Mechanics

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 Joseph-Louis Lagrange

Timeline 1736Born on 25 January in Turin, capital of the Kingdom of Sardinia, as Giuseppe Lodovico Lagrangia. 1753At 17, reads an essay by Edmond Halley and decides to devote himself to mathematics. 1755Writes to Euler with the method that will give rise to the calculus of variations. In September he is appointed professor at the Artillery School in Turin, aged 19. 1757Helps found the scientific society that will become the Turin Academy of Sciences. 1764Wins the Paris Academy prize by explaining why the Moon always shows the same face. 1766Moves to Berlin to take Euler’s place at the Academy of Frederick the Great. 1770Proves that every whole number is a sum of four squares and begins studying the permutations of the roots of an equation. 1772Shares the Paris prize with Euler for the three-body problem, where the Lagrange points appear. 1787After Frederick’s death, moves to Paris, where he lives in the Louvre. 1788Publishes the Analytical Mechanics. 1790Joins the commission that will create the metric system. 1794Lavoisier is guillotined. Lagrange becomes a professor at the new École Polytechnique. 1797Publishes the Theory of Analytic Functions, with the notation $f'(x)$. 1808Napoleon makes him a count of the Empire. 1813Dies on 10 April in Paris and is buried in the Panthéon. ◀ Previous Follow the journey Next ▶ 1736–1766 TurinAt 19 he is a professor at the Artillery School. He writes to Euler the method that will give rise to the calculus of variations. 1766–1787 BerlinHe takes Euler’s place at the Academy of Frederick the Great. He proves the four-square theorem and finds the Lagrange points. 1787–1813 ParisHe lives in the Louvre, publishes the Analytical mechanics, helps create the metric system and teaches at the École Polytechnique. He is buried in the Panthéon. Enable JavaScript to see the map. The stages are listed below. Lagrange in three capitals, Turin, Berlin and Paris. Map: © OpenStreetMap contributors

October 14, 2026 Read

The calculus of variations: a letter from a 19-year-old

The calculus of Newton and Leibniz can find the highest point of a curve: it is where the derivative vanishes. But there are questions of another kind. What is the shortest path between two points? What shape should a slide have so that a ball runs down it fastest? What curve does a hanging chain make? In these questions the answer is not a number but a whole curve, chosen from infinitely many possible curves.

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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.

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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 Analytical Mechanics: a book without figures

Open Newton’s Principia of 1687 and you will see figures on almost every page: triangles, ellipses, arrows, tangents. Open Lagrange’s Analytical Mechanics, published a century later in 1788, and you will see none. The author warns of this himself in the preface, with pride, and adds: Those who love analysis will be pleased to see mechanics become a new branch of it, and will be grateful to me for having thus extended its domain. — Joseph-Louis Lagrange, preface to the Analytical Mechanics (1788)

October 14, 2026 Read

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 Lagrange points: where gravity balances

On 25 December 2021 a rocket lifted off from French Guiana carrying the James Webb Space Telescope. It was not placed in orbit around the Earth, like Hubble. It travelled for a month to an empty point in space, 1.5 million kilometres from here, always on the side opposite the Sun, and stayed there. The address of that point comes from a prize-winning paper by Lagrange, from 1772. It is called L2.

October 14, 2026 Read

Four squares are enough

Take any positive whole number and try to write it as a sum of perfect squares, $0, 1, 4, 9, 16, 25, \ldots$, using as few terms as possible: $$\begin{aligned} 5&=4+1,\\ 6&=4+1+1,\\ 7&=4+1+1+1,\\ 30&=25+4+1,\\ 2026&=2025+1=45^2+1^2. \end{aligned}$$Some numbers are squares, others need two, others three. The number 7 needs four: there is no other way. Could some number need five? In 1770 Lagrange proved that no number does: Every positive whole number is a sum of four squares of integers.

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

Shuffling the roots: Lagrange's theorem

In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5. Lagrange did not try yet another method. He asked a different question: why do the old methods work?

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.

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