Augustin-Louis Cauchy facts for kids

Quick facts for kids Augustin-Louis Cauchy
Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.
Born	(1789-08-21)21 August 1789 Paris, France
Died	23 May 1857(1857-05-23) (aged 67) Sceaux, France
Nationality	French
Alma mater	École Nationale des Ponts et Chaussées
Known for	Continuum mechanics Mathematical analysis Gradient descent Implicit function theorem Intermediate value theorem Spectral theorem Limit (mathematics) See full list
Spouse(s)	Aloise de Bure
Children	Marie Françoise Alicia, Marie Mathilde
Awards	Grand Prize of L'Académie Royale des Sciences
Scientific career
Fields	Mathematics, Physics
Institutions	École Centrale du Panthéon École Nationale des Ponts et Chaussées École Polytechnique
Doctoral students	Francesco Faà di Bruno Viktor Bunyakovsky

Augustin-Louis Cauchy (born August 21, 1789 – died May 23, 1857) was a brilliant French mathematician, engineer, and physicist. He made huge contributions to many areas of mathematics, especially mathematical analysis and continuum mechanics.

Cauchy was one of the first to clearly state and prove important ideas in calculus. He also almost single-handedly started the study of complex analysis (math with imaginary numbers) and permutation groups (how things can be rearranged). He was a very influential mathematician, and many ideas and theorems are named after him. He wrote about 800 research papers and five textbooks!

Biography

Early Life and School

Augustin-Louis Cauchy was born in Paris, France, in 1789. His father, Louis François Cauchy, was a high-ranking official in the Parisian Police. However, the French Revolution started just a month before Augustin-Louis was born, and his family had to leave Paris to stay safe. They lived in a small town called Arcueil, where his father taught him at home.

After the revolution settled down, the family returned to Paris. Cauchy's father got a new job and quickly moved up in the government. He even worked directly with famous mathematicians like Pierre-Simon Laplace and was friends with Joseph-Louis Lagrange.

Following Lagrange's advice, Augustin-Louis went to the best high school in Paris, the École Centrale du Panthéon, in 1802. He was a very smart student and won many awards in subjects like Latin. Even with his success in languages, he decided to become an engineer.

In 1805, he got into the École Polytechnique, a top school for future engineers. He finished there in 1807 and then went on to the École des Ponts et Chaussées (School for Bridges and Roads), where he graduated with the highest honors in civil engineering.

Working as an Engineer

After finishing school in 1810, Cauchy started working as a junior engineer in Cherbourg, where Napoleon wanted to build a naval base. He worked on big projects like canals and bridges for three years. Even though he was very busy, he still found time to write mathematical papers.

In 1812, Cauchy got sick from working too much and returned to Paris. He was also becoming more interested in the beauty of mathematics than in engineering. When he got better, he decided not to go back to Cherbourg. He formally kept his engineering job but spent most of his time working on mathematics.

He tried to join the French Academy of Sciences several times but didn't succeed at first. However, in 1815, after Napoleon was defeated, the new king, Louis XVIII, appointed Cauchy to the Academy. This upset some of his peers, who felt he got the position for political reasons.

Becoming a Professor

In 1815, Cauchy became a professor at the École Polytechnique, teaching mathematics. He was a rising star in math, especially after proving Fermat's polygonal number theorem. He finally left his engineering job to focus entirely on teaching and research.

In 1818, Cauchy married Aloise de Bure, whose family published many of his works. They had two daughters, Marie Françoise Alicia and Marie Mathilde.

During this time, Cauchy was incredibly productive, publishing many important mathematical works. He also got teaching positions at other famous schools like the Collège de France.

Time in Exile

In 1830, another revolution happened in France, and King Charles X was overthrown. Cauchy, who was very loyal to the old government, left Paris. He refused to swear loyalty to the new king, Louis-Philippe, which meant he lost all his teaching jobs in Paris. He kept his membership in the Academy, though, as no oath was required for that.

He went to Turin, Italy, where the King of Sardinia offered him a special professorship in theoretical physics. He taught there from 1832 to 1833.

In 1833, Cauchy moved to Prague to become the science tutor for the young Duke of Bordeaux, Henri d'Artois, who was the grandson of the former king. Cauchy was known for being a difficult lecturer, often assuming his students knew more than they did. The Duke wasn't interested in math or science, so it was a tough match. Cauchy took his job seriously, but the Duke didn't learn much and grew to dislike mathematics. This period lasted five years, and Cauchy did very little research during this time. The only good thing was that he was given the title of baron, which he valued greatly.

In 1834, his wife and daughters joined him in Prague, and the family was finally together again.

Later Years and Return to Paris

Cauchy returned to Paris in late 1838. He still refused to swear an oath of loyalty, so he couldn't get his teaching jobs back.

In 1839, he was elected to the Bureau des Longitudes, an organization focused on astronomy and navigation. It was thought that members of this Bureau might not need to take the oath. However, the king still refused to approve his election without it. For four years, Cauchy was elected but not approved, meaning he didn't get paid or participate in meetings. Still, he refused to take the oath. He focused his research on celestial mechanics (the study of how planets and stars move).

Cauchy was a very religious Catholic and supported the Catholic Church's efforts to create its own schools in France. This made him unpopular with many of his colleagues, who supported the ideas of the Age of Enlightenment.

In 1848, revolutions broke out across Europe, including France. King Louis-Philippe fled, and the oath of allegiance was abolished. This finally cleared the way for Cauchy to get his academic positions back. On March 1, 1849, he was reinstated as a professor of mathematical astronomy at the Faculté de Sciences.

Soon after, Louis Napoleon Bonaparte became Emperor. The idea of requiring loyalty oaths came up again, but a government minister convinced the Emperor to excuse Cauchy from it. Cauchy remained a professor until he died in 1857 at the age of 67.

His name is one of the 72 names carved on the Eiffel Tower in Paris.

Work

Early Discoveries

Cauchy's amazing talent was clear from his early work. In 1805, he found a simple way to solve the problem of Apollonius (how to draw a circle that touches three other circles). In 1811, he expanded on Euler's formula for polyhedra (3D shapes with flat faces).

He also wrote an important paper on how waves travel, which won a major prize from the French Academy of Sciences in 1816.

Waves, Mechanics, and Elasticity

Cauchy worked on the wave theory of light, studying how light spreads and changes direction. He also made contributions to mechanics, which is the study of how things move and interact. He looked at how rods and elastic materials behave and how waves travel through them.

He introduced a special 3x3 grid of numbers, now called the Cauchy stress tensor, which helps describe how forces are distributed within a material. His work on stress in elastic materials is still very important today.

Number Theory

Cauchy was the first to prove the Fermat polygonal number theorem, which is a famous idea in number theory.

Complex Functions

Cauchy is most famous for developing complex function theory. This is a branch of mathematics that deals with numbers that include the imaginary unit i (where i squared equals -1).

One of his most important discoveries is Cauchy's integral theorem, which states that if a function is "well-behaved" (called holomorphic) inside a closed loop, then the integral (a type of sum) around that loop is zero. He first presented ideas for this theorem in 1814, and the full version came out in 1825. Many people see this 1825 paper as his most important work.

In 1826, Cauchy defined the idea of a "residue" for a function. This helps understand functions that have "poles" (points where the function goes to infinity).

In 1831, he presented two more key ideas:

• Cauchy's integral formula: This formula connects the value of a function at a point inside a loop to the integral of the function around that loop.
• Residue theorem: This theorem helps calculate integrals around a loop by summing up the residues of the function's poles inside the loop.

These ideas are still central to complex function theory and are taught to students in physics and engineering today.

Cours d'Analyse

In his book Cours d'Analyse (Course on Analysis), Cauchy emphasized the importance of being very precise and logical in mathematics. He was known for bringing "rigor" (strictness and accuracy) to analysis. This book is often noted as the first place where inequalities and the famous "epsilon-delta" arguments were used in Calculus.

Cauchy defined continuity in a clear way: a function is continuous if a tiny change in the input always leads to a tiny change in the output.

Taylor's Theorem

He was the first to prove Taylor's theorem in a strict way, which helps approximate functions using polynomials. He wrote textbooks for his students at the École Polytechnique, where he explained the basic ideas of mathematical analysis as carefully as possible. In one of these books, he gave the necessary conditions for a limit to exist, which is still taught today. His well-known test for absolute convergence of series also comes from this book.

Argument Principle and Stability

In a paper published in 1855, Cauchy discussed theorems similar to the "Principle of the argument" used in modern complex analysis. This principle is very important in control theory, helping engineers predict the stability of systems like feedback amplifiers. This shows how Cauchy's work impacts both pure math and practical engineering.

Published works

Cauchy was incredibly productive, writing more papers than almost any other mathematician except Leonhard Euler. It took nearly a century to gather all his writings into 27 large books.

His most important contributions to mathematics are found in the strict methods he introduced, mainly in these three major books:

• Cours d'analyse de l'École royale polytechnique (1821)
• Le Calcul infinitésimal (1823)
• Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828)

Some of his other works include:

• Mémoire sur les intégrales définies, prises entre des limites imaginaires (1825)
• Exercices de mathematiques (1826, 1827)
• Leçons sur le calcul différentiel (1829)
• Sur la mecanique celeste et sur un nouveau calcul qui s'applique a un grand nombre de questions diverses etc (1831)
• Exercices d'analyse et de physique mathematique (Volumes 1-4, 1840–1847)
• Analyse algèbrique (1821)
• Nouveaux exercices de mathématiques (1895)

Politics and Religious Beliefs

Augustin-Louis Cauchy grew up in a family that strongly supported the king. His father had to flee Paris during the French Revolution to keep his family safe. Life was tough then, with little food. This experience shaped Cauchy's strong loyalty to the monarchy. He later refused to swear loyalty to any government that came to power after the king was overthrown.

He was also a very devoted Catholic and was involved with religious groups like the Society of Saint Vincent de Paul. He even defended the Society of Jesus (Jesuits) at the Academy, even when it was unpopular to do so. His faith also led him to help Charles Hermite, another mathematician, during an illness, and he encouraged Hermite to become a faithful Catholic. He also spoke out to help the Irish people during the Great Famine.

His strong political and religious views sometimes caused problems with his colleagues. He felt he was treated unfairly because of his beliefs. However, others felt he sometimes provoked people by arguing about religion or defending controversial groups. For example, another mathematician, Niels Henrik Abel, called him a "bigoted Catholic" but also praised his mathematical genius.

When a math position opened up, Cauchy was passed over for someone else, and many believed it was because of his strong views. This led to disagreements with other mathematicians.

Images for kids

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  Cauchy in later life

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  The title page of a textbook by Cauchy.

See also