Augustin-Louis Cauchy: Difference between revisions
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==The works of Cauchy== | ==The works of Cauchy== | ||
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and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions. | and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions. | ||
In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves | In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves | ||
In 1817 Scotland Cauchy filled a temporary post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he defined for the first time a complex function of a complex variable. | In 1817 Scotland Cauchy filled a temporary post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he defined for the first time a complex function of a complex variable. | ||
His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced. These are mainly embodied in his three great treatises, Cours d'analyse de l'École Polytechnique (1821); Le Calcul infinitésimal (1823); Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828); and also in his Courses of mechanics (for the École Polytechnique), Higher algebra (for the Faculté des Sciences), and of Mathematical physics (for the Collège de France). | |||
He wrote numerous treatises and made 789 contributions to scientific journals. These writings covered notable topics including the theory of series (where he developed with perspicuous skill the notion of convergency), the theory of numbers and complex quantities, the theory of groups and substitutions, and the theory of functions, differential equations and determinants. He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. He also contributed significant research in mechanics, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. In optics, he developed the wave theory, and his name is associated with the simple dispersion formula. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Simeon Poisson. | |||
Other significant contributions include being the first to prove the Fermat polygonal number theorem. He created the residue theorem and used it to derive a whole host of most interesting series and integral formulas and was the first to define complex numbers as pairs of real numbers. He also discovered many of the basic formulas in the theory of q-series. His collected works, Œuvres complètes d'Augustin Cauchy, have been published in 27 volumes. | |||
During this period (after 1838) Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique published between 1840 and 1847 proved extremely important. | During this period (after 1838) Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique published between 1840 and 1847 proved extremely important. | ||
Revision as of 04:09, 3 November 2007
Augustin-Louis Cauchy (Paris, August 21 1789 – Sceaux, May 23, 1857) was one of the most prominent mathematicians of the first half of the nineteenth century. He was the first to give a rigorous basis to the concept of limits. He established a convergence criterion for sequences of the type that are now called Cauchy sequences. The Cauchy condition for the convergence of series can be found in any present-day textbook on calculus. Probably Cauchy is most famous for his singlehanded development of complex function theory, with Cauchy's residue theorem as the fundamental result.
Cauchy was a prolific writer, he wrote approximately 800 research articles and five complete textbooks. He was a devout Roman Catholic, strict (Bourbon) royalist, and a close associate of the Jesuit order.
Biography
Youth and education
Cauchy's father (Louis-François Cauchy) was a high official in the Parisian Police of the Old Régime. He lost his position because of the French Revolution (July 14, 1789) that broke out one month before Augustin-Louis was born. This fact is sometimes seen as the cause of the deep hatred of the French Revolution that Cauchy felt all through his life. The Cauchy family survived the revolution and the following Reign of Terror (1794) by escaping to Arcueil, where Cauchy jr. got his first education from his father. After the death of Robespierre (1794) it was safe for the family to return to Paris, where Cauchy sr. found himself a new bureaucratic job and where he quickly moved up the ranks. When Napoleon Bonaparte came to power (1799) Louis-François Cauchy made further promotion and became Secretary-General of the Senate working directly under Laplace, who is now better known for his work on mathematical physics. Also the famous mathematician Lagrange was no stranger to the Cauchy family.
On Lagrange's advice Augustin-Louis was enrolled in the École Central du Panthéon in the fall of 1802. This was the best secondary school of Paris at that time. Most of the curriculum consisted of classical languages and the young ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of these successes, Augustin-Louis decided for an engineering career and prepared himself for the entrance examination to the École Polytechnique. In 1805 he became second out of 293 applicants on this exam, and was, of course, admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young pious Cauchy some problems in adapting. Nevertheless, he finished the Polytechnique in 1807 at the age of 18 and went on to the École des Ponts et Chaussées (School for Bridges and Highways). He graduated in civil engineering with the highest honors.
Engineering days
After finishing school in 1810 Cauchy accepted a job as junior engineer in Cherbourg, where Napoleon intended to build a naval port. Here Augustin-Louis stayed three years and although he had an extremely busy managerial type job, he still found time to prepare three mathematical manuscripts, which he submitted to the Première Classe of the Institut de France. (In the revolutionary years the French Académie des Sciences was known as the "First Class" of the Institut de France.) Cauchy's first two manuscripts (on polyhedra) were accepted, the third one (on directrixes of conic sections) was rejected.
In September 1812, 23 years old, Cauchy returned to Paris after becoming ill from being overworked. Another reason for his return to the capital was that he was losing his interest in his engineering job, being more and more attracted to abstract beauty of mathematics. In Paris he would have a much better chance to find a mathematics related position. Formally he kept his engineering position, although he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior. The next three years Augustin-Louis was mainly on unpaid sick leave, and spent his time quit fruitfully on mathematics (on the related topics of symmetric functions, the symmetric group and the theory of higer-order algebraic equations). He tried to be admitted to the First Class of the Institut, but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo and the newly installed Bourbon king Louis XVIII (a brother of the beheaded Louis XVI) took the restoration in hand. The Académie des Sciences was re-established in March 1816, Lazare Carnot and Gaspard Monge were removed from this Academy for political reasons and the king appointed Cauchy to take the place of one of them. Judgement by Cauchy's peers was harsh, they considered his acceptance of membership of the Academy an outrage and Cauchy created many enemies in scientific circles.
Professor at École Polytechnique
In November 1815, Louis Poinsot, who was an associate professor at the École Polytechnique, asked to be exempted from his teaching duties because of health reasons. Cauchy was by then a rising mathematical star who certainly merited a professorship. One of his great successes at that time was the proof of Fermat's polygonal number theorem. However, the fact that Cauchy was known to be very loyal to the Bourbons, doubtlessly helped him in becoming the successor of Poinsot. So, Augustin-Louis finally quit his engineering job and received a one-year contract for teaching math to second-year students of the École Polytechnique. In 1816 this Bonapartist, a-religious school was reorganized and several liberal professors were fired. The reactionary Cauchy was promoted to full professor.
When Cauchy was twenty eight years old he was still living with his parents. By then Cauchy sr. found it high time that his son would marry. He found him a nice bride, Aloïse de Bure, five years his junior. They married on April 4, 1818 with great Roman Catholic pomp and ceremony in the Church of Saint-Sulpice. In 1819 the couple's first daugther, Marie Françoise Alicia, was born and in 1823 the second and last daughter, Marie Mathilde. It appears that Cauchy gave no important place to his family in his life, his work had higher priority.
The oppressive political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died and was succeeded by his even more reactionary brother Charles X. During these years Cauchy was highly productive and published one important mathematical treatise after the other. He received cross appointments at the Collège de France and the Faculté des Sciences of the University.
In exile
In July 1830 France had another revolution. Charles X fled the country and was succeeded by the non-Bourbon king Louis-Philippe (from the house of Orléans). Riots, in which uniformed students of the École Polytechnique took active part, raged close to Cauchy's home in Paris. These events marked a turning point in Cauchy's life and a break in his mathematical productivity. Cauchy, shaken by the fall of the government and moved by a deep hatred of the liberals that were taking power, left Paris to go abroad, leaving his family behind him. He spent a short time in Switzerland where he had to decide whether he would swear a required oath of allegiance to the new regime. He refused to do this and consequently lost all his positions in Paris, except his membership of the Academy for which an oath was not required. In 1831 Cauchy went to the Italian city of Turin, and after some time there, he accepted an offer from the King of Sardinia (who ruled Turin and the surrounding Piedmont region) of a chair of theoretical physics, created especially for him. He taught in Turin in 1832 and 1833.
In August 1833 Cauchy left Turin for Prague to become the science tutor of the thirteen year old Duke of Bordeaux, the exiled Crown Prince and grandson of Charles X. As a professor of the École Polytechnique Cauchy had been a notoriously bad lecturer, assuming levels of understanding that only a few of his best students could reach and cramming his allotted time with much too much material. The young Duke had taste nor talent for mathematics or science. Although Cauchy took his mission very seriously, he did this with great clumsiness and with surprising lack of authority over the Bourbon Prince.
During his civil engineering days Cauchy once had been briefly in charge of repairing a few of the Parisian sewers, and he made the mistake of telling his pupil this, who with great malice went about saying that Mister Cauchy started his career in the sewers of Paris. The tutorship lasted until the Prince became 18 years old in September 1838. Cauchy did hardly any research during those five years, while the Prince acquired a life-long dislike of mathematics. The only good that came out of this episode was Cauchy's promotion to Baron, a title that Cauchy was very attached to. In 1834 his wife and two daughters moved to Prague and Cauchy was finally reunited with his family after four years of exile.
Last years
Cauchy returned to Paris and his position at the Academy of Sciences late 1838. He could not regain his teaching positions, because he still refused to swear an oath of allegiance. Yet, he desperately wanted to regain a formal position in Parisian science. In August 1839 a vacancy appeared in the Bureau des Longitudes. This Bureau had some resemblance with the Academy, for instance, it had the right to coopt its members. Further, it was believed that members of the Bureau could "forget" about the oath of allegiance, although formally, unlike the Academicians, they were obliged to take it. The Bureau des Longitudes was an organization founded in 1795 to solve the problem of determining position on sea (mainly the longitudinal coordinate, latitude is easily determined from the motion of the sun). Since it was thought that position on sea was best determined by astronomical observations, the Bureau had developed into an organization resembling an academy of astronomical sciences. In November 1839 Cauchy was elected to the Bureau and discovered immediately that the matter of the oath was not easily dispensed with. Without his oath the king refused to approve his election. During four years Cauchy was in the absurd position of being elected but not being approved. Hence he was not a formal member of the Bureau, did not receive payment, could not participate in meetings and not submit papers. Still Cauchy refused to take any oaths. However, he felt loyal enough to direct his research to celestial mechanics. In 1840 he presented a dozen papers on this topic to the Academy. The absurd membership of the Bureau lasted until the end of 1843 when Cauchy was finally replaced by Poinsot.
All through the nineteenth century the French educational system struggled with the separation of Church and State. The Catholic Church strived for freedom of education, that is, the right to establish Catholic schools. The Church found in Cauchy an staunch and famous ally. He was lending his prestige and knowledge to the École Normale Écclésiastique, a school in Paris run by Jesuits for training teachers for their colleges. He also took part in the founding of the Institut Catholique. The purpose of this institute was to soften the effects of the absence of Catholic university education in France. These activities did not make Cauchy popular with his colleagues who, on the whole, supported the enlightenment ideals of the French Revolution. When in 1843 a chair of mathematics became vacant at the Collège de France, Cauchy applied and got three out of forty five votes.
The year 1848 was the year of revolution, all over Europe revolutions broke out, beginning in France. King Louis-Philippe, fearful of the fate of Louis XVI, fled to England. The oath of allegiance was abolished and the road to an academic appointment was clear for Cauchy. On March 1, 1849 he was reinstated at the Faculté de Sciences as a professor of mathematical astrononomy. After political turmoil all through 1848, France chose to become a Republic under presidency of Louis Napoleon Bonaparte, son of the first king of Holland and nephew of Napoleon Bonaparte. Soon (early 1852) the president was promoted to Emperor of France and accepted the name Napoleon III. Not quite unexpectedly, the idea came up in bureaucratic circles that it would be useful to require a loyalty oath from all state functionaries including university professors. Not always does history repeat itself, however, because this time a cabinet minister was able to convince the Emperor to exempt Cauchy from the oath. Cauchy remained a professor at the University until his death at the age of sixty seven. He received the Last Sacraments and died 4 o'clock a.m. during the night of May 23, 1857.
The works of Cauchy
(To be continued)
Reference
Bruno Belhoste, Augustin-Louis Cauchy: a biography, translated from the French by F. Ragland, Springer, New York (1991). ISBN 0-387-97220-X
External links
- Biography at MacTutor History of Mathematics, John J. O'Connor and Edmund F. Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland.