Isaac Newton · Science
Newton’s invention of the calculus - the mathematics of change, motion, and the infinitely small - which gave science the tool to describe a world in continuous flux.
The mathematics Newton inherited - the algebra and geometry of the ancients and the Renaissance - was powerful, but it described a static world of fixed quantities, straight lines, and unchanging shapes. It could find the area of a circle or the length of a line, but it had no way to grasp change itself: how a quantity varies from instant to instant, how a body moving through space changes its speed and position continuously, how a curve bends differently at every point. Yet the new physics was a physics of motion and change - of bodies accelerating, planets sweeping along curved orbits, quantities flowing and varying continuously in time. To describe such a world, Newton needed a new kind of mathematics: a mathematics of change, of motion, of quantities continuously varying. This was the calculus. Newton invented it (he called it the method of fluxions) precisely because the physics he was creating demanded it - to describe a world in continuous flux, where everything is changing at every moment, the old mathematics of fixed quantities would not do. The calculus is, at its heart, the mathematics of change.
The calculus has two great operations, which Newton (and Leibniz) showed to be intimately connected. The first is differentiation - finding the rate of change of a quantity, the derivative. Given how a body’s position changes with time, differentiation finds its velocity (the rate of change of position); differentiate again, and you find its acceleration (the rate of change of velocity). The derivative answers: how fast is this quantity changing, right now? The second operation is integration - finding the accumulated total of a continuously varying quantity, which corresponds geometrically to the area under a curve. Given how a body’s velocity changes with time, integration finds the total distance travelled (the accumulation of all the tiny distances covered in each instant). The integral answers: what is the total, accumulated from all the continuously varying parts? Differentiation breaks a changing quantity down into its instantaneous rate of change; integration builds a total up from infinitely many infinitely small parts. These two operations - the derivative (rate of change) and the integral (accumulated total) - are the twin pillars of the calculus, the tools for analysing a world of continuous change.
With the calculus, Newton gained a tool of almost limitless power for describing the physical world. Armed with it, he could state his laws of motion in their full precision - for the second law, that force produces acceleration (the rate of change of velocity), is a statement in the calculus, relating a force to a rate of change. He could describe the continuously curving orbits of the planets, calculate the areas they sweep out, and derive the elliptical paths from the law of gravitation. He could find the rates of change of any varying quantity, and accumulate the totals of any continuously varying process. The calculus became the language in which the laws of physics are written. Nearly every fundamental law of nature discovered since - in mechanics, electricity and magnetism, fluid flow, heat, quantum mechanics, relativity - is expressed as a differential equation, a statement in the calculus relating quantities to their rates of change. And the calculus reaches far beyond physics: it is the essential tool of engineering, economics, statistics, biology, and every science that deals with continuous change. Newton’s invention of the calculus was thus not merely a contribution to mathematics but the forging of the central tool of modern quantitative science - the mathematics of change, without which the modern understanding of the physical world would be impossible.
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You learned that Newton invented the calculus to describe change and motion. Explain what problem the calculus solves and why it was essential to the new physics.
Leads to Gottfried Leibniz.
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