Bernoulli

The life of James Bernoulli

Timeline 1655Born on 6 January in Basel, into a family of spice merchants. 1671Receives his master’s degree in philosophy. Goes on studying theology, as his father wishes. 1676Finishes theology and sets off on his travels. In Geneva he teaches mathematics to a blind girl. 1681Visits England, where he meets Robert Boyle and Robert Hooke. Writes about the great comet of 1680. 1683Back in Basel, lectures on experimental physics and studies the problem of compound interest. 1684Marries Judith Stupanus. Begins reading Leibniz’s papers on the calculus. 1687Becomes professor of mathematics at the University of Basel, a post he holds until his death. 1689Publishes the first of his dissertations on infinite series, with the harmonic series. 1690Solves the isochrone problem and uses the word integral for the first time. 1694Describes the lemniscate. 1697Breaks publicly with his brother Johann over the brachistochrone. 1705Dies on 16 August in Basel, leaving the Ars conjectandi unfinished. 1713The Ars conjectandi is published by his nephew Nicolaus. ◀ Previous Follow the journey Next ▶ 1655–1676 BaselHe is born into a family of spice merchants. At his father’s wish he studies theology; on his own, mathematics. 1676 GenevaHe tutors Elisabeth von Waldkirch, a blind girl, teaching her to read, write and a little mathematics. 1677–1680 FranceHe studies the philosophy of Descartes and Malebranche. 1681 HollandHe meets the scientists of the new experimental philosophy. 1681 EnglandHe meets Robert Boyle and Robert Hooke and writes about the great comet of 1680. 1683–1705 BaselHe returns home, teaches himself Leibniz’s calculus and is a professor at the university until his death in 1705. Enable JavaScript to see the map. The stages are listed below. Jakob Bernoulli’s years of travel, from Basel to England and back. Map: © OpenStreetMap contributors

October 14, 2026 Read

Interest that never ends and the number e

Imagine a generous bank that pays 100% interest a year. You deposit 1 dollar in January and, in December, you have 2 dollars. Now the bank offers something else: instead of 100% at the end of the year, it pays 50% every six months, and the interest from the first half also earns interest in the second. Is it worth it? And if it paid every month, every day, every second? Could you become infinitely rich?

October 14, 2026 Read

The harmonic series and the call for help

Even as the infinite series is contained by the finite sum, and in the unbounded a bound is discerned, so the soul of immensity dwells in the minute, and in the narrowest limits there are no limits. — Jakob Bernoulli, Tractatus de seriebus infinitis (1689) Between 1689 and 1704, Jakob Bernoulli published in Basel five dissertations on infinite series, sums with infinitely many terms. It was a new and dangerous subject. Some of these sums give a number, like

October 14, 2026 Read

The marvellous spiral and the lemniscate

With Leibniz’s calculus in hand, the Bernoulli brothers went hunting for curves. For each one they asked: what is the tangent, what is the length, what is the area, what is the curvature? Many of these curves had been known since the Greeks, but only now could these questions be answered by a general rule. Two of them kept Jakob’s name forever. The spiral that does not change shape The logarithmic spiral had already been studied by Descartes and by Torricelli. In polar coordinates, where each point is given by its distance $r$ from the centre and by the angle $\theta$, it is

October 14, 2026 Read

The art of conjecturing: counting the chances

To conjecture about something is to measure its probability. Therefore we define the art of conjecturing, or stochastics, as the art of measuring the probabilities of things as exactly as possible, so that in our judgements and actions we may always choose or follow what seems to us better, more satisfactory, safer or more carefully considered. — Jakob Bernoulli, Ars conjectandi (1713), part IV In 1654, Pascal and Fermat exchanged letters on how to divide the stakes of an interrupted game. In 1657, the Dutchman Christiaan Huygens published a fifteen-page booklet on the calculations of dice games. These were parlour problems: dice, cards, bets.

October 14, 2026 Read

The golden theorem: the law of large numbers

Even the most foolish of men, by some instinct of nature, by himself and without any instruction, is convinced that the more observations have been made, the less danger there is of missing the mark. — Jakob Bernoulli, Ars conjectandi (1713), part IV With a die, we know the chance of each face in advance: $\frac16$. But what is the chance that a man of 40 will live another twenty years? That it will rain tomorrow? That a medicine will work? In these cases there are no equal faces to count. Jakob proposed another route: observe many times and use the frequency as an estimate of the probability. If, of many men aged 40, half reached 60, the chance must be close to one half.

October 14, 2026 Read

The Bernoulli numbers: adding a thousand powers in seven minutes

With the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first thousand numbers, added together, give 91,409,924,241,424,243,424,241,924,242,500. — Jakob Bernoulli, Ars conjectandi (1713), part II The story goes that young Gauss added the numbers from 1 to 100 in seconds, pairing the first with the last: $1+2+\cdots+n=\frac{n(n+1)}2$. For squares and cubes there are formulas too, known since Archimedes and the Arab mathematicians:

October 14, 2026 Read

The legacy of James Bernoulli

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.

October 14, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Jakob Bernoulli, painted in 1687 by his brother Niklaus Bernoulli: source page · public domain. Title page of the Ars conjectandi, Basel, 1713: source page · public domain. Jakob Bernoulli’s tombstone in the cloister of Basel Minster, photo by Wladyslaw Sojka: source page · Free Art License. Cross-section of a nautilus shell, photo by Florian Elias Rieser: source page · CC BY-SA 4.0. Blog figures The cover, the drawing of the two spirals and the graph of the proportion of heads in simulated coin tosses are original illustrations, with versions in each of the site’s languages. The table of notations for the integral, in the post on the legacy, was redone from a table in this site’s old calculus course.

October 14, 2026 Read