Isaac Newton created calculus, which he called the “method of fluxions,” and developed it alongside several related breakthroughs in infinite series, numerical methods, and curve classification. Most of this work began in 1665-66, when Newton was in his early twenties, though he didn’t publish much of it for decades. Here’s what he actually built and why it mattered.
The Method of Fluxions: Newton’s Calculus
Newton’s central mathematical invention was calculus, his framework for analyzing quantities that change continuously over time. He called changing quantities “fluents” (meaning smoothly flowing measurable things) and their rates of change “fluxions.” In modern terms, a fluent is a function and a fluxion is its derivative. Newton represented fluxions by placing a dot over the variable, so the rate of change of x was written ẋ. Physicists still use this dot notation today, though the fraction-style notation dx/dt (introduced later by Leibniz) became more common in most other fields.
Newton framed his entire system around two fundamental problems. The first: given a quantity, find how fast it’s changing at any moment. The second: given the rate of change, recover the original quantity. These are what we now call differentiation and integration. By the middle of 1665, Newton had worked out the standard rules for differentiation with the same generality that Leibniz would independently publish two decades later.
Newton used infinitesimals freely in his calculations. He treated a tiny increase in time (which he wrote as a small letter “o”) as the building block for understanding instantaneous change. Multiplying ẋ by this tiny time increment gave dx, the infinitesimal change in x. This let him compute slopes of curves, areas under curves, and the motion of objects with a single unified method.
The Generalized Binomial Theorem
Before Newton, mathematicians could expand expressions like (1 + x) raised to a whole-number power into a finite sum of terms. Newton extended this to any real-number exponent, including fractions and negatives. This was a major leap because it turned a finite formula into an infinite series, opening up entirely new ways to represent and compute with functions.
For example, raising (1 + x) to the power of negative one produces an endless alternating series: 1 − x + x² − x³ + x⁴ and so on. Raising (1 − x²) to the power of negative one-half produces another infinite series. These aren’t just mathematical curiosities. They gave Newton a practical tool for converting complicated expressions into simpler pieces he could work with term by term.
Infinite Series for Key Functions
The generalized binomial theorem was really a starting point. Newton’s deeper achievement was using it to derive infinite series representations for functions that had previously resisted calculation. His method was elegant: expand a function into an infinite series using the binomial theorem, then integrate each term individually (since integrating a power of x is straightforward).
This approach led him to the series for the natural logarithm. By integrating the series for 1/(1 + x) term by term, he found that log(1 + x) equals x − x²/2 + x³/3 − x⁴/4 and so on. By integrating the series for 1/√(1 − x²), he derived the series for the inverse sine function. He then reversed these series to obtain formulas for the exponential function and the sine function itself. The sine series, for instance, gives sin(y) as y − y³/6 + y⁵/120 − y⁷/5040, continuing with alternating terms. These series remain standard tools in mathematics and engineering.
What’s striking is that the only calculus Newton needed for all of this was the ability to differentiate and integrate powers of x. The real work was algebraic manipulation of infinite series, a skill Newton had in extraordinary abundance.
A Method for Solving Equations Numerically
Newton also developed an iterative technique for finding solutions to equations that can’t be solved with a simple formula. Now called the Newton-Raphson method, it starts with a rough guess at the answer, then uses the function and its derivative to refine that guess repeatedly. Each iteration roughly doubles the number of correct digits, a property called quadratic convergence.
This method is still one of the most widely used algorithms in scientific computing. It shows up in financial modeling (calculating options pricing), in engineering simulations, and even inside computer chips that need to compute square roots quickly. Any time software needs to solve an equation that has no neat closed-form answer, there’s a good chance some version of Newton’s method is running behind the scenes.
Classification of Cubic Curves
Newton made a less famous but still significant contribution to geometry by systematically classifying cubic curves, equations where the highest power of the variables is three. He organized them into four main classes based on the form of their equations, then subdivided further based on the nature of their roots. For one class alone, he identified five distinct species depending on whether the roots of the underlying cubic expression are all different real numbers, have two equal roots, have all three equal, or include complex roots. Each species produces a differently shaped curve: some have separate oval-shaped loops, some pinch into pointed nodes, some form smooth bell-like shapes. This classification, published in 1704, was one of the earliest systematic studies of algebraic curves.
The Publication Gap
One of the most notable things about Newton’s mathematics is how long it stayed unpublished. He developed the core of his calculus in 1665-66 but didn’t publish his major mathematical treatises until 1704, nearly four decades later. In the meantime, Gottfried Wilhelm Leibniz independently developed his own version of calculus in the 1680s and published it first. This led to one of the bitterest priority disputes in the history of science. Newton claimed he had communicated key ideas to Leibniz through letters in 1676, and the controversy consumed both men for years. Modern historians generally credit both with independent discovery, though Newton’s version came first chronologically while Leibniz’s notation proved more practical for most purposes.
Where Newton’s Math Lives Today
Calculus is now so thoroughly embedded in science and engineering that it’s almost invisible. In aerospace engineering, it underlies trajectory prediction, orbital mechanics, thrust optimization, and stability control of aircraft and spacecraft. In robotics, Newtonian models guide force control, motion planning, and autonomous navigation. Mechanical engineers use it for vibration analysis, machine design, and impact modeling. Structural engineers rely on it to analyze how buildings and bridges respond to loads.
Beyond these direct applications, Newton’s infinite series remain essential in computational physics, where complex functions are routinely approximated by their series expansions. His numerical method for solving equations is a core algorithm in everything from weather modeling to computer graphics. The mathematics Newton created in his twenties, working largely alone during a plague year, became the language in which most of modern engineering and physical science is written.