Omar Khayyam - Solution of the cubic: A cube and sides are equal to a number.
From Omar Khayyan - (I added the numbering for clarity)
A cube and sides are equal to a number.
1.Let the line AB [see figure] be the side of a square equal to the given number of roots, [that is, (AB)^2=a, the coefficient of "x".]
2.Construct a solid whose base is equal to the square on AB, equal in volume to the given number, [ b ]. The construction has been shown previously. Let BC be the height of the solid. [i.e. BC·(AB)^2 = b, so BC=b/a.] Let BC be perpendicular to AB ...
3.Construct a parabola whose vertex is the point B ... and parameter AB.
4.Then the position of the conic HBD will be tangent to BC. Describe on BC a semicircle. It necessarily intersects the conic. Let the point of intersection be D; drop from D, whose position is known, two perpendiculars DZ and DE on BZ and BC. Both the position and magnitude of these lines are known.