Isaac Newton published his basic theories to calculus in 1669.

 

Isaac Newton, one of the most influential scientists in history, is often credited with laying the groundwork for modern mathematics and physics. In 1669, he made one of his most important contributions to mathematics by publishing his early work on calculus. Newton’s development of calculus—referred to at the time as “the method of fluxions”—was a revolutionary advancement that allowed mathematicians to solve problems involving change and motion that had previously been unsolvable.

Before Newton, mathematics was largely focused on static problems, such as geometry and algebra, where quantities remained constant. However, the physical world is dynamic, constantly involving change. To understand nature more deeply, it was essential to develop a way to study changing quantities, such as the motion of objects, the growth of populations, or the spread of diseases. Calculus provided the tools to deal with such problems by introducing concepts like limits, derivatives, and integrals, which allow for the precise calculation of rates of change and areas under curves.

Newton’s early work on calculus was not published in a formal book in 1669, but it circulated as a manuscript titled De Analysi per Aequationes Numero Terminorum Infinitas (On Analysis by Infinite Series). This document was shared privately among a few colleagues, most notably Newton’s mentor, Isaac Barrow, who passed it on to John Collins, a mathematician who helped spread Newton’s ideas. In this manuscript, Newton laid out some of the key principles of what would later become calculus, including the notion of series expansions for functions and methods for calculating tangents to curves (a precursor to derivatives).

At the time, Newton’s work was groundbreaking but also difficult to comprehend. He approached the subject in a way that was fundamentally different from the methods of ancient Greek mathematics or Renaissance-era algebra. Newton’s introduction of “fluxions” represented quantities that were changing over time, while his “fluents” were the quantities themselves. This concept of fluxions was, in essence, the foundation for what we now call derivatives, a core component of calculus.

Interestingly, Isaac Newton’s calculus was developed independently, but not in isolation. Around the same time, the German mathematician Gottfried Wilhelm Leibniz was also formulating his own version of calculus. While Newton used his method of fluxions to solve various problems in physics, particularly in his studies of motion and planetary orbits, Leibniz approached calculus in a more formal and symbolic way. Leibniz’s notation, including the familiar ∫ symbol for integrals and d for differentials, is the one that is most commonly used today.

Despite the similarities in their discoveries, a bitter priority dispute erupted between Newton and Leibniz over who had invented calculus first. Newton claimed that his work predated Leibniz’s publications, while Leibniz argued that he had arrived at his results independently. The controversy divided the mathematical community of Europe for many years, with British mathematicians siding with Newton and much of the rest of Europe supporting Leibniz.

Nevertheless, Newton’s contributions to calculus were essential to the advancement of science. His mathematical innovations allowed him to develop his famous laws of motion and the law of universal gravitation. Calculus became the tool that could describe the changing world with mathematical precision, influencing countless fields such as physics, engineering, economics, and biology.

Though Newton’s formal publications on calculus came later, particularly in his seminal work Principia Mathematica (1687), his 1669 manuscript marked the beginning of a new era in mathematics.