Free The King of Infinite Space by David Berlinski

geometry, the countercurrent.
    Writing in the seventeenth century, René Descartes created analytic geometry in a work titled La Géométrie . Descartes was not quite sure what he was doing. His great work he left almost as an afterthought. In analytic geometry, the Euclidean plane is made accessible, and so it is opened up,by means of a coordinate system. A point is chosen arbitrarily, the origin. Since all points are in the end the same, it hardly matters which point is chosen. Whatever the point, it corresponds to the number zero. Thereafter, the point is bisected by two straight and perpendicular lines, the coordinate axes of the system. The positive natural numbers run from the origin out to infinity, the negative numbers run out the other way, back-street boys to the end, and precisely the same scheme is repeated for the vertical axis, making four line segments starting at zero and proceeding inexorably to the edge of the blackboard and the space beyond.
    Any point in the plane may now be identified by a pair of numbers ( Figure VIII.1 ). Hidden previously in the sameness of space, a point acquires a vivid arithmetical identity. It is the point corresponding to two numbers, where before it was some drab or other, anonymous. Once points have acquired their numerical identity, the mathematician can deploy the magnificent machinery of algebraic analysis to endow Euclidean geometry with a second and incomparably more vivid form of life.

    F IGURE VIII.1. Euclidean coordinate system
    A RITHMETIC IS THE place where the numbers are found, and algebra, the place where they are treated in their most general aspects. The points and straight lines of Euclidean geometry—make of them what you will. They are undefined. Now that a geometrical point has been identified with a pair of numbers, a straight line can be defined by the equation Ax + By + C = 0, where A , B , and C are numerical parameters, fixed place markers, and x and y variables denoting points resident on the line.
    Did Euclid have circles to command? He did. A circle whose center is at the point ( a , b ), and whose radius is R , is perfectly and completely described by the formula ( x–a ) 2 + ( y–b ) 2 = R 2 . The identity of the circle has been dominated by a numerical regime: its center is a pair of numbers; its radius, another number, and its circumference, an endless succession of numbers.
    Analytic geometry has the power to depict a great many familiar geometrical shapes, such as the parabola, the ellipse,and the hyperbola. There is also the cardioid , its penciled heart emerging from the billet-doux of ( x 2 + y 2 + ax ) 2 = a 2 ( x 2 + y 2 ) ( Figure VIII.2 ).

    F IGURE VIII.2. The cardioid
    There are curves that look like a woman’s smile, or the valley between hills, or the exuberant petals of some tropical flower.
    There is an abundance.
    I N A LITTLE book titled The Coordinate Method , a troika of Russian mathematicians (I. M. Gelfand, E. G. Glagoleva, and A. A. Kirillov) offers this account of analytic geometry: “By introducing coordinates, we establish a correspondencebetween numbers and points of a straight line.” They then add: “In doing so, we exploit the following remarkable fact : There is a unique number corresponding to each point of the line and a unique point of the line corresponding to each number” (emphasis added). The remarkable fact to which they appeal is often described as the Cantor-Dedekind axiom, although how a fact could be an axiom, they do not say.
    It hardly matters. There is no such fact, and neither is there any such axiom. There are some numbers that no accessible magnitude can express. In Book X of the Elements , Euclid offers a proof that this is so, one based on an earlier proof attributed to the Pythagorean school. He fails only to notice that what he has done constitutes an act of immolation.
    The hypotenuse of a right triangle whose two sides are both 1 is, by the Pythagorean