Read Online Why Beauty is Truth by Ian Stewart - Free Book Online Page B

= square
    Each listed term would have a positive numerical coefficient. You may be wondering why this list does not include cases like
    cube + square = side
    The reason is that in these cases we can divide both sides of the equation by the unknown, reducing it to a quadratic.

    Omar did not entirely invent his solutions but instead built on earlier Greek methods for solving various types of cubic equation using conic sections. He developed these ideas systematically, and solved all fourteen types of cubic by such methods. Previous mathematicians, he noted, had discovered solutions of various cases, but these methods were all very special and each case was tackled by a different construction; no one before him had worked out the whole extent of possible cases, let alone found solutions to them. “Me, on the contrary—I have never ceased to wish to make known, with exactitude, all of the possible cases, and to distinguish among each of the cases the possible and impossible ones.” By “impossible” he meant “having no positive solution.”
    To give a flavor of his work, here is how he solved “A cube, some sides, and some numbers are equal to some squares,” which we would write as
    x 3 + bx + c = ax 2 .
    (Since we don’t care about positive versus negative, we would probably move the right-hand term to the other side and change a to – a as well: x 3 – ax 2 + bx + c = 0).
    Omar instructs his readers to carry out the following sequence of steps. (1) Draw three lines of lengths c / b ,, and a , with a right angle. (2) Draw a semicircle whose diameter is the horizontal line. Extend the vertical line to cut it. If the solid vertical line has length d , make the solid horizontal line have length c d /. (3) Draw a hyperbola (solid line) whose asymptotes (those special straight lines that the curves approach) are the shaded lines,passing through the point just constructed. (4) Find where the hyperbola cuts the semicircle. Then the lengths of the two solid lines, marked x , are both (positive) solutions of the cubic.

    Omar Khayyám’s solution of a cubic equation.
    The details, as usual, matter much less than the overall style. Carry out various Euclidean constructions with ruler and compass, throw in a hyperbola, carry out some more Euclidean constructions—done.
    Omar gave similar constructions to solve each of his fourteen cases, and proved them correct. His analysis has a few gaps: the points required in his construction sometimes fail to exist when the sizes of the coefficients a, b, c are unsuitable. In the construction above, for example, the hyperbola may not meet the semicircle at all. But aside from these quibbles, he did an impressive and very systematic job.
    Some of the imagery in Omar’s poetry is mathematical and seems to allude to his own work, in the self-deprecatory tone that we find throughout:
    For “Is” and “Is-Not,” though with Rule and Line
And “Up-And-Down” by logic I define,
Of all that one should care to fathom, I
Was never deep in anything but—Wine.
    One especially striking stanza reads:
    We are no other than a moving row
Of Magic Shadow-shapes that come and go
Round with the Sun-illumined Lantern held
In midnight by the Master of the Show.
    This recalls Plato’s celebrated allegory of shadows on a cave wall. It serves equally well as a description of the symbolic manipulations of algebra, and the human condition. Omar was a gifted chronicler of both.

4
THE GAMBLING SCHOLAR
    â€œI swear to you, by God’s holy Gospels, and as a true man of honor, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them.”
    This solemn oath was—allegedly—sworn in 1539.
    Renaissance Italy was a hotbed of innovation, and mathematics was no