot
be obtained by arithmetic" (Amir-Moez, 1963, p. 336).
5.4. Sharaf al-Din al-Muzaffar al-Tusi. A generation after the death of Omar
Khayyam, another Muslim mathematician, Sharaf al-Tusi (ca. 1135-1213, not to
be confused with Nasir-Eddin al-Tusi, whose work was discussed in Section 4 of
Chapter 11), wrote a treatise on equations in which he analyzed the cubic equation
using methods that are surprisingly modern in appearance. This work has been
analyzed by Hogendijk (1989). Omar Khayyam had distinguished between the
eight types of cubic equations that always have a solution and the five that could
fail to have a solution. Al-Tusi provided a numerical method of solution for the
first eight types that was essentially the Chinese method of solving cubic equations.