# Mathematician:Augustin Louis Cauchy

## Mathematician

French Engineer and mathematician, from a suburb of Paris, which at the time was home to many leading mathematicians.

Wrote seven books and more than 700 papers in various fields of mathematics.

Made significant contributions in number theory, the theory of determinants, eigenvalues, ordinary and partial differential equations, permutation groups, and the foundation of calculus.

Famous for founding the theory of functions of a complex variable.

Argued by some as the *founder of group theory*.

A devout Roman Catholic, also strongly devoted to the Bourbon kings who ruled France after Napoleon's defeat. When Charles X was exiled in $1830$, Cauchy willingly followed the former king into exile in Prague.

Some make much of what are perceived as his personal shortcomings with respect to his religious beliefs, political leanings and moral attitudes, but it is at least as equally often suggested that they are ultimately unimportant in comparison with to his legacy.

## Nationality

French

## History

- Born: 21 Aug 1789, Paris, France
- 1805: Entered the École Polytechnique
- 1807: Graduated from the École Polytechnique, entered the École des Ponts et Chaussées
- 1810: Moved to Cherbourg: worked on port facilities for Napoleon's English invasion fleet
- 1811: Proved that the angles of a convex polyhedron are determined by its faces
- September 1812: Returned to Paris suffering from depression
- 1815: Appointed assistant professor of analysis at the École Polytechnique
- 1816: Won the Grand Prix of the French Academy of Sciences for a work on waves
- 1817: Replaced Jean-Baptiste Biot at the Collège de France
- September 1830: Left Paris after the July revolution, and spent a short time in Switzerland where he helped to set up the Académie Helvétique
- 1831: Went to Turin, accepted an offer from the King of Piedmont of a chair of theoretical physics, where he taught from 1832
- 1833: To Prague, in order to follow Charles X and to tutor his grandson (with not much success)
- 1838: Returned to Paris and regained his position at the Academy, but not his teaching positions because he had refused to take an oath of allegiance
- 1848: Regained his university positions on overthrow of Louis Philippe
- Died: 23 May 1857, Sceaux (near Paris), France

## Theorems and Definitions

- Cauchy Criterion
- Cauchy Determinant
- Cauchy Distribution
- Cauchy Equivalent
- Cauchy Form of Remainder of Taylor Series
- Cauchy Horizon
- Cauchy Matrix
- Cauchy Principal Value
- Cauchy Product
- Cauchy Sequence
- Cauchy Surface

- Cauchy-Crofton Formula (with Morgan William Crofton)
- Cauchy-Euler Equation (with Leonhard Paul Euler)
- Cauchy-Green Tensor (with George Green)
- Cauchy-Riemann Equations (with Bernhard Riemann)
- Navier-Cauchy Equations (with Claude-Louis Navier)

- Cauchy Argument Principle
- Cauchy Boundary Condition
- Cauchy Condensation Test
- Cauchy's Convergence Criterion
- Cauchy's Convergence Test (or Cauchy Convergence Test, or Cauchy's Test) (also known as the Nth Root Test or just the Root Test)
- Cauchy's Equation
- Cauchy Formula for Repeated Integration
- Cauchy Functional Equation
- Cauchy's Group Theorem
- Cauchy's Lemma (Group Theory) (on groups with elements of prime order)
- Cauchy's Lemma (Number Theory) (on representation of integers as sums of $4$ integers or squares)
- Cauchy's Inequality
- Cauchy's Integral Formula
- Cauchy Integral Test
- Cauchy's Mean Theorem (also known as Cauchy's Formula)
- Cauchy Mean Value Theorem
- Cauchy Momentum Equation
- Cauchy Problem
- Cauchy's Radical Test
- Cauchy Ratio Test (also known as D'Alembert's Test, for Jean le Rond d'Alembert, or just the Ratio Test)
- Cauchy's Theorem (Geometry)

- An elegant proof of what is now called the Nyquist Stability Criterion.

- Binet-Cauchy Identity (with Jacques Philippe Marie Binet) (also known as
**Binet's Formula**) - Cauchy-Bunyakovsky-Schwarz Inequality (with Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz)
- Cauchy-Binet Formula (with Jacques Philippe Marie Binet) (also known, confusingly, as the
**Binet-Cauchy Identity**) - Cauchy-Frobenius Lemma (with Ferdinand Georg Frobenius)
- Cauchy-Goursat Theorem (with Édouard Jean-Baptiste Goursat) (also known as the
**Cauchy Integral Theorem**) - Cauchy-Hadamard Theorem (with Jacques Salomon Hadamard)
- Cauchy-Kovalevsky Theorem (with Sofia Vasilyevna Kovalevskaya)
- Cauchy-Lipschitz Theorem (with Rudolf Otto Sigismund Lipschitz)
- Cauchy-Peano Theorem (with Giuseppe Peano)
- Maclaurin-Cauchy Test (with Colin Maclaurin)

Results named for **Augustin Louis Cauchy** can be found here.

Definitions of concepts named for **Augustin Louis Cauchy** can be found here.

## Publications

- 1815:
*Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions operees entre les variables qu'elles renferment*(*J. l'École Polytechnique***Vol. 10**: pp. 29 – 112)

- 1821:
*Cours d'analyse* - 1823:
*Le Calcul infinitésimal* - 1826:
*Sur un nouveau genre de calcul analogue au calcul infinitésimal* - 1829:
*Leçons sur le Calcul Différentiel* - 1840-47:
*Exercices d'analyse et de physique mathématique*

- 1843:
*Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire, et sur diverses transformations de produits composés d'un nombre indéfini des facteurs*(*Comptes rendus de l'Académie des Sciences***Vol. 17**: pp. 523 – 531)

## Critical View

*His scientific production was enormous. For long periods he appeared before the Academy once a week to present a new paper, so that the Academy, largely on his account, was obliged to introduce a rule restricting the number of articles a member could request to be published a year.*- -- Øystein Ore

## Also known as

Some sources render his name as **Augustin-Louis Cauchy**.

## Sources

- John J. O'Connor and Edmund F. Robertson: "Augustin Louis Cauchy": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{XV}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 55$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Cauchy, Augustin Louis, Baron**(1789-1857) - 1991: David Wells:
*Curious and Interesting Geometry*... (previous) ... (next): A Chronological List Of Mathematicians - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($\text {1789}$ – $\text {1857}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Cauchy, Augustin Louis, Baron**(1789-1857) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Cauchy, Augustin Louis, Baron**(1789-1857) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Cauchy, Augustin Louis**(1789-1857)