Algebra

The life of Évariste Galois

Timeline 1811Born on 25 October in Bourg-la-Reine, near Paris. 1823Enters the Louis-le-Grand college in Paris. 1827Discovers mathematics and abandons everything else. 1828Fails the École Polytechnique entrance examination for the first time. 1829Publishes his first paper. Sends work to the Academy. His father commits suicide. Second failure at the Polytechnique. Enters the École Normale. 1830The paper entered for the Grand Prize is lost when Fourier dies. July Revolution. 1831Expelled from the École Normale. Poisson rejects his memoir. Arrested twice. 1832Dies on 31 May, aged 20, after being wounded in a duel. 1846Joseph Liouville publishes his works. ◀ Previous Follow the journey Next ▶ 1811–1823 Bourg-la-ReineHe is born in the town where his father will be mayor. His mother teaches him at home until he is 12. 1823–1829 Louis-le-GrandAt school he discovers mathematics at 15, “the frenzy of mathematics dominates him”. Having failed the Polytechnique twice, he enters the École Normale in 1829 and is expelled in 1831. 1831 Pont NeufOn 14 July he is arrested on the bridge, wearing the uniform of the disbanded artillery and armed. 1831–1832 Sainte-Pélagie prisonHe spends eight months in prison. In July 1831 he receives Poisson’s rejection. 1832 Cochin hospitalWounded in the duel of 30 May, he is taken to the hospital, where he dies on the morning of 31 May, aged 20. Enable JavaScript to see the map. The stages are listed below. Galois’s Paris, from school to prison and hospital, within a few kilometres. Map: © OpenStreetMap contributors

October 15, 2026 Read

The life of Niels Henrik Abel

Timeline 1802Born on 5 August near Stavanger, on the west coast of Norway. 1804The family moves to Gjerstad, where his father is the pastor. 1815Enters the Cathedral School of Christiania, today’s Oslo. 1818Bernt Holmboe becomes his mathematics teacher. 1820His father dies. Abel begins to support part of the family. 1821Enters the university. Believes he has solved the quintic and finds the mistake. 1824Publishes, at his own expense, the proof that there is no formula for the fifth degree. 1825Travels to Berlin, where he meets August Leopold Crelle. 1826In Paris, hands the Academy his great memoir on integrals. Cauchy loses it. 1827Returns to Norway, jobless and in debt. Publishes elliptic functions. 1829Dies of tuberculosis at Froland on 6 April, aged 26. 1830Receives, posthumously, the Grand Prize of the Paris Academy, with Jacobi. ◀ Previous Follow the journey Next ▶ 1802 near StavangerHe is born on the west coast of Norway, the second of seven children of a pastor. 1804–1815 GjerstadHe grows up in his father’s parish in the south of the country. 1815–1825 ChristianiaAt the Cathedral School his teacher Holmboe discovers his talent. In 1824 he publishes the six-page pamphlet on the quintic. 1825–1826 BerlinHe meets August Leopold Crelle, who publishes seven of his papers in the first volume of his journal. 1826 ParisHe hands the Academy his great memoir on integrals. Nobody pays attention to him, and Cauchy loses the manuscript. 1827–1828 ChristianiaHe returns jobless and in debt. He publishes elliptic functions and races Jacobi. 1828–1829 FrolandHe travels by sleigh at Christmas to see his fiancée, falls ill and dies of tuberculosis on 6 April. Two days later the appointment from Berlin arrives. Enable JavaScript to see the map. The stages are listed below. Abel’s great journey, from Norway to Berlin and Paris, and the return to die at Froland. Map: © OpenStreetMap contributors

October 15, 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

Shuffling the roots: the group of an equation

To jump with both feet over the calculations; to group the operations, to classify them according to their difficulties and not according to their forms: such is, in my view, the mission of future geometers. — Évariste Galois, preface to a manuscript (1831) Sixty years before Galois, Lagrange had looked at expressions made from the roots of an equation and counted how many values they take when the roots change places. Galois went one step further. Instead of looking at the expressions, he looked at the swaps. And not at all of them: only at those the equation allows.

October 15, 2026 Read

The fifth-degree equation: the formula that does not exist

Geometers have been much occupied with the general solution of algebraic equations, and several of them have sought to prove its impossibility. But, if I am not mistaken, nobody has succeeded so far. — Niels Henrik Abel, Memoir on algebraic equations (1824) Everyone who has been to school knows one formula for solving equations: the quadratic formula. The equation $ax^2+bx+c=0$ has the roots $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$The recipe uses only the four operations and a square root. The Babylonians could already solve problems of this kind almost four thousand years ago, and Euclid solved them with squares and rectangles. The natural question is: are there recipes like this for equations of higher degree?

October 15, 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.

October 14, 2026 Read

The binomial series and the devil's series

Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. — Niels Henrik Abel, letter to Holmboe (January 1826) In 1665 the young Newton discovered that the binomial formula also held for exponents that are not whole numbers: $$(1+x)^m=1+mx+\frac{m(m-1)}{2}x^2+\frac{m(m-1)(m-2)}{6}x^3+\cdots$$For a positive integer $m$ the sum stops after $m+1$ terms. For $m=\frac12$ or $m=-1$ it never stops: it is a series, a sum with infinitely many terms. Newton, Euler and all the mathematicians of the eighteenth century used this series with enormous success. Almost nobody asked when it works.

October 15, 2026 Read

What a group is: the six symmetries of the triangle

Cut an equilateral triangle out of paper, number its corners 1, 2 and 3 and place it over its outline drawn on the table. In how many ways can you pick the triangle up and put it back so that it covers exactly the same outline? There are six. You can do nothing; turn it by $120°$ or $240°$; or flip the triangle over about one of its three axes, each passing through a corner and the midpoint of the opposite side.

October 15, 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

Inverting the integral: elliptic functions

Start with a familiar function, seen in an odd way. The integral $$\arcsin x=\int_0^x\frac{dt}{\sqrt{1-t^2}}$$gives the length of an arc of a circle of radius 1, measured from the lowest point, as a function of the height $x$. It is a complicated function. But its inverse, which gives the height as a function of the arc, is the sine, the simplest function in trigonometry. And the sine has a property the integral hides: it is periodic. Since a quarter turn measures

October 15, 2026 Read

When an equation can be solved by radicals

With the idea of the group of an equation, Galois answered the question left open after Abel: which equations can be solved by radicals? The answer does not depend on any trick. It depends only on the shape of the group. Each root cuts the group Solving an equation by radicals means adding roots, one at a time, to the numbers already known. At the start we know only the rational numbers, and the group of the equation is whole. Each time we add a root, such as $\sqrt3$ or $\sqrt[3]{5}$, new true relations appear, and fewer permutations respect them. The group shrinks. When the group becomes just the identity, all the roots of the equation are written with known numbers: the equation is solved.

October 15, 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