From Cosmos magazine
https://cosmosmagazine.com/mathematics/seduced-calculus
In 1665, Isaac Newton, recently graduated from Cambridge, returned to live with his mother in their Lincolnshire farmhouse. The Great Plague was devastating towns across the country. The university had closed down to protect its staff and students. Newton made himself a small study and started to fill a giant jotter he called the Waste Book with mathematical thoughts. Over the next two years the solitary scribbler, undistracted, devised new theorems that became the foundations of the Philosophiæ Naturalis Principia Mathematica, his 1687 treatise that, more than any work before or since, transformed our understanding of the physical universe. The Principia established a system of natural laws that explained why objects, from apples falling off trees to planets orbiting the Sun, move as they do. Yet Newton’s breakthrough in physics required an equally fundamental breakthrough in maths. He formalized the previous half-century’s work on infinity and infinitesimals into a general system with a unified notation. He called it the method of fluxions, but it became better known as the “calculus of infinitesimals’, and now, simply, “calculus’. A body that moves changes its position, and its speed is the change in position over time. If a body is travelling with a fixed speed, it changes its position by a fixed amount every fixed period. A car with constant speed that covers 60 miles between 4pm and 5pm is travelling at 60 miles per hour. Newton wanted to solve a different problem: how does one calculate the speed of a body that is not travelling at a constant speed? For example, let’s say the car above, rather than travelling consistently at 60mph, is continually slowing down and speeding up because of traffic. One strategy to calculate its speed at, say, 4.30pm, is to consider how far it travels between 4.30pm and 4.31pm, which will give us a distance per minute. (We just need to multiply the distance by 60 to get the value in mph.) But this figure is just the average speed for that minute, not the instantaneous speed at 4.30pm. We could aim for a shorter interval – say, the distance travelled between 4.30pm and 1 second later, which would give us a distance per second. (We’d then multiply by 3,600 to get the value in mph). But again this value is the average for that second. We could aim for smaller and smaller intervals, but we are never going to get the instantaneous speed until the interval is tinier than any other – when it is zero, in other words. But when the interval is zero, the car does not move at all! This line of reasoning should sound familiar, because I used it two paragraphs ago when explaining how to calculate the gradient of a tangent. To find the gradient we divide an infinitesimally small quantity (length) by another infinitesimally small quantity (another length). To get the instantaneous speed we also divide an infinitesimally small quantity (distance) by another infinitesimally small quantity (time). The problems are mathematically equivalent. Newton’s method of fluxions was a method to calculate gradients, which enabled him to calculate instantaneous speeds. Calculus allowed Newton to take an equation that determined the position of an object, and from it devise a secondary equation about the object’s instantaneous speed. It also allowed him to take an equation determining the object’s instantaneous speed, and from it devise a secondary equation about position, which, as it turned out, was equivalent to the calculation of areas using infinitesimals! Calculus, therefore, gave him the mathematical tools to develop his laws of motion. In his equations, he called the variables x and y “fluents” and the gradients “fluxions’, written by the “pricked letters” ẋ and ẏ.
