Calculus

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 Johann Bernoulli

Timeline 1667Born on 27 July in Basel, the tenth child of Nicolaus Bernoulli. 1682Sent to Neuchâtel to learn the trade. He has no talent for it. 1683Enters the University of Basel. Studies medicine and, with his brother Jakob, mathematics. 1691Solves the catenary problem. Travels to Geneva and then to Paris. 1692Begins teaching the calculus to the Marquis de l’Hôpital. 1694Takes his doctorate in medicine. Signs a contract with l’Hôpital and marries Dorothea Falkner. 1695Becomes professor of mathematics in Groningen, in Holland. 1696Issues the challenge of the brachistochrone. L’Hôpital’s book, made from his lessons, comes out. 1705Returns to Basel. Jakob dies, and Johann inherits his chair. 1717States the principle of virtual work in a letter to Varignon. 1720Begins guiding the young Leonhard Euler. 1734Shares a Paris Academy prize with his son Daniel and throws him out of the house. 1748Dies on 1 January in Basel, aged 80. ◀ Previous Follow the journey Next ▶ 1667–1682 BaselHe is born, the tenth child of Nicolaus Bernoulli, who wants to make him a merchant. 1682 NeuchâtelHe is sent to learn trade. He has no talent for it. 1683–1691 BaselHe studies medicine and, with his brother Jakob, Leibniz’s calculus. He solves the problem of the catenary. 1691 GenevaHe passes through Geneva on his way to Paris. 1691–1692 ParisHe meets Malebranche’s circle and starts teaching calculus to the Marquis de l’Hôpital, also at the château of Oucques. 1695–1705 GroningenProfessor of mathematics in Holland. In 1696 he issues the brachistochrone challenge. 1705–1748 BaselHe returns home, inherits Jakob’s chair, guides the young Euler and quarrels with his son Daniel. He dies in 1748, aged 80. Enable JavaScript to see the map. The stages are listed below. Johann Bernoulli between Basel, Paris and Groningen. Map: © OpenStreetMap contributors

October 14, 2026 Read

The life of Gottfried Wilhelm Leibniz

Timeline 1646Born on 1 July in Leipzig, in the Holy Roman Empire. 1661Enters the University of Leipzig, aged 14. 1666Publishes De arte combinatoria. Leipzig refuses him a doctorate in law; Altdorf awards it the following year. 1672Goes to Paris on a diplomatic mission. Begins to study mathematics seriously with Christiaan Huygens. 1673Presents his calculating machine in London and is elected to the Royal Society. 1675In manuscripts of October and November, creates the symbols $\int$ and $d$. 1676Moves to Hanover, where he will work as librarian and counsellor for 40 years. 1684Publishes the first paper on differential calculus in the journal Acta Eruditorum. 1700Founds the Berlin Academy of Sciences and becomes its first president. 1703Publishes his explanation of binary arithmetic. 1716Dies on 14 November in Hanover. ◀ Previous Follow the journey Next ▶ 1646–1666 LeipzigHe is born and studies in Leipzig. The university refuses him a doctorate in law, saying he is too young. 1666–1667 AltdorfHe obtains his doctorate within a few months, near Nuremberg, and turns down the professorship he is offered. 1668–1672 MainzHe enters the service of the Prince-Archbishop of Mainz as a jurist and adviser. 1672–1676 ParisHe arrives on a diplomatic mission and learns mathematics from Christiaan Huygens. In 1675 he creates the symbols $\int$ and $d$. 1673 LondonHe shows his calculating machine at the Royal Society and is elected a fellow. 1676 The HagueOn his way to Hanover he visits the philosopher Baruch Spinoza. 1676–1716 HanoverLibrarian and adviser to the dukes for forty years. He publishes the calculus (1684) and binary arithmetic (1703). He dies in 1716. 1687–1690 RomeHe travels for three years through archives in Germany, Austria and Italy to write the history of the ducal family. 1700 BerlinHe founds the Berlin Academy of Sciences and becomes its first president. Enable JavaScript to see the map. The stages are listed below. The journeys of Leibniz, diplomat, librarian and mathematician. Map: © OpenStreetMap contributors

October 11, 2026 Read

The life of Isaac Newton

Timeline 1643Born on 4 January at Woolsthorpe, England. In English records of the time the date is 25 December 1642. 1661Enters Trinity College, Cambridge, as a poor student. 1665–1667With the university closed by the plague, he returns home. He develops series, fluxions, the theory of colours and his first ideas on gravity. 1669Aged 26, becomes Lucasian Professor of Mathematics, succeeding Isaac Barrow. 1672Presents his reflecting telescope and publishes his theory of light and colours. 1687Publishes the Principia, paid for out of Edmond Halley’s own pocket. 1696Moves to the Royal Mint in London, first as Warden and, from 1699, as Master. 1703Becomes President of the Royal Society. 1704Publishes Opticks. 1727Dies on 31 March in London. He is buried in Westminster Abbey. ◀ Previous Follow the journey Next ▶ 1643–1661 WoolsthorpeHe is born on a farm, the posthumous son of a farmer, and goes to school in nearby Grantham. 1661–1665 CambridgeHe enters Trinity College as a poor student who pays his way by serving other students. 1665–1667 WoolsthorpeWith the university closed by the plague, he goes home. He develops series, fluxions, the theory of colours and his first ideas on gravity. 1667–1696 CambridgeHe holds the Lucasian chair, builds the reflecting telescope and writes the Principia (1687). 1696–1727 LondonHe runs the Royal Mint, at the Tower of London, and presides over the Royal Society. He dies in 1727 and is buried in Westminster Abbey. Enable JavaScript to see the map. The stages are listed below. Newton’s three places, from Woolsthorpe to Cambridge and London. 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

The catenary: the hanging chain

Ut pendet continuum flexile, sic stabit contiguum rigidum inversum. (As hangs the flexible line, so but inverted will stand the rigid arch.) — Robert Hooke, anagram deciphered in 1705 Take a thin chain by its two ends and let it hang. It forms a beautiful curve that looks like a parabola. Galileo, in 1638, thought it really was a parabola, at least approximately. The Dutchman Christiaan Huygens, at 17, showed that it could not be. But what curve it was, nobody knew.

October 14, 2026 Read

Leibniz's notation: a language for change

In signs one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labour of thought is wonderfully diminished. — Leibniz, letter to Tschirnhaus (1678) Newton reached calculus first, but it was Leibniz who decided how it would be written. And the writing made all the difference. The letter d On 11 November 1675, in a manuscript written in Paris, Leibniz began using the letter $d$ for an infinitely small difference. The $d$ comes from the Latin differentia, from the verb differre, “to carry apart”. So:

October 11, 2026 Read

L'Hôpital's rule: the lessons sold to a marquis

I acknowledge that I owe much to the insights of Messrs Bernoulli, above all to those of the younger, now professor at Groningen. I have made free use of their discoveries and of those of Mr Leibniz. — Marquis de l'Hôpital, preface to the Analysis of the Infinitely Small (1696) In 1691 the Marquis Guillaume de l’Hôpital was 30 years old and had a title, a castle and a passion: to understand Leibniz’s calculus, which had barely been published and which almost nobody in France could read. When the young Swiss Johann Bernoulli, who knew the calculus better than almost anyone, turned up in Paris, the marquis hired him as a private tutor.

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

Maxima and minima: the rectangle with the largest area

You have 24 metres of fencing and want to enclose a rectangular plot with the largest possible area. Which dimensions should you choose? A long thin rectangle wastes fencing; a square seems better. But how do you prove it? Fermat invented a method for questions like this around 1636, more than thirty years before Newton and almost fifty before Leibniz published calculus. Fermat’s problem Fermat’s first example was exactly this: divide a segment of length $b$ into two parts, $x$ and $b-x$, so that the product $x(b-x)$ is as large as possible. That is our plot: with 24 m of fencing, two adjacent sides add up to $12$, and the area is

October 11, 2026 Read

The product rule: two changes in one calculation

If you know how $u$ changes and how $v$ changes, how does the product $uv$ change? The natural guess is to multiply the changes: $d(uv)=du\cdot dv$. It looks so symmetrical, so elegant… Leibniz made the same guess. Leibniz tests and fails In the same manuscript of 11 November 1675 in which he began using the letter $d$, Leibniz asked himself whether $d(xy)$ would equal $dx\,dy$, and whether $d\!\left(\frac xy\right)$ would equal $\dfrac{dx}{dy}$. Instead of trusting elegance, he tested it with an example and found it did not. A few days later he found the right rule.

October 11, 2026 Read

Fluxions: how to measure an instantaneous change

A car leaves one city and reaches another, 90 km away, in one hour. Its average speed was 90 km/h. But at every instant the speedometer showed a different number. What exactly does it measure? Speed is distance divided by time, but in an instant no distance is covered and no time passes. Dividing zero by zero gives no answer. That was the question Newton wanted to answer in 1665. His answer was the method of fluxions.

October 11, 2026 Read