Probability

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

The life of Blaise Pascal

Timeline 1623Born on 19 June in Clermont, in the Auvergne region of France. 1626His mother, Antoinette Begon, dies. Blaise is 3. 1631His father, Étienne Pascal, moves to Paris with his three children. 1635At 12, he rediscovers on his own that the angles of a triangle add up to two right angles. 1640At 16, he publishes the Essay on conics, with the hexagon theorem. 1642Starts building a calculating machine to help his father, a tax official in Rouen. 1647Publishes the New experiments concerning the vacuum. Descartes visits him. 1648Florin Périer, his brother-in-law, measures the weight of air by climbing the Puy de Dôme. 1654Exchanges letters with Fermat about games of chance and writes the Treatise on the arithmetic triangle. On 23 November he has his “night of fire”. 1656Begins publishing the Provincial letters under a pseudonym. 1658Solves the problems of the cycloid and launches a contest about them. 1662Opens a line of public coaches in Paris. Dies on 19 August, aged 39. 1670His friends publish the Pensées, put together from his notes. ◀ Previous Follow the journey Next ▶ 1623–1631 ClermontHe is born in Clermont, in Auvergne. His mother dies when he is three. 1631–1639 ParisHis father teaches him at home and takes him to Mersenne’s gatherings. At 16 he writes the Essay on conics. 1640–1647 RouenHis father is a tax collector in Normandy. To help him, Blaise builds the Pascaline and carries out his experiments on the vacuum. 1648 Puy de DômeAt Pascal’s request his brother-in-law Florin Périer climbs the mountain with a barometer and shows that the air weighs less at the top. 1647–1662 ParisThe letters to Fermat on games of chance, the night of fire, the Provincial Letters, the cycloid and the public coaches. He dies in 1662, aged 39. 1655 Port-Royal-des-ChampsAfter the night of fire he spends periods at the convent where his sister Jacqueline lives and defends Port-Royal against the Jesuits. Enable JavaScript to see the map. The stages are listed below. Pascal between Clermont, Paris and Rouen, and the mountain that weighed the air. Map: © OpenStreetMap contributors

October 11, 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

Pascal's triangle: build it by adding

Write a $1$. Below it, two $1$s. From then on, each row starts and ends with $1$, and each number in between is the sum of the two numbers just above it: $$\begin{array}{ccccccccccc} &&&&&1&&&&&\\ &&&&1&&1&&&&\\ &&&1&&2&&1&&&\\ &&1&&3&&3&&1&&\\ &1&&4&&6&&4&&1&\\ 1&&5&&10&&10&&5&&1 \end{array}$$That is all. A rule a child learns in a minute produces one of the richest tables in mathematics. Today it is called Pascal’s triangle, but Pascal was one of the last to find it.

October 11, 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

Combinations: counting without listing

In Brazil’s Mega-Sena lottery you choose 6 numbers out of 60. How many different tickets are there? Listing them all would take a lifetime: there are more than fifty million. But you can count without listing, and Pascal showed how. Choosing without order Start with a small case: in how many ways can you choose 2 people out of a group of 5 to form a committee? If order mattered, there would be $5$ options for the first person and $4$ for the second: $5\cdot4=20$. But on a committee “Ana and Bruno” is the same as “Bruno and Ana”, and every pair has been counted twice. There are $20/2=10$ committees.

October 11, 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 Chevalier de Méré's dice and the fair split

Antoine Gombaud, who styled himself the Chevalier de Méré, was a writer at the French court and a passionate gambler. Around 1654 he brought Pascal two questions about bets. Pascal found them so good that he wrote to Pierre de Fermat, and probability theory was born from those letters. About his friend, Pascal remarked to Fermat: He has a very good mind, but he is not a geometer; and that, as you know, is a great defect. — Blaise Pascal, letter to Fermat (29 July 1654)

October 11, 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 Pascaline: a sum made of gears

In 1640 Étienne Pascal was appointed tax commissioner for Normandy. The job meant adding up column after column of figures late into the night, and his son Blaise, then 17, helped. To make matters worse, French money was not decimal: $$1\ \text{livre}=20\ \text{sols},\qquad 1\ \text{sol}=12\ \text{deniers}.$$Adding $3$ livres, $15$ sols and $8$ deniers to $2$ livres, $7$ sols and $9$ deniers needs three kinds of carry: deniers: $8+9=17$, which is $1$ sol and $5$ deniers; sols: $15+7+1=23$, which is $1$ livre and $3$ sols; livres: $3+2+1=6$. Result: $6$ livres, $3$ sols and $5$ deniers. Do that a thousand times a night and you will make mistakes. In 1642 Blaise began building a machine that would not.

October 11, 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 mystic hexagram: six points on a conic

In February 1640 a sheet printed on one side only, like a poster, circulated in Paris under the title Essay on conics. It had no proofs, only definitions, three lemmas and a list of results. The author was 16 years old and his name was Blaise Pascal. When Father Mersenne showed the sheet to Descartes, the reaction was suspicion: Descartes thought the young man had taken everything from Girard Desargues, the architect and geometer Pascal admired. Desargues’s influence was real, and Pascal himself acknowledged it. But the central result on the sheet was new.

October 11, 2026 Read