Augustin-Louis Cauchy

How to use this course

Augustin-Louis Cauchy published about 800 papers, more than anyone of his time except Leonhard Euler. Students meet his name everywhere: Cauchy sequences, the Cauchy–Schwarz inequality, Cauchy’s integral theorem, Cauchy’s theorem on groups, the Cauchy stress tensor, Cauchy’s dispersion formula. Behind the names lies a single programme. Cauchy wanted every result derived from clear definitions, whether the object was a limit, an integral of a complex function, or the force acting across a surface inside a solid.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches or derivations, and exercises with folded solutions. Lessons 04–07 cover pure mathematics; Lessons 08–09 cover his mechanics of continua and his optics; Lesson 10 returns to two early results in geometry and number theory.

Timeline

A child of the Revolution, 1789–1805

Augustin-Louis Cauchy was born in Paris on 21 August 1789. His father, Louis-François Cauchy, was a lawyer and an official of the Paris police lieutenancy under the old monarchy. The Revolution ended that career. During the Terror the family left for their country house at Arcueil, south of Paris, where food was scarce and the children were often hungry. Augustin was the eldest of several children; two brothers, Alexandre Laurent and Eugène François, became jurists.

The father educated his sons himself, with a strong Catholic and classical emphasis. After Napoleon came to power the family’s fortunes recovered, and Louis-François became secretary-general of the Senate. Among the senators were the two greatest French mathematicians of the age, Pierre-Simon Laplace and Joseph-Louis Lagrange. Both noticed the boy. Lagrange advised the father to keep him away from mathematics books until he had a solid grounding in languages, so that his talent would not run ahead of his general education.

Cauchy entered the École centrale du Panthéon in 1802 and won prizes in Latin and Greek. In 1804 he began serious mathematics, and in 1805 he sat the entrance examination of the École polytechnique, examined by Jean-Baptiste Biot. He placed second in France.

Polytechnique and Cherbourg, 1805–1812

At the École polytechnique Cauchy studied analysis with Sylvestre-François Lacroix and mechanics with Gaspard de Prony; his tutor in analysis was André-Marie Ampère. The school trained engineers for the state, and in 1807 he moved on to the École des ponts et chaussées, the school of civil engineering. He did his practical training on the Ourcq canal, which was to bring water to Paris.

In 1810 he was posted as a junior engineer to Cherbourg, where Napoleon was building a naval port for a planned invasion of England. He carried a small library: the Mécanique céleste of Laplace, Lagrange’s Théorie des fonctions analytiques, Virgil and a book of devotions. He worked long days on the harbour and spent his free hours on mathematics.

In February 1811 Cauchy sent the Institute his first paper. A second followed in 1812. In it he proved that a convex polyhedron is determined by its faces (Lesson 10). Adrien-Marie Legendre and Étienne-Louis Malus encouraged him, and the work made his name. In September 1812 he returned to Paris exhausted and depressed, officially on sick leave. He never returned to Cherbourg.

The breakthrough years, 1812–1816

Back in Paris, Cauchy produced a run of papers that would have made several careers. In November 1812 he read a memoir on functions that take only two values, equal and opposite, under permutations of their variables. It contains the first systematic theory of determinants, including the multiplication theorem (Lesson 07), and was published in 1815 together with a study of how many values a function can take under permutations. In 1813 he proved Fermat’s claim that every positive integer is a sum of \(k\) polygonal numbers of order \(k\) (Lesson 10). In 1814 he wrote a long memoir on definite integrals, in which integrals of complex functions first appear in his work.

Positions came more slowly. He was passed over for the Bureau des longitudes in favour of Legendre, for the geometry section of the Institute in favour of Louis Poinsot, and in a 1814 mechanics election he received no votes. In 1815 he became assistant professor of analysis at the École polytechnique.

The fall of Napoleon changed everything. In 1816 the restored Bourbon monarchy reorganized the Institute and expelled Lazare Carnot and Gaspard Monge from the Académie des sciences for political reasons. Cauchy, a fervent royalist, was named by royal ordinance to one of the vacated seats. Many colleagues resented the manner of his appointment. The same year he won the Academy’s Grand Prix for a memoir on the propagation of waves on the surface of a heavy liquid, a problem that led him to early forms of the Fourier transform. At 26 he was a member of the Academy and a professor at the leading school of science in Europe.

Rigour: the Cours d’analyse

For the general reader. Eighteenth-century calculus worked spectacularly but rested on intuitions about “infinitely small” quantities and on formal manipulation of infinite series, which sometimes gave nonsense. In 1821 Cauchy wrote a textbook for his students that defined limits, continuity and convergence in words that can be checked, and proved theorems from those definitions. Modern analysis courses still follow its plan.

Definitions

Cauchy defined a variable quantity to have a limit when its successive values approach a fixed value so that they differ from it by as little as one wishes. An infinitely small quantity was simply a variable with limit zero. A function \(f\) was continuous between two bounds if, for each \(x\) between them, an infinitely small increment of \(x\) produces an infinitely small increment \(f(x+\alpha)-f(x)\). In modern language: \(f(x+\alpha)-f(x)\to0\) as \(\alpha\to0\). The \(\varepsilon\)–\(\delta\) form came later, in the lectures of Karl Weierstrass, but Cauchy’s proofs often used inequalities in exactly that spirit.

Cauchy proved the necessity and took the sufficiency as evident. It cannot be proved without a precise construction of the real numbers, which Richard Dedekind and Georg Cantor supplied in 1872. Bernard Bolzano had stated the criterion in 1817.

The Cours also proves the intermediate value theorem by repeated subdivision, and the theorem that the product of two absolutely convergent series converges to the product of their sums (the Cauchy product). In Note II it proves an inequality for finite sums.

Proof. Lagrange’s identity gives \(\sum a_i^2\sum b_i^2-(\sum a_ib_i)^2=\sum_{i<j}(a_ib_j-a_jb_i)^2\ge0\). Cauchy argued this way.

The sum theorem

The Cours asserts that a convergent series of continuous functions has a continuous sum. This is false. In 1826 Niels Henrik Abel pointed out an exception from Fourier series, and the repair, uniform convergence, was clarified by Seidel, Stokes and Weierstrass in the 1840s. Cauchy himself added a corrected hypothesis in 1853.

The integral and the differential equation

For the general reader. Before Cauchy, an integral was usually defined as the reverse of a derivative. Cauchy turned this around. He defined the integral as a limit of sums of areas of thin rectangles, proved that the limit exists for every continuous function, and then proved that differentiating it gives back the function. He used the same habit of mind to prove that differential equations have solutions, instead of assuming it.

The integral (1823)

In the Résumé des leçons sur le calcul infinitésimal, Cauchy took a function \(f\) continuous on \([x_0,X]\), a partition \(x_0<x_1<\cdots<x_n=X\), and the sum

Cauchy’s proof uses the fact that a continuous function on a closed interval changes by little over short intervals, uniformly along the interval. He took this for granted; Eduard Heine proved uniform continuity in 1872. Bernhard Riemann in 1854 extended the definition to discontinuous functions.

Proof. Apply Rolle’s theorem to \(h(x)=\big(f(b)-f(a)\big)g(x)-\big(g(b)-g(a)\big)f(x)\), which satisfies \(h(a)=h(b)\).

Existence for differential equations

In his lectures at the École polytechnique in the 1820s, Cauchy proved that the initial value problem \(y'=f(x,y)\), \(y(x_0)=y_0\), has a solution, by taking the limit of Euler’s polygonal approximations. The proof circulated through his teaching and was later published by his disciple the Abbé François Moigno. From the 1830s he also developed the calcul des limites, the method of majorant series, to prove existence of analytic solutions; Sofia Kovalevskaya extended it to partial differential equations in 1875.

Complex integration and residues

For the general reader. Euler and Laplace had evaluated hard real integrals by passing through complex numbers, without justification. Cauchy made the step legitimate. He showed that the integral of a well-behaved complex function between two points does not depend on the path, so long as the path does not cross a singularity. When it does cross one, the integral jumps by an amount he called the residue. Out of this came the most powerful computing tool in analysis.

Three steps, 1814–1831

The 1814 memoir on definite integrals studied how a double integral can depend on the order of integration and contains the first traces of residues. The 1825 Mémoire sur les intégrales définies prises entre des limites imaginaires states that the integral of \(f(z)\) between two complex points is the same along all paths that can be deformed into each other without meeting a point where \(f\) becomes infinite. In 1826 the Exercices de mathématiques introduced the residue as a new calculus. In Turin in 1831 Cauchy derived the integral formula and from it the expansion of a function in power series, convergent up to the nearest singularity.

Proof sketch (for \(f\) continuously differentiable). Write \(f=u+iv\) and \(dz=dx+i\,dy\). For \(\gamma\) bounding a region \(D\), Green’s theorem gives

because complex differentiability is equivalent to the Cauchy–Riemann equations \(u_x=v_y,\ u_y=-v_x\). Édouard Goursat removed the hypothesis of continuous derivative in 1900.

The integral theorem lets one shrink \(\gamma\) to small circles around the singular points, as in the figure at the top of this page. On a small circle \(|z-a|=r\), each term \(c_n(z-a)^n\) integrates to 0 except \(n=-1\), which gives \(2\pi i\,c_{-1}\). For a simple pole, \(\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z)\).

Determinants and permutations

For the general reader. Shuffling variables is the oldest source of group theory. In 1812, at 23, Cauchy studied expressions that change sign when two variables are swapped, and from them built the theory of determinants. Thirty years later, prompted by questions of Joseph Bertrand, he returned to permutations and proved one of the first general theorems about finite groups.

Alternating functions (1812, published 1815)

Cauchy considered the product \(\prod_{i<j}(a_j-a_i)\), which changes sign under any transposition of the variables, and from it defined the determinant of an \(n\times n\) array \((a_{i,j})\) as the alternating sum \(\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\,a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}\). He introduced the word “determinant” in this sense, the square array notation with double indices, and proved the product rule. Jacques Binet proved it independently the same year.

The companion memoir of 1815 asked how many distinct values a function of \(n\) variables can take when its variables are permuted. Cauchy proved that if the number of values is greater than 2, it is at least the largest prime \(p\le n\). This is a statement about the possible indices of subgroups of \(S_n\), in the line of Lagrange and Ruffini.

The theorem of 1845

In 1844–46 Cauchy wrote a long series of papers on systems of “conjugate substitutions”, closed sets of permutations that we call permutation groups. Among his results is the following.

Proof (J. H. McKay, 1959). Let \(X=\{(g_1,\dots,g_p)\in G^p : g_1g_2\cdots g_p=1\}\). Choosing \(g_1,\dots,g_{p-1}\) freely determines \(g_p\), so \(|X|=|G|^{p-1}\), which is divisible by \(p\). The cyclic group \(\mathbb Z/p\) acts on \(X\) by rotating the tuple (if \(g_1\cdots g_p=1\) then \(g_2\cdots g_pg_1=1\)). Orbits have size 1 or \(p\), and the orbits of size 1 are the constant tuples \((g,\dots,g)\) with \(g^p=1\). So the number of such \(g\) is divisible by \(p\). It is at least 1 (the identity), hence at least \(p\), and some \(g\ne1\) has \(g^p=1\).

Stress and elasticity

For the general reader. How does a bridge carry a load? Every small piece of the material pushes and pulls on its neighbours across every imaginary surface. Before 1822 engineers described this with forces between molecules, or with special theories for beams and plates. Cauchy replaced them with one idea: at each point, the force per unit area across a surface depends linearly on the direction the surface faces. The object that encodes this, the stress tensor, is the basis of solid mechanics, fluid mechanics and geophysics.

The memoir of 1822

On 30 September 1822 Cauchy read to the Academy a memoir on the equilibrium and motion of solid and fluid bodies, elastic or not. It had been prompted by Claude-Louis Navier’s work on elastic plates. A summary appeared in the Bulletin de la Société philomathique in 1823, and full treatments in the Exercices de mathématiques of 1827 and 1828. Cauchy postulated that the force per unit area exerted across a small surface element with unit normal \(\mathbf n\), the traction \(\mathbf t(\mathbf n)\), depends only on the point and on \(\mathbf n\), and that it may have a tangential component. Earlier writers had considered only pressures normal to the surface.

The tetrahedron argument

Take a small tetrahedron at \(\mathbf x\) with three faces in the coordinate planes, with outward normals \(-\mathbf e_j\) and areas \(A_j\), and a slanted face of area \(A\) with normal \(\mathbf n\). Projection gives \(A_j=n_jA\). The volume is \(\tfrac13hA\), where \(h\) is the height. Balance of momentum reads

Divide by \(A\) and let \(h\to0\): \(\mathbf t(\mathbf n)=-\sum_j n_j\,\mathbf t(-\mathbf e_j)\). A thin slab argument gives \(\mathbf t(-\mathbf e_j)=-\mathbf t(\mathbf e_j)\), so \(\mathbf t(\mathbf n)=\sum_jn_j\,\mathbf t(\mathbf e_j)\). Setting \(\sigma_{ij}=t_i(\mathbf e_j)\) gives the theorem.

Equations of motion

For any region \(V\) with boundary \(\partial V\), the divergence theorem turns \(\int_{\partial V}\sigma_{ij}n_j\,dA\) into \(\int_V\partial_j\sigma_{ij}\,dV\). Since \(V\) is arbitrary,

Balance of angular momentum, applied the same way, forces \(\sigma_{ij}=\sigma_{ji}\). Being real and symmetric, \(\boldsymbol\sigma\) has three orthogonal principal directions with real principal stresses, a point Cauchy made explicitly; across those planes the traction is purely normal.

Strain and the isotropic law (1828)

For small displacements \(\mathbf u\), Cauchy measured deformation by the strain \(\varepsilon_{ij}=\tfrac12(\partial_iu_j+\partial_ju_i)\). For an isotropic body with stress proportional to strain he obtained a law with two constants. In today’s notation, with Lamé’s constants \(\lambda,\mu\),

Navier’s 1821 molecular theory had produced the same equation with a single constant, the special case \(\lambda=\mu\). Cauchy and Siméon-Denis Poisson also derived elasticity from central forces between molecules in 1828–29, which again yields only one constant for an isotropic body. The debate between one constant and two lasted half a century, until experiments by Wertheim and others settled it in favour of two. Cauchy also treated the “perfectly inelastic” case, in which stress depends on the rate of deformation; for an incompressible body this resembles the equation of viscous flow that Navier had found and George Gabriel Stokes later established.

Elastic waves and the theory of light

For the general reader. In the 1820s Augustin Fresnel showed that light behaves as a transverse wave, vibrating across its direction of travel. Waves of that kind need a medium that resists shearing, like a solid. Cauchy, with his new theory of elastic solids, set out to derive the laws of optics from the mechanics of an elastic aether. The attempt failed in its central aim, but it produced the wave theory of elastic solids used by seismologists and a formula for the colours of glass that opticians still use.

Two kinds of wave

Take the equation of Lesson 08 without body force and look for plane waves \(\mathbf u=\mathbf a\,e^{i(\mathbf k\cdot\mathbf x-\omega t)}\). Substituting,

Poisson found the two speeds in 1829. Seismologists now call them P and S waves. For optics the result was a difficulty: light shows only the transverse kind. Cauchy tried several ways out, including aethers in which the longitudinal wave has zero or infinite speed. George Green in 1838 and James MacCullagh in 1839 criticized the stability or the boundary conditions of such models, and no elastic aether ever reproduced all of Fresnel’s laws of reflection and double refraction.

Dispersion

Glass bends blue light more than red: its refractive index depends on wavelength. A continuous elastic medium cannot explain this, because its wave speed does not depend on wavelength. Cauchy’s insight, published in his Prague memoir on the dispersion of light (1836), was that if the aether consists of molecules whose spacing is not negligible compared with the wavelength, the speed of light in a medium must depend on the wavelength. Expanding in powers of the ratio, he obtained a series for the refractive index.

A lattice model

The mechanism can be seen in one dimension. Masses \(m\) at spacing \(a\), joined by springs of stiffness \(K\), obey \(m\ddot u_j=K(u_{j+1}-2u_j+u_{j-1})\). Trying \(u_j=e^{i(kja-\omega t)}\) gives

The phase speed falls as the wavelength \(2\pi/k\) shrinks. Since the refractive index is inversely proportional to the speed, \(n\approx n_0\big(1+\frac{k^2a^2}{24}\big)\), a constant plus a multiple of \(1/\lambda^2\): Cauchy’s first two terms.

Polyhedra and polygonal numbers

For the general reader. Cauchy made his name at 22 with a theorem a child can state: if you build a convex polyhedron from rigid faces, it cannot flex. A year later he proved a claim Pierre de Fermat had left without proof, that every whole number is a sum of three triangular numbers, four squares, five pentagonal numbers, and so on. Both results show his early gift for combinatorial arguments.

Proof outline. Label each edge \(+\), \(-\) or \(0\) according to whether its dihedral angle is larger, smaller or equal in \(Q\) than in \(P\). Suppose some edge is non-zero. At each vertex, going around the edges with non-zero labels, count sign changes. A geometric lemma about convex spherical polygons, the arm lemma, shows that every vertex with a non-zero label has at least four sign changes. Counting the same sign changes face by face and using Euler’s formula \(V-E+F=2\) shows that the total number is less than \(4V\). This contradiction proves all labels are 0. Cauchy’s proof of the arm lemma had a gap, repaired by Ernst Steinitz in the 1930s.

Convexity matters: Robert Connelly built a flexible non-convex polyhedron in 1977.

The case \(k=4\) is Lagrange’s four-square theorem (1770) and \(k=3\) is Gauss’s triangular theorem (1796). Cauchy reduced the general case to these with a lemma: if \(a,b\) are odd positive integers with \(b^2<4a\) and \(3a<b^2+2b+4\), then there are non-negative integers \(s,t,u,v\) with \(a=s^2+t^2+u^2+v^2\) and \(b=s+t+u+v\). Melvyn Nathanson gave a short modern proof based on this lemma in 1987.

Professor, academician, exile, 1816–1838

From 1816 to 1830 Cauchy was at the centre of Parisian science. He taught analysis and mechanics at the École polytechnique, substituted for Jean-Baptiste Biot at the Collège de France from 1817, and taught at the Faculté des sciences. His students found his lectures demanding; colleagues on the school’s council complained that his rigorous course left too little time for applications. In 1818 he married Aloïse de Bure, from a family of publishers. They had two daughters.

His productivity overwhelmed the journals. In 1826 he started the Exercices de mathématiques, a private journal that he wrote alone. He was a hard referee and a forgetful one. The memoir that Niels Henrik Abel submitted to the Academy in 1826 was assigned to Cauchy and Legendre and mislaid; Cauchy reported on it only after Abel’s death in 1829, and it was published in 1841. The memoirs of Évariste Galois of 1829 also went unreported. Cauchy was also a devout Catholic, active in charitable and religious societies, and his religious and royalist zeal made him enemies.

Exile

The July Revolution of 1830 replaced Charles X with Louis-Philippe. Cauchy refused to swear the oath of allegiance to the new king and left Paris in September 1830, first for Switzerland. He thereby lost all his posts except his seat in the Academy, which required no oath. In 1831 he settled in Turin, where King Charles Albert of Sardinia created a chair of theoretical physics for him, and where he wrote the memoir on power series of Lesson 06. In 1832 he was elected a Fellow of the Royal Society.

In 1833 the exiled Charles X summoned him to Prague to tutor his grandson Henri d’Artois, duc de Bordeaux, the legitimist heir. It was not a success: the prince was uninterested, and Cauchy was not a gifted teacher of thirteen-year-olds. Cauchy’s wife and daughters joined him in 1834. In Prague he met Bernard Bolzano, whose ideas on rigour were close to his own, and wrote his memoir on the dispersion of light (Lesson 09).

Return, last years, legacy, 1838–1857

Cauchy returned to Paris late in 1838 and resumed his place in the Academy, but without the oath he could not teach in state institutions. In November 1839 the Bureau des longitudes elected him; the government refused to confirm the election, and he was replaced in 1843. The same year the Collège de France preferred Guglielmo Libri to him for a chair. He kept writing. When the weekly Comptes rendus of the Academy began in 1835 he filled it with so many notes that a page limit was soon imposed on contributions. Between 1840 and 1847 he published the four volumes of the Exercices d’analyse et de physique mathématique, including his group theory of 1844–46.

The Revolution of 1848 abolished the oath. On 1 March 1849 Cauchy became professor of mathematical astronomy at the Faculté des sciences of Paris. In 1850 he lost an election to the Collège de France to Joseph Liouville, which strained their relations, and his last years included a priority dispute with Jean-Marie Duhamel over the theory of inelastic collisions in which he was in the wrong. He died at Sceaux on 23 May 1857, of a bronchial illness, aged 67. His collected works, the Œuvres complètes, fill 27 volumes, the first published in 1882.

Legacy

• Analysis: definitions of limit, continuity, convergence and integral that, sharpened by Weierstrass, became the standard of rigour. Cauchy sequences define completeness in every metric space.
• Complex analysis: the integral theorem, residues, the integral formula and power series; Riemann and Weierstrass built function theory on these foundations.
• Differential equations: existence theorems and the method of majorants, extended in the Cauchy–Kovalevskaya theorem; the initial value problem is still called the Cauchy problem.
• Algebra: determinants, permutations and the theorem on elements of prime order, a forerunner of Sylow’s theorems (1872).
• Mechanics and physics: the stress tensor, strain, and the equations of linear elasticity, the language of solid mechanics, seismology and fluid dynamics; the dispersion formula of optics.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Augustin-Louis Cauchy, MacTutor History of Mathematics.
2. Augustin-Louis Cauchy, Wikipedia (with extensive references).
3. B. Belhoste, Augustin-Louis Cauchy: A Biography, Springer, 1991.
4. C.-A. Valson, La vie et les travaux du baron Cauchy, Gauthier-Villars, 1868.

Mathematics and physics

1. A.-L. Cauchy, Cours d’analyse de l’École royale polytechnique, 1821. English translation by R. E. Bradley and C. E. Sandifer, Springer, 2009.
2. A.-L. Cauchy, Œuvres complètes, 27 vols., Gauthier-Villars, from 1882; scans on Gallica.
3. J. V. Grabiner, The Origins of Cauchy’s Rigorous Calculus, MIT Press, 1981.
4. F. Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge, 1997.
5. O. Darrigol, Between hydrodynamics and elasticity theory: the first five births of the Navier–Stokes equation, Archive for History of Exact Sciences 56 (2002).
6. H. Maleki Moghadam et al., Cauchy tetrahedron argument and the proofs of the existence of stress tensor, arXiv:1706.08518.
7. C. Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638–1788 and Essays in the History of Mechanics, Springer, 1960 and 1968.
8. J. Z. Buchwald, Cauchy’s theory of dispersion anticipated by Fresnel, in Archimedes 30, Springer, 2011.
9. J. H. McKay, Another proof of Cauchy’s group theorem, American Mathematical Monthly 66 (1959), 119.
10. M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer (chapter on Cauchy’s rigidity theorem).
11. M. B. Nathanson, A short proof of Cauchy’s polygonal number theorem, Proc. AMS 99 (1987).