The study of curves was a central fascination of seventeenth-century mathematics, and Newton had plunged into the field when he read a translation of René Descartes’s Geometrya year before the plague hit. Descartes’s approach helped link together the approach of classical geometry, which explored shapes and their properties, and the ideas of algebra, with equations whose solutions could be mapped onto a particular curve.^*

One of the key advances Newton encountered in his copy of Geometry—the Latin translation that spread Descartes’s ideas throughout learned Europe—was the coordinate system now known as Cartesian coordinates. It provided a way to map any point in two dimensions with just two numbers, corresponding to its horizontal and vertical positions. Using two lines perpendicular to each other—the familiar cross of every graph in grade school math classes—and a standard unit length applied to both axes, Descartes created a systematic way to measure and map any shape a geometer wanted to study—including the classical curves, circles, ellipses, and the rest.

When Newton came to study this work, and then more contemporary mathematics, he soon turned one of its approaches on its head. In classical geometry, the starting point for most European mathematicians, the curve or the shape is the object of inquiry. Even though Cartesian coordinates offered a new and powerful way of representing equations as shapes on his coordinate system, many of Newton’s contemporaries saw such equations as a property of a given figure, a line or a circle or some more complicated form. But it took Newton just a few months after encountering Descartes to realize, as his biographer, Richard Westfall reports, “The equation is more basic than the curve; the equation defines, or as Newton put it, expresses the nature of the curve.”

That sounds like a technical point, or even one of taste: some folks think in pictures and, if they are mathematical, dive into the relationships between shapes and volumes, while others play the game of manipulating those abstractions. But Newton’s insight—starting first with the equation, rather than the shape—was foundational because it would, first hesitantly and then through centuries of development, yield a new way of seeing the world through mathematics. For his predecessors, the classical geometers, the curve was there, complete, a synoptic view of the object. But in Newton’s work, in the early years of what is now known as analytic geometry, a curve is built up as a calculation reveals the solutions to the equations that generate any given geometrical object. The accumulation of specific answers to these calculations—points on a curve, plotted on the page to produce a geometric object—can be interpreted in various ways. The interpretation that Newton would develop focused on arguably the most important implication: equations describe the evolution of a system—how its solutions build a picture on a page. That picture is a map of the relationship of variables—things that can change. If one of those variables is the passage of one moment into the next, then the abstract play of symbols and shapes becomes a portrait of change in action.

Ultimately, this mathematical insight is at the heart of modern physics, the science that Newton, more than any other single thinker, would create. In its simplest form, the idea is this: the full picture, the complete geometrical representation of all the available solutions to a system of equations, can be understood as all the possible outcomes for a given phenomenon described by that mathematics. Each specific calculation, fed with observations of the current state of whatever you’re interested in, the flight of a cannonball, the motion of a planet, how a curveball swerves, how rapidly an outbreak of the plague might spread, makes a prediction for what will happen next. In his twenties, working on his own, with almost no systematic experience of the study of the real world, Newton did not yet grasp the full power of the ideas implied by the way he had begun to think about math. That would come in time. But what made his annus mirabilisso miraculous was the speed and depth with which Newton forged the foundations of his ultimately revolutionary way of comprehending the world.

THE NEXT STEP in that revolutionary path came as Newton worked on new ways to analyze and solve mathematical problems. Only the simplest algebraic equations can be solved just by plugging in numbers and doing the arithmetic. Seventeenth-century attempts to analyze more complicated expressions often employed a particular mathematical tool, the infinite series—endless sequences of terms (for example, 1, ½, ¼, ⅛…and so on).

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