The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)

Part II. Hope and Monsters

Chapter 32. A Fly on the Wall

The mathematical patterns that Kepler had found in the heavens looked different from those Galileo had found on Earth. Perhaps that was to be expected. What did falling rocks have to do with endlessly circling planets, which plainly were not falling at all?

Isaac Newton’s answer to that question would make use of mathematical tools that Kepler and Galileo did not know. Both astronomers were geniuses, but everything they found might conceivably have been discovered in Greece two thousand years before. To go further would require a breakthrough the Greeks never made.

The insight that eluded Euclid and Archimedes (and Kepler and Galileo as well) supposedly came to René Descartes when he was lying in bed one morning in 1636, idly watching a fly crawl along the wall. (“I sleep ten hours every night,” he once boasted, “and no care ever shortens my slumber.”) The story—so claimed one of Descartes’ early biographers—was that Descartes realized that the path the fly traced as it moved could be precisely described in numbers. When the fly first caught Descartes’ eye, for instance, it was 10 inches above the floor and 8 inches from the left-hand edge of the wall. A moment later it was 11 inches above the floor and 9 inches from the left edge. All you needed were two lines at right angles—the horizontal line where the wall met the floor, say, and the vertical line from floor to ceiling where two walls met. Then at any moment the fly’s position could be pinpointed—this many inches from the horizontal line, that many from the vertical.

Pinpointing a location was an old idea, as old as latitude and longitude. The new twist was to move beyond a static description of the present moment—the fly is 11 inches from here, 9 inches from there; Athens is at 38˚N, 23˚E—and to picture a moving point and the path it drew as it moved. Take a circle. It can be thought of in a static way, as a particular collection of points—all those points sitting precisely one inch from a given point, for instance. Descartes pictured circles, and other curves, in a more dynamic way. Think of an angry German Shepherd tethered to a stake and straining to reach the boys teasing him, just beyond his reach. The dog traces a circle—or, more accurately, an arc that forms part of a circle—as he moves back and forth at the end of his taut leash. A six-year-old on a swing, pumping with all his might, traces out part of a circle as the swing arcs down toward the ground and then up again.

From the notion of a curve as a path in time, it was but a step to the graphs that we see every day. The key insight was that the two axes did not necessarily have to show latitude and longitude; they could represent any two related quantities. If the horizontal axis depicted “time,” for instance, then a huge variety of numerical changes suddenly took on pictorial form.

The most ordinary graph—changes in housing prices over the last decade, rainfall this year, unemployment rates for the past six months—is an homage to Descartes. A table of numbers might contain the identical information, but a table muffles the patterns and trends that leap from a graph. We have grown so accustomed to graphs that show how something changes as time passes that we forget what a breakthrough they represent. (Countless expressions take this familiarity for granted: “off the charts,” “steep learning curve,” “a drop in the Dow.”) Any run-of-the-mill illustration in a textbook—a graph of a cannonball’s position, moment by moment, as it flies through the air, for example—is a sophisticated abstraction. It amounts to a series of stop-action photos. No such photos would exist for centuries after Descartes’ death. Only familiarity has dulled the surprise.41

Even in its humblest form (in other words, even aside from thinking of a curve as the trajectory of a moving point), Descartes’ discovery provided endless riches. With his horizontal and vertical axes in place, he could easily construct a grid—he could, in effect, tape a piece of graph paper to any spot he wanted. That assigned every point in the world a particular address: x inches from this axis, y inches from that one. Then, for the first time, Descartes could approach geometry in a new way. Rather than think of a circle, say, as a picture, he could treat it as an equation.

A circle consisted of all the points whose x’s and y’s combined in a particular way. A straight line was a different equation, a different combination of x’s and y’s, and so was every other curve. A curve was an equation; an equation was a curve. This was a huge advance, in the judgment of John Stuart Mill “the greatest single step ever made in the progress of the exact sciences.” Now, suddenly, all the tools of algebra—all the well-developed arsenal of techniques for manipulating equations—could be enlisted to solve problems in geometry.

But it was not simply that algebra could be brought to bear on geometry. That would have been a huge practical breakthrough, but Descartes’ insight was a conceptual revolution as well. Algebra and geometry had always been seen as independent subjects. The distinction wasn’t subtle. The two fields dealt with different topics, and they looked different. Algebra was a forest of symbols, geometry a collection of pictures. Now Descartes had come along and showed that algebra and geometry were two languages that described a shared reality. This was completely unexpected and hugely powerful, as if today someone suddenly showed that every musical score could be converted into a scene from a movie and every movie scene could be translated into a musical score.