## 9.6. Cauchy, Augustin (1789-1857)

Augustin Cauchy was the mathematician that set the foundation of rigor in modern analysis. A product of the revolutions in France during the eighteenth and nineteenth centuries, he provided the revolutionary ideas that set this branch of mathematics on its present course. He is also known as being one of the most prolific writers in the history of the science.Augustin Louis Cauchy was born in Paris, France, on August 21, 1789, only a month after the storming of the Bastille. His father, a government official and staunch royalist, recognized the coming revolution and quickly moved his family to a country cottage in Arcueil. Having escaped the guillotine, the family was poor and the young boy was generally malnourished. For the rest of his life, this early poverty left the future mathematician in a state of ill-health. During his eleven year stay at the cottage, Augustin received a classical education and a strong disposition for the monarchy from his father, who wrote his own textbooks in verse, and strict Catholic religious training from his mother. This training would influence the rest of his life. His zealous political and religious beliefs would alienate this great mathematician from the majority of his countrymen.

In 1800, the Cauchy's family returned to Paris after the political situation stabilized. During this early period in his life, Augustin's talent was recognized by two great mathematicians, Marquis Laplace and Joseph Lagrange. Both, after seeing the young boy's work, encouraged him to continue in mathematics. As Lagrange once predicted, he would eventually outdo both of them. Cauchy's education, however, was in engineering. After attending the Ecole Polytechnique, a military engineering school taught by some of the country's greatest mathematicians, and the Ecole des Ponts et Chaussees, he took a position as an engineer in Napoleon's army at Cherbourg. Somehow during his busy schedule, he found time to dabble in mathematics. During his three years there, he produced several significant mathematical papers, including one on determinants that gave the term its modern meaning. All this mathematical output also accomplished to ruin his health and ended his career in military engineering.

Now with all his efforts focused on mathematics, Cauchy became a star on the mathematics scene. After a slew of achievements, he became a professor at the Polytechnique in 1816. Then at the unheard of age of 27, he was elected to the Academy of Sciences in Paris. To be selected for the Academy was one of the highest honors that could be given to a scientist. Unfortunately, his acceptance of the position was filled with controversy. At the time, Napoleon had just been overthrown and the Bourbons had been returned to the throne. The new king immediately went about removing the former emperor's supporters including Gaspard Monge, a member of the Academy. Despite his political views, Monge was one of the greatest mathematicians in France and his removal was considered an outrage. However, when Cauchy was offered his seat, he accepted without reserve. Being a staunch royalist, he saw nothing wrong with the removal of an enemy of the king and saw it as his duty to the monarchy to take the position. This action did not bode well with many of countrymen and made him many enemies. Nevertheless, the mathematician continued his work and somehow found time to be married two years later.

This would not be the last time his political views would get him into trouble. In 1830 after the overthrow of King Charles X, all members of the Academy were obligated to swear an oath of allegiance to the new king. Having already taken an oath to Charles, Cauchy refused. He was removed from his position and self-exiled to Switzerland without his family. There he became a professor at the University of Turin and planned to spend the rest of his life working on mathematics. That was not to be. Two years later, Charles X, now in exile, asked the professor to supervise the education of his heir Henri. Being a good royalist, he agreed and was joined with his family in Vienna. His new duties overwhelmed him and his mathematical work lessened to a trickle. He found his escape in 1838 when he returned to Paris. Before he left, the king had given him the impressive sounding but practically useless title of baron. He still refused to take the oath and constantly struggled to find and hold a position. Finally in 1848, the oath was abolished and he resumed his old posts. Recognizing his value to Academy, he was exempted when the oath was reestablished in 1852.

Augustin Cauchy died on May 23, 1857, after contracting a fever on a trip to the country to help restore his health. His last words were, "Men die but their works endure."

Cauchy's life was one as unusual and complicated as the times he lived in. Brought up as a devout Catholic in a time most Frenchmen were opposed to the Church, he suffered prejudice from many people. However, the discrimination did not discourage him from engaging in his life's favorite hobby, charity. When he was not involved in some math problem, he was often working on some new mercy mission for the less fortunate. On the other hand, he could be bigoted against those who did not hold his religious views. For example, part of the reason Cauchy delayed the publication of fellow mathematician Niels Abel's masterpiece was because the latter called him a "bigoted Catholic."

He also tended to be just as opinionated in matters of politics. A supporter of the monarchy, he came into direct conflict with the supporters of both the republic and Napoleon. Again, he was both discriminated against and prejudiced against others. On one hand, his life was put into constant turmoil because of the affair with the oath. On the other hand, he helped repress the mathematical work of Evariste Galois because the latter was a radical republican. Certainly, Cauchy led a very complicated and intricate life.

Cauchy is famous in the field of mathematics for two main reasons: his numerous contributions to the science and his immense publishing. His works spanned every branch of mathematics and are simply too long to list. He is especially famous for his works with convergent series and rigor in analysis. Early in his career, Cauchy developed the criteria for determining if an infinite series is convergent or divergent. While attending a lecture on the subject, it is told that Laplace became panicked and rushed home. He had just finished his masterpiece that used infinite series as its backbone and desperately checked each one for convergence, which they did. Cauchy second great contribution was setting the groundwork for rigor in analysis and all of mathematics. Rigor is discovery of the logical foundations of a science. Over the previous centuries, mathematicians had tried in vain to discover what were the underlying principles of calculus and many had asserted that Newton's discovery was flawed. Cauchy took the first step toward unifying the science. First, he defined continuity and derivative in terms of the limit. Second, he gave the first good definition of the limit as : "When the values successively assigned to the same variable indefinitely approach a fixed value, so as to end by differing from it as little as desired, this fixed value is called the limit of all the others." Though this is not a mathematical definition, it is a good approximation of the idea, which would be further clarified by future mathematicians. Other important works include determinants, polygonal numbers, complex numbers and the theory of substitutions.

Cauchy is also famous for his writings. Simply put, he overwhelmed the mathematics world with the number and size of his works. All in all, his total output included 789 full length papers, one of the largest outputs ever. It was not uncommon for him to finish two such papers in one week. In addition, these works tended to be rather long, sometimes extending for over 300 pages. In fact, after submitting several large papers to be published in the weekly bulletin, the Academy was forced to limit submissions to four pages to save their small budget from Cauchy's pen. However, all this writing did get his work out into the public and spread his ideas. A lot of his fame can be assessed to the fact that he simply overwhelmed all his competitors on the bookshelves. Because of this fact, his name is prominent in almost any analysis textbook.

For related information on Cauchy, see: Niels Abel , Cauchy condensation test for series, Cauchy criteria, Cauchy criterion for series, Cauchy sequence, Cauchy sequence and convergence.

**Sources**

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- Bell, E.T.
*Men of Mathematics.*New York: Simon and Schuster, Inc., 1937. - Wilson, John H. "Cauchy."
*Encyclopedia of World Biography.*New York: McGraw-Hill, Inc., 1973. vol. 2, pp. 433-435.

**Paul Golba**