Calculus

The brachistochrone: the challenge of 1696

Given two points A and B in a vertical plane, to find the curve AMB along which a moving body M, descending under its own weight, passes from A to B in the shortest possible time. — Johann Bernoulli, Acta Eruditorum (June 1696) In June 1696, in the Acta Eruditorum, Johann Bernoulli challenged “the most acute mathematicians of the whole world”. The question sounds like something from a funfair: what is the shape of the fastest slide between two points? The name of the problem comes from Greek: brachistos, shortest, and chronos, time. Brachistochrone, the curve of least time.

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 ∫ sign: from sum to accumulation

On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.

October 11, 2026 Read

Areas and integration: the inverse of fluxions

Measuring the area of curved figures is one of the oldest problems in mathematics. Around 250 BC Archimedes showed that the area of a parabolic segment is $\frac43$ of the area of a triangle inscribed in it. It was a stroke of genius, but made to measure: every new curve needed a new idea. By the 17th century, Fermat, Cavalieri, Wallis and others could already compute the area under $y=x^n$ for many values of $n$. What was missing was a general rule. And what was missing was the realization that Newton and Leibniz each reached in their own way: computing areas and computing rates of change are inverse operations.

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 exponential calculus: differentiating x to the power x

Leibniz’s calculus knew how to differentiate powers with a fixed exponent, like $x^3$ or $x^{1/2}$: the derivative of $x^n$ is $nx^{n-1}$. But in 1694, in a letter to Leibniz, Johann Bernoulli put a strange creature on the table: the expression $$y=x^x,$$in which the base and the exponent vary at the same time. The rule for $x^n$ does not work, because the exponent is not fixed. Nor does the rule for $a^x$, because the base is not fixed. In 1697 Johann published in the Acta Eruditorum the “principles of the exponential calculus”, with the method for handling these powers, which he called percurrent, because the exponent runs through values.

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 principle of virtual work

In every equilibrium of any forces whatever, in whatever way they are applied, the sum of the positive energies will be equal to the sum of the negative energies. — Johann Bernoulli, letter to Pierre Varignon (26 January 1717) Since Archimedes, each simple machine had had its own explanation. The lever obeyed the law of the arms; the inclined plane, a decomposition of forces invented by Stevin; the pulley, an argument about the ropes; the screw, yet another. In 1717, in a letter to his French friend Pierre Varignon, Johann Bernoulli showed that all these laws are cases of a single principle.

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 Johann Bernoulli

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.

October 14, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Johann Bernoulli, engraving in the Rijksmuseum collection: source page · CC0. Portrait of Guillaume de l’Hôpital: source page · public domain. Title page of the Analyse des infiniment petits, Paris, 1696: source page · public domain. Portrait of Daniel Bernoulli, engraving by Johann Jakob Haid after Johann Rudolf Huber: source page · public domain. The chain and the arch, drawings by Giovanni Poleni, Memorie istoriche della gran cupola del Tempio Vaticano, 1748: source page · public domain. Blog figures The cover and the diagram of the cycloid and the straight line between the same points are original illustrations, with versions in each of the site’s languages. The diagram is to scale: both axes use the same unit. The table of function notations, in the post on the legacy, was redone from a table in this site’s old course on functions.

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