The calculus war

Leibniz published calculus in 1684 and 1686. Newton had reached the same ideas in 1665–1666 but kept everything in manuscripts. For fifteen years nobody disputed anything.

In 1699 the Swiss mathematician Nicolas Fatio de Duillier, a friend of Newton, hinted in a book that Leibniz had drawn on Newton. In 1708 the Scot John Keill went further and accused Leibniz of plagiarism. The argument: in 1676 Leibniz had visited London and seen a copy of Newton’s De analysi.

Leibniz, a Fellow of the Royal Society, asked the Society for a fair inquiry. In 1712 a committee concluded that Newton was the first inventor. Leibniz was never asked to present his case, and the report had been written, in secret, by Newton himself, the Society’s President.

Leibniz replied with an anonymous pamphlet and tried to turn the dispute into a mathematical challenge. He died in 1716 with the matter unresolved. It is said that Newton, on hearing of his death, remarked that he had “broken Leibniz’s heart”.

Today historians’ conclusion is clear: the two invented calculus independently. Leibniz’s manuscripts show step by step how he reached his ideas, by a path different from Newton’s. Newton got there first; Leibniz published first, and with better notation.

See the same story told from Newton’s side in Newton’s legacy.

The notation that won

The quarrel had an unexpected winner: notation. Mathematicians on the Continent, such as the Bernoullis, Euler and Lagrange, used Leibniz’s $dx$ and $\int$ and carried calculus very far in the 18th century. In England, out of loyalty to Newton, people kept the dots, and English mathematics fell behind. In 1812 Cambridge students founded the Analytical Society precisely to adopt Leibniz’s notation, in a campaign they jokingly described as promoting “the principles of pure D-ism” against the “Dot-age” of the university.

Leibniz gave mathematics:

  • the symbols $dx$, $\dfrac{dy}{dx}$ and $\int$;
  • the words function and coordinates, in their mathematical sense;
  • the dot “$\cdot$” for multiplication and the colon “:” for division;
  • the sign of similarity $\sim$ and of congruence $\cong$, in versions close to today’s.

Beyond mathematics

Leibniz was also one of the great philosophers in history, author of the Monadology and the Theodicy. In physics he argued with Descartes’s followers about how to measure the “force” of a moving body, and his vis viva ($mv^2$) is an ancestor of the idea of kinetic energy. In computing, the binary system and the dream of calculating with reasoning anticipated ideas that would only be realized in the 20th century.

Norbert Wiener, the founder of cybernetics, called Leibniz the field’s “patron saint”. In Germany, the leading prize for scientific research is called the Gottfried Wilhelm Leibniz Prize.

Key points

  • Leibniz lived from 1646 to 1716 and was a jurist, diplomat, librarian and philosopher as well as a mathematician.
  • He learned mathematics seriously in Paris, with Huygens, as an adult.
  • He created the symbols $d$ and $\int$ in 1675 and published calculus in 1684 and 1686.
  • The chain rule, $\dfrac{dz}{dx}=\dfrac{dz}{dy}\cdot\dfrac{dy}{dx}$, looks like cancelling thanks to the notation.
  • The derivative of a product is $u\,dv+v\,du$, not $du\,dv$.
  • $\int$ and $d$ are inverse operations: adding up differences gives back the total.
  • He organized binary arithmetic and published it in 1703.
  • He built a machine that multiplied using the stepped drum.
  • He found $\dfrac\pi4=1-\dfrac13+\dfrac15-\cdots$, already known to Madhava and Gregory.
  • Newton and Leibniz invented calculus independently.

Sources: life and work, MacTutor · Leibniz collection at the Gottfried Wilhelm Leibniz Bibliothek.

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