Augustin Louis Cauchy

Augustin Louis, baron Cauchy , born with Paris on August 21st, 1789 and died in Seals (Hauts-de-Seine) on May 23rd, 1857, is a Mathématicien French, member of the Academy of Science and professor with the Polytechnic school. Enthusiastic catholic, he is the founder many Christian works, of which the Work of the Schools of the East. Royalist legitimist, it voluntarily exiled himself at the time of the advent of Louis-Philippe, after the Glorious Three. Its political position and ecclesiastic were worth many oppositions to him.

It was one of the most prolific mathematicians, behind Leonhard Euler, with nearly 800 publications and seven works; its research covers the whole of the mathematical fields of the time. One owes analyzes of it him in particular the introduction of the holomorphic functions and the criteria of convergence of the series S and the whole series. Its work on the Permutation S was precursory Théorie of the groups. In Optical, one owes him of work on the electromagnetic wave propagation.

Its work strongly influenced the development of mathematics at the 19th century. The negligence whose Cauchy towards work of Welsh Évariste showed and Niels Abel, losing their manuscripts, however sullied its prestige.

Biography

Born on August 21st, 1789 with Paris, Augustin Louis Cauchy is the oldest son of Louis François Cauchy (1760 - 1848) and of Marie-madeleine Desestre (1767 -?). His/her father was first clerk of the general Lieutenant of police force of Paris Louis Thiroux de Crosne in 1789; following the excécution of this last in April 1794, Louis François was withdrawn with Arcueil to flee the denunciation and the Terreur. Its family undergoes nevertheless the Loi of the maximum and knew the famine. It turned over to occupy of the various administrative stations in July and was named general secretary of the Senate on January 1st, 1800. It obtained an apartment of function to the Palais of Luxembourg under the Empire. It was close to the Minister of Interior Department and mathematician Pierre-Simon Laplace (1749 - 1827) and senator and mathematician Joseph-Louis Lagrange (1736 - 1813) .

Augustin Louis receives the first Christian education of his father; he learns Latin, the literature and science. He attends then the central École of the Pantheon and sees himself decreeing in 1803 and 1804 various prices in the literary tests of the open Competition. He attends the college Napoleon and has in particular as a professor Jacques Binet. At 16 years, in 1805, it is received second with the Polytechnic school; he is questioned by Jean-Baptiste Biot. Family friends, Berthollet, Lagrange, and Laplace, supported it during its secondary studies.

Under the First Empire

It is received first with the prestigious body of the National school of the Highways Departments in 1807. Become Suction engineer, it has to take part in the construction of the Canal of Ourcq then bridge of Saint-Cloud. Engineering then seemed the natural field of application of mathematics. January 18th 1810, it is named to deal with the building site of the port of Cherbourg, which was to become a strategic military position of the First Empire. Free Cauchy this station in March. During its stay in Cherbourg, it begins its first work in mathematics during its spare time, independently of the academic institutions. After a first writing is mislaid by Gaspard de Prony (1755 - 1839) , it publishes, encouraged by Lagrange, its two first reports, bearing on the Polyhèdre S, in February 1811 and in January 1812. It gives also semi-official hours of teaching to prepare students with the examinations of entry, and is impassioned for the Natural history.

During a serious disease (of which the causes can be allotted to an overwork or the after-effects of the famine that he knew during his childhood), he goes back in autumn 1812 to Paris, and takes a few months of vacation. After a station of professor-assistant is refused to him, it has by its former professor Pierre-Simon Girard to take part again in March 1813 in the building site of Ourcq. At that time, under the influence of Lagrange and Laplace, it expresses the wish to give up its work of engineer to devote itself to mathematics. Two requests near the Academy of Science, called then the Institute, were supported by Laplace and Siméon Denis Poisson (1741 - 1840) , in May 1813 and in November 1814 after the death of Lagrange and Lévêque; but they were both rejected. Cauchy receives temporarily a post office with the philomatic Société in December 1814. In 1816, it gains the price of mathematics for work over the Propagation of the waves.

Member of the Congregation of the Holy-Virgin since his Polytechnique studies, Cauchy can profit from the importance which this movement at the beginning of the takes Second Restoration. He becomes professor assisting at the Polytechnic school in November 1815, then professor of analysis and mechanics in December. Following an ordinance of March 21st, 1816 restoring the Academies, it integrates the Academy of Science under royal nomination, parallel to the reference of important mathematicians known for their republican and liberal positions, Lazare Carnot (1753 - 1823) and Gaspard Monge (1746 - 1818) . Cauchy is shown hard by its pars: “It accepted without hesitating, not by interest, never it was not sensitive to a similar reason, but by conviction. ”

In 1818, it marries Aloïse de Bure, with which it will have two girls, Alicia (1819) and Mathilde (1823).

It given each year to the Polytechnic school a course of analyzes until 1830. Confrontations with his/her colleagues, François Arago (1786 - 1853) and Small Alexis Therese 1791 - 1820) , occur, having for object the supposed insufficiency of its systems design courses, and he is also criticized by certain pupils for the time overload of his courses. Invited to write them, it publishes various treaties during this period: a first part of the notes of course under the algebraic title Analysis in 1821; then complete notes under the title Lessons on differential calculus in 1829, without taking account of the requirements of his/her colleagues and the ministry.

Exile

At the conclusion of the glorious Three (July 1830), its asserted clericalism and its position antilibérale force it with the exile. Indeed, royalist devoted to Charles X, it refuses to lend oath to the new king Louis-Philippe as the law of August 30th, 1830 requires it. Consequently, it loses its station at the Polytechnic school in November. Because of its attachment to the dynasty of the Bourbons and by reaction to the support of the students of the Polytechnic school for the Revolution, Cauchy voluntarily exiles with Freiburg in Suisse in September 1830, his wife and her children remaining in Paris. It vainly tries to found an Academy there where the emigrated scientists could teach. On invitation of the king of Piedmont, Charles-Albert, it occupies the pulpit lately created of sublime physics at the university of Turin in January 1832. It accomplishes a voyage to Rome and is received by the pope Gregoire XVI. After the removal of his younger brother Amédée Cauchy at 26 years, Augustin makes two consecutive voyages to Paris.

Refusing to return to France in spite of the reiterated requests of its family, it agrees the invitation of the king in exile Charles X to become the tutor of the duke of Bordeaux Henri d' Artois (1820 - 1883) . It is selected for its scientific knowledge and its attachment with the religion. It settles in 1833 with Prague, soon joined by his wife in 1834. Become member of the Academy of Prague, it remains in 1835 with Toeplitz, then in 1836 with Budweitz, Kirchberg, and Gloritz. In thanks for his devotion, Charles X creates it Baron in 1839.

Return in France

There regains fine Paris 1838, wishing to remain politically neutral, and takes again its place with the Academy. However, it does not recover its post of teacher at the Polytechnic school. Whereas it had published little during its stay in Germany, it publishes close to an article per week of 1839 in February 1848, except in 1844. In November 1839, it is elected to succeed Gaspard de Prony with the Bureau of longitudes. But, because he refuses to lend oath, its nomination is officially rejected by the government in 1843. It makes the public affair in December. The same year, he is candidate with the pulpit of mathematics of the Collège de France, left vacant after the death of Sylvestre-François Lacroix (1765 - 1843) ; he sees it refusing with the profit of the count Libri.

The insurrection in February 1848 led to the temporary suppression of the political oath. After the escape of the count Libri for legal proceedings for flights and illegal sale of books, Cauchy postulates again with the pulpit of mathematics of the Collège de France, but withdraws with the profit of Joseph Liouville (1809 - 1822) , finally elected in January 1851. In 1849, Cauchy becomes, after Urbain the Glassmaker (1811 - 1877) , holder of the mathematical pulpit of astronomy to the Faculty of Science of Paris. Victor Puiseux, one of his/her friends and pupils, will succeed to him his death. He takes also a pulpit in the Sorbonne.

Cauchy refuses to lend oath to Napoleon III (1808 - 1873) , restored in 1852. It is however not maintained less of it in its functions, following the intervention of Hippolyte Fortoul (1811 - 1856) .

In 1857, take place of the quarrels on mechanics implying Cauchy. May 23rd towards 4:00 of the morning local time, he dies of a Rhume in the family home of his wife with Sceaux. He is buried with the cemetery of Seals. Its last wish was that its work is the subject of an integral publication. During the 19 last years of its life, it had published more than 500 memories.

Position

Ecclesiastical engagement

Convinced Christian, near to the Jesuit S, Augustin Cauchy engaged in a brotherhood, the Congregation of the Holy-Virgin, at the time of his studies. He was criticized as of his stay in Cherbourg for his use to request morning and evening: “It is said that my devotion will make me turn the head. ” Of return to Paris, it on several occasions used its position with the Academy to promulgate its thought. It defended the Créationnisme openly. In 1824, it condemned research in neurology of Franz Joseph Gall (1758 - 1828) . Its standpoint was regarded as nonscientist and strongly condemned in the written press by Stendhal in two articles successifs (1783 - 1842) .

It tested an antipathy for the liberal ideas resulting from the 18th century. However, it engaged for freedom of teaching by defending the schools of the Jésuite S as of his return in France in 1838. Removed in 1772 and restored under the Restoration, they were called into question under the Monarchy of July. Engaged at the sides of Xavier de Ravignan, priest of Notre-Dame, Cauchy called upon the Institute: “Catholic, I cannot remain indifferent to the interests of the religion; geometrician, I cannot remain indifferent to the interests of Science. You do not regard as enemies of civilization, these even which lit and civilized so many various people. ” Pierre-Antoine Berryer (1790 - 1868) , Charles de Montalembert (1810 - 1870) and of Vatisménil supported it in its step. It is probable that the reasons for which it could not enter to the Collège de France into 1843 are its engagement with the sides of the Jesuits and the strong opposition of the count Libri. Only certain establishments of the Jesuits were finally closed in 1845. The business ended in 1848: the Second Republic ensured the independence of teaching.

Cauchy founded various Christian works:

• It gave an active support since 1838 for the Société of Saint-Vincent-of-Paul, Christian work founded in 1833 to bring a help to stripped.
• It founded in 1842 the catholic Institut, or Centers of Luxembourg, from which it chaired the science section.
• It proposed in 1843 an opuscule on the Prévention of the crimes sent to Alexis de Tocqueville (1805 - 1859) .
• Under a request signed by the Institute, was founded in 1846 the work of Ireland aiming at fighting the famine in Ireland.
• In 1854, it founded work for the observation of Sunday, requiring the closing of the trade the Sunday.
• In 1855, Cauchy is one of the founders of the work of the Schools of the East, whose objective is to consolidate the emancipation by education. Lenormant and Cauchy became the vice-presidents of work. The first president was the rear-admiral Mathieu, colleague of Cauchy at the Office of longitudes.

Political commitment

Cauchy is a monarchist antilibéral, near to Andre-Marie Ampère (1775 - 1836) . It used its position with the Academy to promote the royalist thought, and was voluntarily exiled in 1830 to be opposed to the new mode. He considered the dynasty of the Bourbons as “the supports of the religion and Christian civilization, the defenders of the ideas and the principles to which he had dedicated early his heart and his heart. ”

Its political commitment was worth strong oppositions within the Institute to him, then of the Academy, coming in particular from Poinsot or Arago. However, Arago gave its support in 1839 to Cauchy for its candidature for the Office of longitudes. He knew also oppositions with the ministries, by his refusal reiterated to lend an oath of fidelity to each new mode.

Scientific position

The genius of Cauchy was recognized as of its more young age. As of 1801, Lagrange had this comment: “You see this small man, eh well! He will replace us all as long as we are geometricians. ” The prevalence of Cauchy in sciences is explained by the multitude of its fields of studies: its work “embraces about all the branches of mathematical sciences, since the theory of the numbers and the pure geometry until astronomy and optics. ”

Although its talents of mathematician were applauded, the favors from which it profited during the Second Restoration were not appreciated. Criticizing Laplace and Poisson openly, he quickly knew conflicts with his old supports with which he owed his first publications. Its relationship with Poisson was degraded with time and a competition between them settled. Its votes with the Academy were regarded as directed. In spite of the influence of Cauchy on the new generations, its last years were darkened by a quarrel of priority in mechanics, where he refused to recognize his error.

As a member of the Academy, Cauchy was to read and correct the articles sent. It made a negligence towards work of Niels Henrik Abel (1802 - 1829) and of Welsh Évariste (1811 - 1832) . Its opinion on the report of Abel delayed and the report/ratio provided in June 1829 was finally unfavourable; the searchs for Welshman to him had been subjected in May and did not have any answer. Such an attitude was violently reproached to him. In its biography, Valson gives an explanation: “One must excuse it not to have always had time to deal with the publications of others, when it did not find in the course of its own life the leisure necessary to connect and classify its personal work. ”

Work

The whole of work of Cauchy were published of 1882 to 1974 at Gauthier-Villars, in the complete Œuvres in 27 volumes which gather approximately 800 articles covering the analysis, the Algèbre, the Mécanique and the Probabilité S. During the preparation of its courses and conferences, Cauchy reflects on the bases of the analysis and introduced rigorous definitions of concepts only intuitively used before him. An important part of its work relates to the introduction of the holomorphic functions and the convergent series.

Analyzes

Before work of Cauchy in analysis, the series S and series of functions were usually used in calculations, without a precise formalism not being developed. Current errors were made: the mathematicians did not put a question about the possible divergence of the series used, which was mentioned by Cauchy. In its Systems design course , it defines the convergence rigorously series, and studies in particular the series with positive terms: the partial sums converge if and only if they are limited. It gives results of Comparaison of series. He deduces from the convergence of the trigonometrical series a criterion of convergence which bears its name today, the Critère of Cauchy: if the higher Limit of the continuation $|a\_n|^\; \{1/n\}$ is lower than 1, the series of general term $a\_n$ converges. Interested by the whole series (called then series of powers), it highlights the existence of a Rayon of convergence (which it calls circle of convergence), and gives of it a method of calculating, consequence of its criterion of convergence. It shows that under certain assumptions, the product of the sums of two convergent series can be obtained as the sum of a series, called thereafter Produit of Cauchy. It gives a version for the whole series of it.

A regular function was wrongly regarded as the sum of its series of MacLaurin: in other words, it was thought wrongly that an indefinitely derivable function was thereafter given of its derivative successive in a point. Cauchy raises two problems: on the one hand, the ray of convergence of this whole series can be null, and on the other hand, on the intersection of the fields of definition, the function and the sum of its series of MacLaurin are not necessarily equal. However solutions of differential equations linear had been expressed in the form of whole series without any justification. After having exhibé of the examples of functions punts, Cauchy is interested of close with the Développement of Taylor, and evaluates the remainder in the form of the principal determination. It gives sufficient conditions thus to obtain positive responses with the raised questions.

Always in its Systems design course , it states and shows the Théorème of the intermediate values, demonstration finalized by Bolzano. It specifies the concepts of Limite; and formalizes in terms of limits the Continuité and the derivability. It is stopped in its work by a nuance which it does not perceive: the difference between simple Convergence and uniform Convergence. However, simple convergence (convergence of a succession of functions in each point of evaluation) is not a sufficient condition to preserve continuity by passage in extreme cases. It is the first to give a serious definition of the integration. It defines the integral of a function of a real variable on an interval as a limit of a succession of sums of Riemann taken on an increasing succession of subdivisions of the interval considered. Its definition makes it possible to obtain a theory of integration for the continuous functions. In its algebraic Analysis , it defines the Logarithme S and the Exponentielle S like single continuous functions respectively checking the functional equations $f\; (x+y)\; =f\; (X)\; F\; (there)$ and $f\; (xy)\; =f\; (X)\; +f\; (there)$. Although he endeavoured to give rigorous bases to the analysis, he did not wonder about the existence of the body of the real numbers, established later by Georg Cantor.

In its course of Polytechnic, Lesson of differential and integral calculus , it brings clearness and rigor with the resolutions of the differential equations linear of order one and was interested in the partial derivative equations (Théorème of Cauchy-Lipschitz).

Complex analysis

One must in Cauchy the introduction of the bases of the Analyze complexes. Under the influence of Laplace, it presents in the report On the definite integrals (1814) the first writing of the equations of Cauchy-Riemann like condition of Analycité for a function of a complex variable. In this article, it is interested in integration of an analytical function of a variable complexes on the contour of a Rectangle, gives the definition of Résidu, and provides the first calculation of residue. In On the definite integrals taken between imaginary limits (1825), it gives the first curvilinear definition of Intégrale, shows invariance by homotopy (formulated in terms of analysis), and precisely states the Remainder theorem for the analytical functions like tool for the calculation of integrals.

In 1831, Cauchy proposes an expression of the complex number of roots of a polynomial in an area of the complex plan. If F and P is polynomials, it shows:

$\backslash \; int\_\; \{\backslash \; partial\; U\}\; F\; (Z).\; \backslash \; frac\; \{P\text{'}\; (Z)\}\{P\; (Z)\}dz=\; \backslash \; sum\; F\; (z\_i),$ where the integral is taken on the contour of the field U , and where the sum relates to the roots of P pertaining to the field U .

During its stay in Turin, he deduces from the formula of Cauchy previously stated an expression of the coefficients of the Taylor series of an analytical function of a complex variable like integrals. It from of deduced the inequalities known as of Cauchy and the results on the convergence of the analytical functions of a complex variable. Its work will be published in 1838 and will be continued by Laurent, which provides like generalization of the whole series the series of Laurent.

Towards 1845, Cauchy takes as a starting point the work of the German mathematicians on the imaginary numbers, and in particular the trigonometrical writing. It initially pushes back this geometrical aspect for then using it in its own work. It defines the concept of derived from a function from a complex variable; it establishes then equivalence between derivability and analycity, thus melting the definition of the holomorphic functions. All its preceding results on the subject relate to the holomorphic functions; the formula of Cauchy became a central tool in the study of the holomorphic functions, and he then studies again the equations of Cauchy-Riemann.

Algebra

Lagrange had shown that the solution of a general algebraic equation of degree N passes by the introduction of an intermediate equation: its Resolvent of which the degree is the number of functions with N variable obtained by permutation of the variables in the expression of a polynomial function. This number is a divider of $n!$: this result is seen today like a consequence of current the theorem of Lagrange. In 1813, Cauchy improves this estimate and shows that this number is higher than the smallest prime factor of N . Its result was generalized then in current the theorem of Cauchy.

It was the first to make a study of the permutations like objects (called then substitutions ). It introduces the writings still used today to note the permutations; it defines the produced, the order, and establishes the existence and the unicity of the decomposition of the permutations in product of cycles ( circular substitutions ) to disjoined supports. Work of Cauchy and Lagrange on the subject is regarded as precursors of the Théorie of the groups. However, Cauchy did not know the theory of the groups and gave without the knowledge a first study of the symmetrical Groupe.

In Linear algebra, it wrote a treaty on the determinant containing the main part of the properties of this application. He studied the Diagonalisation symmetrical endomorphisms real and that he showed in dimension two and three and if the characteristic Polynôme does not have any multiple root. Lastly, it formalized the concept of polynomial characteristic.

Geometry

In 1811, it is interested in its first report in the equality of convex polyhedrons whose faces are equal. He proposes a demonstration of the Théorème of Descartes-Euler, concerning the numbers of tops, faces and edges of a convex Polyèdre. Its proof consists in projecting the polyhedron in a following graph planar what is today called a stereographic Projection. However, Cauchy made an error, by not making a clear assumption on the studied polyhedrons.

In its second report in 1812, it gave formulas to calculate the plane angles.

Mechanics and optics

In Mechanical, Cauchy proposed to describe the matter to oppose to continuity matter a system of material points whose movements are continuous. According to Cauchy, the forces between these particles must become negligible at the estimable distances. Cauchy stated laws on the variations of tension, condensation and dilation. It made a study on the elasticity of the bodies.

Being interested in the variation of the molecules of ether, Cauchy establishes the equations of light propagation in 1829. It establishes the modes of polarization plane waves, highlighted by former work of Fresnel. Being interested in the limiting conditions on the level of an interface, Cauchy showed the laws of the reflection and the Réfraction of the light. It found the results of Brewster on the variation of the angle of polarization at the time of a reflection or a refraction. Lastly, it showed the existence of waves évanescentes, checked in experiments by Jasmin.

Under the influence of Coriolis, Cauchy studied the dispersion of the light. Its work on the shades rejected one of the objections to the undulatory theory of the light. It highlighted the phenomenon of Diffraction.

In Astronomy, its research on the series enabled him to revise the Théorie of the disturbances installation by Lagrange, Laplace, and Poisson to study the stability of the Solar system. Cauchy was interested more closely in astronomical calculations starting from its election at the Office of Longitudes in 1839. In 1842, he proposed methods of calculating of primitives of rational expressions as a cosine and sine; these methods were justified by the development of the perturbative function. In 1845, the report of the Glassmaker on the planet Pallas is checked in a few hours by Cauchy.

Probabilities

Work of Cauchy on the principle of the minimax made it possible to develop the statistical decision theory. In 1853, he studied a family of even distributions via their characteristic functions answering a variational problem; among these laws must be mentioned the normal Loi and the Loi of Cauchy, discovered by Poisson. Making use of the characteristic functions, it published a proof of the Central limit theorem.

Principal publications

• Systems design course (1821)
• Lessons on the applications of the infinitesimal calculus to the geometry (1826)
• Exercises of mathematics (1827)
• complete Works (28 volumes, 1882-1974) complete Text in line

; Memories

• Theory of the waves
• Memories on the polarization of the light
• Theory of the numbers

Homages

• Its name is registered on the Eiffel Tower;
• a street bears its name in the 15th district of Paris.

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