Zero - Charles Seife [52]
Point and Counterpoint
One will then see the simplicity with which these concepts lead to properties already known and to an infinity of others which ordinary geometry does not seem to touch easily.
—JEAN-VICTOR PONCELET
The first development—projective geometry—was born in the turmoil of war. In the 1700s, France, England, Austria, Prussia, Spain, the Netherlands, and other countries were vying for power. Alliances formed and broke over and over again; new territorial disputes erupted over colonies, and countries struggled to dominate trade to and from the New World. France, England, and other countries skirmished throughout the first half of the eighteenth century, and roughly a quarter century after Newton died, a full-scale war erupted. France, Austria, Spain, and Russia fought England and Prussia for nine years.
In 1763 the French capitulated and the Seven Years’ War was over. (Two years of fighting occurred before war was officially declared.) The victory made England the preeminent power in the world, but it came at a great cost. Both France and England were exhausted and in debt—and they would both suffer the consequences: revolutions. A little more than a decade after the end of the Seven Years’ War, the American Revolution began; the revolt would strip England of its richest colony. In 1789, just as George Washington was sworn into office in the newly founded United States, the French Revolution began. Four years later the revolutionaries removed the French king’s head.
A mathematician, Gaspard Monge, signed the revolutionary government’s record of the king’s execution. Monge was a consummate geometer, specializing in three-dimensional geometry. He was responsible for the way architects and engineers draw buildings and machines: they project the design onto a vertical plane and a horizontal plane, preserving all the information needed to reconstruct the object. Monge’s work was so important to the military that much of it was made into a state secret by the revolutionary government and by the Napoleonic government that succeeded it soon afterward.
Jean-Victor Poncelet was a student of Monge’s who learned about three-dimensional geometry as he trained to become an engineer for Napoleon’s army. Unluckily for Poncelet, he entered the army just as Napoleon set off for Moscow in 1812.
While retreating from Moscow, Napoleon’s army was whittled down to almost nothing by a harsh winter and an equally harsh Russian army. At the battle of Krasnoy, Poncelet was left for dead on the battlefield. Still alive, he was captured by the Russians. Moldering in a Russian prison, Poncelet founded a new discipline: projective geometry.
Poncelet’s mathematics was the culmination of the work begun by the artists and architects of the fifteenth century, like Filippo Brunelleschi and Leonardo da Vinci, who discovered how to draw realistically—in perspective. When “parallel” lines converge at the vanishing point in a painting, observers are tricked into believing that the lines never meet. Squares on the floor become trapezoids in a painting; everything gets gently distorted, but it looks perfectly natural to the viewer. This is the property of an infinitely distant point—a zero at infinity.
Johannes Kepler, the man who discovered that planets travel in ellipses, took this idea—the infinitely distant point—one step further. Ellipses have two centers, or foci; the more elongated the ellipse, the farther apart these foci are. And all ellipses have the same property: if you had a mirror in the shape of an ellipse and you placed a lightbulb at one focus, all the beams of light would converge at the other focus, no matter how stretched-out the ellipses are (Figure 29).
In his mind Kepler stretched an ellipse out more and more, dragging one focus farther and farther away. Then Kepler imagined that the second