Newton-Leibniz Calculus Controversy: Who Invented Calculus?

The Newton–Leibniz calculus controversy was a bitter dispute over who invented calculus, the branch of mathematics that deals with rates of change and areas under curves. Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany each developed calculus independently in the 1660s and 1670s, but from 1699 their supporters accused each other of plagiarism, and the quarrel ran until Leibniz’s death in 1716. Today historians credit both men, but the feud shaped mathematics in Britain and Europe for a century.

Two independent inventions

Newton developed what he called the “method of fluxions” during his annus mirabilis of 1665–1666, when plague closed Cambridge University. He treated quantities as flowing in time and worked out their rates of change, called fluxions. He wrote up his results in an October 1666 tract, in De analysi (1669), which circulated in manuscript among a few mathematicians, and in a longer treatise of 1671. None of these was printed until much later.

Leibniz, a German lawyer, diplomat and philosopher, lived in Paris from 1672 to 1676. Guided by Christiaan Huygens, he studied the latest mathematics and began working on tangents, areas and infinite series. In the autumn of 1675 he wrote notes introducing the symbol ∫ for summing and d for differences, the notation still used today.

Leibniz published first. His paper “Nova Methodus pro Maximis et Minimis”, a new method for maxima and minima, appeared in the Leipzig journal Acta Eruditorum in October 1684, followed by a paper on integration in 1686. Newton’s methods first appeared in print in his Principia of 1687, in geometric form, and more directly only from 1704.

The letters of 1676

The seeds of the quarrel lay in contacts between the two men. Leibniz visited London in 1673 and again briefly in October 1676. On the second visit he was shown papers held by the mathematician John Collins, including a copy of Newton’s De analysi, and took notes on its sections dealing with infinite series.

In 1676 Newton also wrote two letters to Leibniz through Henry Oldenburg, secretary of the Royal Society. These are known as the Epistola prior (June 1676) and the Epistola posterior (October 1676). Newton described his results on series but hid the core of his fluxion method in an anagram, a scrambled Latin sentence that only he could decode. Leibniz replied in 1677, openly explaining his own differential method.

Modern scholars who have studied Leibniz’s notes conclude that his calculus was already well developed before he saw Newton’s manuscript, and that his ideas came from a different route, through sums and differences of sequences.

How the quarrel began

For about fifteen years there was no open dispute. In the first edition of the Principia (1687), Newton included a passage acknowledging that Leibniz had a similar method. Leibniz’s calculus spread rapidly on the Continent, especially through the Swiss brothers Jakob and Johann Bernoulli and the Marquis de l’Hôpital, whose 1696 textbook was the first on differential calculus.

The first public accusation came in 1699 from Nicolas Fatio de Duillier, a Swiss mathematician close to Newton, who suggested that Leibniz had borrowed from Newton. A more serious charge followed in a paper by the Scottish mathematician John Keill, written in 1708 and published in the Royal Society’s Philosophical Transactions in 1710, which openly accused Leibniz of plagiarism.

Leibniz, a Fellow of the Royal Society, appealed to the society in 1711 for a retraction. Newton had been its President since 1703.

The Royal Society’s verdict

In March 1712 the society appointed a committee to examine the matter. Within about seven weeks it produced a report concluding that Newton was the “first inventor”. The report, with a selection of old letters and papers, was printed in early 1713 as the Commercium Epistolicum.

The committee never asked Leibniz for his side of the story, and Newton himself drafted much of the report. In 1715 an anonymous summary of it appeared in the Philosophical Transactions; it too was written by Newton.

Leibniz responded with an anonymous broadsheet in 1713, the Charta Volans, which cited an unnamed “leading mathematician”, in fact Johann Bernoulli, suggesting that Newton had not understood higher derivatives and might have borrowed from Leibniz. The argument widened into philosophy and theology, including an exchange of letters in 1715–1716 between Leibniz and Newton’s ally Samuel Clarke about space, time and God.

Leibniz’s death and the aftermath

Leibniz died in Hanover on 14 November 1716, isolated and out of favour with his employer, the Elector of Hanover, who had become King George I of Great Britain in 1714. Newton continued to defend his priority. In the third edition of the Principia (1726) he removed the passage that had acknowledged Leibniz.

British mathematicians, loyal to Newton, kept his dot notation and geometric methods. Meanwhile Continental mathematicians such as Leonhard Euler, the Bernoullis and Joseph-Louis Lagrange used Leibniz’s more flexible notation and pushed analysis far ahead. Many historians judge that British mathematics fell behind for much of the 18th century. Reform came in 1812, when Cambridge students including Charles Babbage, John Herschel and George Peacock founded the Analytical Society to promote Continental methods.

Two styles of calculus

Beyond priority, the two men thought about calculus in different ways. Newton’s approach was rooted in motion and geometry. He pictured a curve as the path of a moving point and spoke of “fluents”, quantities that change, and “fluxions”, their rates of change. He marked fluxions with a dot over a letter and often presented his results in the language of classical geometry, which he considered more rigorous.

Leibniz started from the idea of tiny differences and sums. He thought of a curve as made of infinitely many infinitely small segments and wrote dx and dy for the differences in each direction. Integration was a sum of such small pieces, hence his long S. His notation made rules such as the chain rule look almost obvious on the page, and he took care to choose symbols that would help others calculate.

Both approaches raised philosophical problems. Critics, most famously the Irish bishop George Berkeley in The Analyst (1734), mocked the idea of quantities that were neither zero nor something, calling them “ghosts of departed quantities”. A fully rigorous foundation based on limits came only in the 19th century, through the work of Augustin-Louis Cauchy, Karl Weierstrass and others.

Myths and facts

• Myth: one of them stole calculus from the other. The weight of evidence shows independent invention, by different routes and with different notations.
• Fact: Newton was first, Leibniz published first. Newton’s methods date from 1665–1666; Leibniz’s from 1675, published in 1684.
• Fact: neither started from nothing. Both built on earlier work by Fermat, Descartes, Wallis, Barrow, Pascal and others on tangents, areas and infinitesimals.

Timeline

• 1665–1666: Newton develops his method of fluxions.
• 1669: Newton’s De analysi circulates in manuscript.
• 1675, autumn: Leibniz introduces his notation for calculus in Paris.
• 1676: Newton’s Epistola prior and Epistola posterior; Leibniz visits London in October.
• 1684, October: Leibniz publishes “Nova Methodus” in Acta Eruditorum.
• 1687: Newton’s Principia published.
• 1699: Fatio de Duillier accuses Leibniz of borrowing from Newton.
• 1710: Keill’s charge of plagiarism published.
• 1712–1713: Royal Society committee and the Commercium Epistolicum.
• 1716, 14 November: Leibniz dies in Hanover.

Why it matters

The calculus controversy is a classic case of simultaneous discovery and of how pride, nationalism and institutional power can distort scientific debate. Its outcome left the world two founders of calculus, with Leibniz’s notation dominating mathematics and Newton’s ideas dominating physics through the Principia.