Augustin-Louis Cauchy

The life of Augustin-Louis Cauchy

Timeline 1789Born on 21 August in Paris, five weeks after the fall of the Bastille. 1794During the Terror, the family takes refuge in Arcueil, next door to Laplace. 1805Enters the École Polytechnique, second in the entrance examination. 1810Goes to Cherbourg as an engineer on Napoleon’s military harbour. 1813Returns to Paris in poor health and proves Fermat’s polygonal number theorem. 1816Appointed to the Academy of Sciences by royal decree, in place of Monge. 1821Publishes the Cours d’analyse of the École Polytechnique. 1825Presents the memoir on integrals between imaginary limits. 1830Refuses to swear allegiance to Louis-Philippe and goes into exile. 1833In Prague, becomes tutor to the Duke of Bordeaux, grandson of Charles X. 1838Returns to Paris. 1848The new republic abolishes the oath; the following year he takes up the chair at the Sorbonne. 1857Dies on 23 May at Sceaux, aged 67. A child of the Revolution Augustin-Louis Cauchy was born in Paris on 21 August 1789. The Bastille had fallen on 14 July, and his father, Louis-François Cauchy, was a senior official of the Paris police, an Ancien Régime post that became dangerous overnight. In 1794, during the Terror, the family fled to their country house in Arcueil, south of the city. There they knew hunger and poverty; Augustin was a frail child, and poor health stayed with him all his life.

October 25, 2026 Read

The polygonal number theorem

The Greeks already arranged pebbles into figures. With 1, 3, 6, 10 pebbles you make triangles; with 1, 4, 9, 16, squares; with 1, 5, 12, 22, pentagons. These are the polygonal numbers. The formula The $n$-th number with $k$ sides is obtained by adding up an arithmetic progression that starts at 1 and grows in steps of $k - 2$: $$P_k(n) = 1 + (k-1) + (2k-3) + \cdots = \frac{n\,\big((k-2)n - (k-4)\big)}{2}.$$For $k = 3$ we get the triangular numbers $\frac{n(n+1)}{2}$: 1, 3, 6, 10, 15, … For $k = 4$, the squares $n^2$. For $k = 5$, the pentagonal numbers $\frac{n(3n-1)}{2}$: 1, 5, 12, 22, 35, … We will also count zero, $P_k(0) = 0$.

October 27, 2026 Read

Determinants and permutations

On 30 November 1812, the young engineer from Cherbourg read to the Institut de France a memoir with a long title: on functions that can take only two values, equal and of opposite signs, when the variables are interchanged. Behind the title was the first general theory of determinants. Functions that change sign Take three variables $a_1, a_2, a_3$ and the product of the differences $$\Delta = (a_2 - a_1)(a_3 - a_1)(a_3 - a_2).$$If we swap $a_1$ and $a_2$, each factor either stays the same or changes sign, and altogether $\Delta$ becomes $-\Delta$. Any swap of two variables does the same. A permutation of the variables, which is a sequence of swaps, multiplies $\Delta$ by $+1$ or by $-1$: it is even or odd.

October 27, 2026 Read

Limits and continuity

In the eighteenth century calculus was a formidable and somewhat mysterious machine. Euler summed divergent series and drew correct results from them; Lagrange tried to base everything on power series; books spoke of “infinitely small” quantities that were zero when convenient and non-zero when not. It worked, but nobody could explain exactly why. The Cours d’analyse In 1821 Cauchy published the Cours d’analyse de l’École royale polytechnique, with the lessons he gave to future engineers. In the preface he announced his intention of giving analysis “all the rigour that is demanded in geometry”. Everything begins with a definition:

October 30, 2026 Read

The convergence of series

What is $1 - 1 + 1 - 1 + \cdots$? Grouped one way, $(1 - 1) + (1 - 1) + \cdots = 0$. Grouped another, $1 - (1 - 1) - (1 - 1) - \cdots = 1$. Euler and Leibniz thought the “right” answer was $\frac{1}{2}$, the average. In the eighteenth century such sums were handled freely, and the habit sometimes led to absurd results. Partial sums In chapter 6 of the Cours d’analyse, Cauchy cut the problem off at the root. A series $u_0 + u_1 + u_2 + \cdots$ has partial sums

October 31, 2026 Read

Cauchy's inequality

The 1821 Cours d’analyse ends with nine notes, a kind of appendix with results Cauchy did not want to put in the main text. The second, on inequalities, contains two gems that every olympiad student knows. The inequality Cauchy’s theorem says that, for any real numbers $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, $$(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \le (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2),$$with equality only when the two lists are proportional.

October 31, 2026 Read

The derivative and the integral

Two years after the Cours d’analyse, Cauchy published the Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal (1823). In forty short lessons he rebuilt differential and integral calculus on the idea of limit. It is, in essence, the outline that calculus textbooks still follow today. The derivative For Lagrange, the derivative of $f$ was the coefficient of $i$ in the power series of $f(x + i)$. Cauchy defined it as a limit:

October 31, 2026 Read

The integral in the complex plane

At the end of the eighteenth century, Euler and Laplace computed hard real integrals with a daring trick: they replaced the real variable with a complex one, did the calculations and returned to the real. Laplace called this “the passage from the real to the imaginary”. The results checked out, but nobody could say when the trick was legitimate. It was to answer this question that Cauchy, at 25, submitted a memoir on definite integrals to the Academy in 1814. It was only published in 1827, and it was only the beginning.

October 31, 2026 Read

Stress in elastic solids

Cauchy never stopped being an engineer. In 1822 Claude-Louis Navier had presented to the Academy a theory of elastic solids based on forces between molecules, and Augustin Fresnel was explaining light as a transverse vibration of an elastic medium, the ether. Cauchy, a colleague of both at the Academy, followed this work closely. On 30 September 1822 he announced to the Academy a new theory, one that needed no molecules.

October 31, 2026 Read

The legacy of Augustin-Louis Cauchy

Cauchy is mad, and there is nothing to be done with him; and yet he is, at present, the one who knows how mathematics ought to be done. — Niels Henrik Abel, letter to Holmboe (1826) Cauchy published about 800 papers and seven books. His Œuvres complètes, edited by the Academy of Sciences, came out between 1882 and 1974 in 27 thick volumes. Among mathematicians, only Euler and very few others left a body of work comparable in size.

October 31, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the files or thumbnails provided by Commons, reduced to at most 800 pixels wide, with no other editing. Portrait of Augustin-Louis Cauchy, lithograph by Rudolf Hoffmann (1856) after a photograph by Bertsch: source page · public domain. Oil portrait of Cauchy, around 1840, attributed to Jean Roller, Château de Sceaux: source page · public domain. Lithograph of Cauchy by Louis Grégoire and Deneux, around 1840, Dibner collection of the Smithsonian Libraries: source page · public domain. Bronze medallion by Pierre-Jean David d’Angers (1843): source page · CC0. Title page of the Cours d’analyse (1821), cleaned version (converted from GIF to JPEG): source page · CC0. Cauchy’s name on the Eiffel Tower, photo by Juanedc: source page · CC BY 2.0. Cauchy’s grave in the cemetery of Sceaux, photo by Antoine Rex: source page · CC0. Figures made for the blog The cover and the four interactive figures (bisection, the band of width ε in Cauchy’s criterion, the integral of 1/(1 + z²) around a circle and the stress vector on a rotating cut) are original illustrations, with text in each of the site’s languages.

October 31, 2026 Read