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of doubling the cube. Archimedes developed the theory of these curves, and Apollonius of Perga systematized and extended the subject in his book Conic Sections. What particularly interested Omar Khayyám was the Greek discovery that conic sections could be used to solve certain cubic equations.
    Conic sections are so named because they can be obtained by slicing a cone with a plane. More properly, a double cone, like two ice-cream cones joined at their sharp ends. A single cone is formed by a collection of straight-line segments, all meeting at one point and passing through a suitable circle, the “base” of the cone. But in Greek geometry you can always extend straight-line segments as far as you wish, and the result is to create a double cone.
    The three main types of conic section are the ellipse, parabola, and hyperbola. An ellipse is a closed oval curve that arises when the cutting plane passes through only one half of the double cone. (A circle is a special kind of ellipse, created when the plane is exactly perpendicular to the cone’saxis.) A hyperbola consists of two symmetrically related open curves, which in principle extend to infinity, that arise when the cutting plane passes through both halves of the double cone. The parabola is a transitional form, a single open curve, and in this case the cutting plane must be parallel to one of the lines lying on the surface of the cone.

    Conic sections.
    At great distances from the tip of the cone, the curves of a hyperbola become ever closer to two straight lines, which are parallel to the lines where a parallel plane through the tip would cut the cone. These lines are called asymptotes.
    The Greek geometers’ extensive studies of conic sections constituted their most significant area of progress beyond the ideas codified by Euclid. These curves remain vitally important in today’s mathematics, but for quite different reasons from those that interested the Greeks. From the algebraic point of view, they are the next simplest curves after the straight line. They are also important in applied science. The orbits of planets in the solar system are ellipses, as Kepler deduced from Tycho Brahe’s observations of Mars. This elliptical orbit is one of the observations that led Newton to formulate his famous “inverse square law” of gravity. This in turn led to the realization that some aspects of the universe exhibit clearmathematical patterns. It opened up the whole of astronomy by making planetary phenomena computable.

    The majority of Omar’s extant mathematics is devoted to the theory of equations. He considered two kinds of solution. The first, following the lead of Diophantus, he called an “algebraic” solution in whole numbers; a better adjective would be “arithmetic.” The second kind of solution he called “geometric,” by which he meant that the solution could be constructed in terms of specific lengths, areas, or volumes by geometrical means.
    Making liberal use of conic sections, Omar developed geometric solutions for all cubic equations, and explained them in his Algebra , which he completed in 1079. Because negative numbers were not recognized in those days, equations were always arranged so that all terms were positive. This convention led to a huge number of case distinctions, which nowadays we would consider to be essentially the same except for the signs of the numbers. Omar distinguished fourteen different types of cubic, depending on which terms appear on each side of the equation. Omar’s classification of cubic equations went like this:
    cube = square + side + number
    cube = square + number
    cube = side + number
    cube = number
    cube + square = side + number
    cube + square = number
    cube + side = square + number
    cube + side = number
    cube + number = square + side
    cube + number = square
    cube + number = side
    cube + square + side = number
    cube + square + number = side
    cube + side + number