The Calculus Wars

In the history of mathematics, there is no dispute more bitter, more petty, or more fascinating than the priority dispute between Isaac Newton and Gottfried Wilhelm Leibniz over the invention of calculus.

Both men developed calculus independently. Both made fundamental contributions. The argument over who did it first consumed decades, enlisted national pride, derailed careers, and left a shadow over European mathematics for a generation. And the answer, insofar as there is one, is: Newton first, Leibniz better.

Newton: the first but secretive

Isaac Newton developed his method of "fluxions" — what we now call calculus — in the years 1665–1666, while Cambridge was closed due to the plague and he had retreated to his family farm in Woolsthorpe. In eighteen months of extraordinary isolation, he invented calculus, the theory of gravity, and optics.

Newton was notoriously reluctant to publish. His method of fluxions circulated in manuscript form among a small circle of correspondents for years before he published it officially. The Principia Mathematica (1687), which described the laws of planetary motion and gravity, was written using geometric arguments rather than calculus — partly to make it accessible, partly out of caution.

Newton's notation: $\dot{x}$ for the derivative of $x$ with respect to time. This "dot notation" survives today in physics for time derivatives.

Leibniz: independent discovery, better notation

Gottfried Leibniz came to calculus independently in the mid-1670s, about a decade after Newton. He published first: Nova Methodus appeared in 1684, three years before Newton's Principia.

Leibniz's contributions were different in character from Newton's. Where Newton was geometrically motivated (fluxions as rates of flow), Leibniz was algebraically and symbolically oriented. His notation — $\frac{dy}{dx}$ for derivatives, $\int$ for integrals — was designed to be manipulable as symbols, and it was far superior to Newton's.

The notation $\frac{dy}{dx}$ makes the chain rule look like fraction cancellation ($\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$) and the fundamental theorem look natural. Leibniz's $dx$ notation survived because it captures how infinitesimals actually behave in calculations. Nearly all the notation in modern calculus textbooks is Leibniz's, not Newton's.

The dispute begins

For a decade or so after Leibniz's publication, mathematicians on the continent used Leibniz's methods and notation; Newton's English circle used Newton's. There was no overt conflict.

Then in 1699, a Swiss mathematician named Nicolas Fatio de Duillier — a former associate of Newton's — accused Leibniz of plagiarism. He claimed Leibniz had seen Newton's unpublished manuscripts during a visit to London in 1676 and appropriated the ideas.

Leibniz responded indignantly. The debate escalated. In 1712, the Royal Society (of which Newton was president) appointed a commission to adjudicate the priority dispute — and unsurprisingly, concluded in Newton's favour. Newton himself wrote most of the commission's report, anonymously.

The damage

The calculus wars had real consequences beyond the personal. British mathematicians, rallying around Newton's notation and methods, cut themselves off from the mathematical developments happening on the continent. For most of the 18th century, European mathematicians — the Bernoulli family, Euler, Lagrange, Laplace — extended calculus enormously using Leibniz's notation. British mathematics stagnated in comparison.

When the Cambridge Analytical Society finally pushed for the adoption of Leibniz's $d$ notation in Britain around 1815, they described their goal as the promotion of "the principles of pure d-ism" over "the dot-age of the university" — a pun on Newton's dots and the word "dotage."

The verdict of history

Most historians of mathematics now believe both Newton and Leibniz developed calculus independently. Leibniz did correspond with Newton before his own discovery, but the letters he received contained nothing that should have enabled the invention of calculus, and his surviving notebooks show an independent development.

Newton was first chronologically. Leibniz was first in publication. And Leibniz's notation so thoroughly won the battle of presentation that we essentially use his language today, not Newton's.

The real lesson of the calculus wars is not about priority — it is about how mathematical ideas develop. Two extraordinarily gifted people, working on the same problems (areas, tangents, rates of change) at roughly the same time, arrived at the same fundamental insights independently. This suggests that calculus was, in a sense, waiting to be discovered. The mathematical needs of the 17th century — celestial mechanics, optics, the theory of curves — demanded it.

It also suggests that the credit for mathematical discoveries may be somewhat ill-posed. The ideas themselves are what endure.