James Bernoulli lived only fifty years and published little in his lifetime, almost all of it in short papers in the Acta Eruditorum. Even so, few mathematicians left so many words, symbols and ideas that we still use every day: integral, Bernoulli trial, binomial distribution, Bernoulli numbers, law of large numbers, stochastic.

The word integral

Leibniz called the operation inverse to the derivative calculus summatorius, the calculus of sums, and wrote it with $\int$, an elongated S for summa. In May 1690, solving the isochrone problem, Jakob wrote that the two expressions had “equal integrals”. It was the first time the word integral appeared in print in a mathematical text. Johann adopted the term, and in 1696 Leibniz agreed to swap his “summatory calculus” for the Bernoullis’ integral calculus. The symbol stayed Leibniz’s; the name, Jakob’s.

Year Author Notation or name Where
1659 Pietro Mengoli omnia $x$ Geometriae speciosae elementa
1675 Leibniz $\int x$ manuscript of 29 October 1675
1690 Jakob Bernoulli “integral” Acta Eruditorum
1696 Leibniz and Johann Bernoulli calculus integralis letters
1822 Fourier $\int_a^b f(x)\,dx$ Théorie analytique de la chaleur

The hanging chain

In the same 1690 paper, Jakob asked what shape a chain takes when hung by both ends. Galileo had thought it was a parabola. In 1691, Johann, Leibniz and Huygens published the correct answer, the catenary, which today is written with the hyperbolic cosine, $y=a\cosh\frac xa$. Jakob lost the race to his brother, but the question was his. The story of the catenary is in Johann’s track.

De Moivre and the normal curve

The golden theorem said that frequency approaches probability, but Jakob’s estimate of the number of observations was greatly exaggerated. In 1733 the Frenchman Abraham de Moivre, a refugee in London, looked for a better approximation to the terms of the binomial when $n$ is large and found the bell-shaped curve

$$y=\frac1{\sqrt{2\pi}}\,e^{-x^2/2}.$$

It was the first appearance of the normal curve, which Laplace and Gauss would place at the centre of statistics. With it, Jakob’s urn needs about 6,500 draws, not the 25,550 he calculated.

The challenge won

Jakob died without knowing the value of $1+\frac14+\frac19+\frac1{16}+\cdots$. Thirty years later, Leonhard Euler, a pupil of Johann’s, showed that the sum is $\frac{\pi^2}6$. Later still, Euler discovered something even more surprising: the numbers Jakob had found for the sums of powers reappear there. The sum of the reciprocals of the even powers is always $\pi$ raised to that power times a fraction made from a Bernoulli number. For example,

$$\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6,\qquad\sum_{n=1}^\infty\frac1{n^4}=\frac{\pi^4}{90}.$$

Jakob’s two problems, the one on series and the one on powers, were deep down the same problem.

What remained

  • Probability. The Ars conjectandi turned calculations about games into a theory of certainty. The binomial distribution and the Bernoulli trial are in every statistics course.
  • Statistics. The golden theorem is the reason opinion polls, clinical trials and quality control work: many independent observations reveal the hidden probability.
  • Series. The divergence of the harmonic series and the question about the squares opened Euler’s path.
  • Analysis. The Bernoulli numbers appear in the series for $\tan x$, in the Euler–Maclaurin formula and in the Riemann zeta function.
  • Calculus. The word integral, the isochrone, the catenary and the isoperimetric problem, which prepared the calculus of variations of Euler and Lagrange.
  • Curves. The logarithmic spiral and the lemniscate, with the integral that opened the theory of elliptic functions.

To remember

  • James (Jakob) Bernoulli lived from 1655 to 1705, in Basel. He studied theology out of obedience and mathematics out of love, and learned Leibniz’s calculus by himself.
  • Compound interest leads to the number $e=\lim\left(1+\frac1n\right)^n\approx2.718$, which lies between 2 and 3.
  • The harmonic series $1+\frac12+\frac13+\cdots$ passes any number; the sum of the reciprocals of the squares stays below 2.
  • The logarithmic spiral $r=ae^{b\theta}$ grows without changing shape and cuts every radius at the same angle.
  • The chance of $k$ successes in $n$ trials is $\binom nk p^kq^{n-k}$.
  • The more observations, the closer the frequency comes to the probability: that is the law of large numbers.
  • The sums $1^m+2^m+\cdots+n^m$ are polynomials in $n$ whose coefficients come from the Bernoulli numbers.

Sources: biography of Jacob Bernoulli, MacTutor · Florian Cajori, A History of Mathematical Notations, vol. II (1929), §§ 541–544 · Stephen M. Stigler, The History of Statistics (Harvard, 1986).

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