Johann Bernoulli did not write a great book, as his brother wrote the Ars conjectandi or Newton the Principia. His most famous book came out under someone else’s name. What he left was a way of working: turning every problem of physics or geometry into an equation and solving it with Leibniz’s calculus. And he left pupils.

Leibniz’s champion

In the dispute over who had invented the calculus, Newton or Leibniz, Johann was the most combative soldier on Leibniz’s side. When the Royal Society, presided over by Newton, published at the beginning of 1713 a report giving priority to Newton, Leibniz replied with an anonymous broadsheet, the Charta volans, which quoted the judgement of “an eminent mathematician”: it was Johann. In the years that followed, Johann challenged the English mathematicians with hard problems, which he thought impossible with Newton’s method. Leibniz died in 1716, and Johann carried on the fight alone for years. The dispute cut English mathematics off from that of the Continent for almost a century.

The word function

The word function, in the mathematical sense, was born in an exchange of letters between Leibniz and Johann in the 1690s. In 1718 Johann published the first definition:

I call a function of a variable magnitude a quantity composed in any way whatever of this variable magnitude and of constants.

— Johann Bernoulli, Mémoires de l'Académie des Sciences (1718)

He wrote the function of $x$ with the Greek letter $\varphi$, sometimes as $\varphi x$, without brackets. It was his pupil Euler who, in 1734, began to write $f(x)$. This site’s old course on functions has the chronology of the notations:

Year Author Notation
1694 Johann Bernoulli a letter, such as $n$, for a quantity depending on $x$
1698 Johann Bernoulli $X$ or $\xi$ for a function of $x$
1718 Johann Bernoulli $\varphi x$
1734 Leonhard Euler $f\left(\frac xa+c\right)$

A school

Johann taught more important mathematicians than any other teacher of his century. Among them:

  • The Marquis de l’Hôpital, who turned his lessons into the first calculus book.
  • His sons Nicolaus II, Daniel and Johann II. Daniel founded hydrodynamics, with the principle that bears the Bernoulli name; Johann II inherited his father’s chair.
  • Leonhard Euler, the greatest mathematician of the century, whom he guided on Saturdays.
  • Pierre-Louis Moreau de Maupertuis, who proposed the principle of least action, and Alexis Clairaut, who studied the shape of the Earth.

Through Euler, the line continues to Lagrange, who succeeded Euler in Berlin, and from there to much of the mathematics of the nineteenth and twentieth centuries.

What remained

  • Calculus. L’Hôpital’s rule, logarithmic differentiation, the exponential calculus and the definition of a function.
  • Curves. The catenary and the brachistochrone, with the analogy between motion and light.
  • Calculus of variations. The challenge of the brachistochrone and the quarrel with Jakob over isoperimetric problems opened the way for Euler and Lagrange.
  • Mechanics. The principle of virtual work, the basis of statics and of the Analytical Mechanics.
  • Teaching. A generation of mathematicians, with Euler at its head.

To remember

  • Johann Bernoulli lived from 1667 to 1748: Basel, Paris, Groningen and Basel again, where he succeeded his brother Jakob.
  • The catenary, the curve of a hanging chain, is $y=a\cosh\frac xa$. Inverted, it gives the ideal shape of an arch.
  • If $f(a)=g(a)=0$, the limit of $\frac fg$ at $a$ is the limit of $\frac{f'}{g'}$: this is l’Hôpital’s rule, which came from Johann’s lessons.
  • The brachistochrone, the fastest slide, is a cycloid. Johann found it by treating the ball as a ray of light.
  • The derivative of $x^x$ is $x^x(\ln x+1)$; the minimum is at $x=\frac1e$.
  • In equilibrium, the sum of the virtual work of all the forces is zero.

Sources: biography of Johann Bernoulli, MacTutor · Florian Cajori, A History of Mathematical Notations, vol. II (1929), §§ 642–643 · A. Rupert Hall, Philosophers at War: The Quarrel between Newton and Leibniz (Cambridge, 1980).

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