He wasn’t completely free of the older view as he pondered the tricks that schoolmasters used to construct the canonical curves without bothering with any algebra: a string attached to a peg that could be used to generate a circle; the same string fitted to two pegs to trace out an ellipse; and so on. He thought about more elaborate “mechanical” ways complex curves can emerge—the cycloid, for example, a form traced by a point on the rim of a wheel that rolls in a straight line—and others, still more complicated.
The making of a cycloid, from an eighteenth-century encyclopedia
In all those ways of ending up with a curving line on a page, there was one common theme: every curve was a map of motion, a mathematician’s travelogue. A point travels through space, and its trail, its trace, creates the stuff of geometry. Crucially, sometime during these months, Newton realized that this approach, the “generation of figures by motion,” could apply not just to abstract travel, the path of points in Cartesian spaces, but to the real stuff of the real world. In other words, motion in the universe, and not just in the mind’s eye of the geometer, could be expressed in the mathematics he was inventing.
Newton did not grasp the full implication of this work all at once. He understood at least the mathematical side of his breakthrough by as early as November 13, 1665. In the paper he wrote then, he described the “infinitely little lines” that accumulated at each infinitely brief instant of time as his figures evolved. His breakthrough came when he realized that his two seemingly separate questions—how a curve bends and how much of the Cartesian plane it encloses—are actually twin faces of the same problem. Every change in the slope of a curve affects how much that shape encloses beneath it, and the same is true in reverse: the accumulation of territory under a curve reflects the shifting trace of that geometrical figure.
With that insight Newton arrived at a discovery that, on its own, would have made him one of the most famous thinkers who ever lived. Figuring out how to characterize how the shape of a curve is changing at any point in time is the core of what is now called differential calculus, which he then extended to integral calculus, which addresses the questions relating to the areas bounded by such curves. Taken together, those two interrelated ideas, as developed and extended, remain the foundational mathematics of material experience.
Newton never underestimated his own powers. He had to have grasped the importance of his accomplishment in those few months of enforced seclusion on his farm. Yet for most of the next two decades, he kept this new mathematical insight almost entirely to himself.
Still, this was the inflection point, after which the way humankind understood its circumstances would be irreversibly altered from what had been known before. What is motion but change over time? And what is the world but matter in motion, an ever-transforming flux, continuously transforming as the instants pass into seconds, hours, years?
THE INVENTION OF the calculus, the mathematics of change, was one of the keys to what we now call the Newtonian revolution—and Newton in his miracle year put his breakthrough into almost immediate use. As 1666 began and the plague continued to rage, Newton turned from pure math to questions of material experience. At the heart of his inquiry lay the problem of gravity. As he told the story sixty years later, the essential clue to his ultimate theory came to him during the summer of 1666. One day he found himself in his garden “in a contemplative mood.” The tree in front of him was heavy with fruit. Suddenly, an apple fell—an utterly ordinary occurrence. And yet, it nagged at him. “Why should that apple always descend perpendicularly to the ground?” he recalled asking himself. “Why should it not go sideways or upwards? but constantly to the earth’s center?”
Why indeed? The myth of genius has asserted that this was all it took: in the not-quite-infinitesimal slice of time it took for the apple to drop to the ground, Newton grasped the ultimate prize, his theory of gravity. In the moment, so the story goes, he knew that matter attracts matter in proportion to the mass contained in each body divided by the square of the distance between them; that the tug is between the center of each mass; and—the ultimate prize—that the power “like that we here call gravity…extends its self thro’ the universe.”
This much is true: the tree itself was real.