Augustin-Louis Cauchy

A profound mathematician, Cauchy had a great influence over his contemporaries and successors;[5] Hans Freudenthal stated:

"More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."^[6]

Cauchy was a prolific worker; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.

Biography

Youth and education

Cauchy was the son of Louis François Cauchy (1760–1848) and Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, and Eugene François Cauchy (1802–1877), a publicist who also wrote several mathematical works.

Cauchy married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. They had two daughters, Marie Françoise Alicia (1819) and Marie Mathilde (1823).

Cauchy's father was a highly ranked official in the Parisian police of the

When Cauchy was 28 years old, he was still living with his parents. His father found it time for his son to marry; he found him a suitable bride, Aloïse de Bure, five years his junior. The de Bure family were printers and booksellers, and published most of Cauchy's works.[9] Aloïse and Augustin were married on April 4, 1818, with great Roman Catholic ceremony, in the Church of Saint-Sulpice. In 1819 the couple's first daughter, Marie Françoise Alicia, was born, and in 1823 the second and last daughter, Marie Mathilde.^[10]

The conservative political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died, and was succeeded by his even more conservative brother Charles X. During these years Cauchy was highly productive, and published one important mathematical treatise after another. He received cross-appointments at the Collège de France, and the Faculté des sciences de Paris [fr].

In exile

In July 1830, the

These events marked a turning point in Cauchy's life, and a break in his mathematical productivity. Shaken by the fall of the government and moved by a deep hatred of the liberals who were taking power, Cauchy left France to go abroad, leaving his family behind.

Cauchy returned to Paris and his position at the Academy of Sciences late in 1838.[11] He could not regain his teaching positions, because he still refused to swear an oath of allegiance.

In August 1839 a vacancy appeared in the Bureau des Longitudes. This Bureau bore some resemblance to the Academy; for instance, it had the right to co-opt 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 at sea — mainly the longitudinal coordinate, since latitude is easily determined from the position of the sun. Since it was thought that position at 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 that the matter of the oath was not so easily dispensed with. Without his oath, the king refused to approve his election. For four years Cauchy was in the position of being elected but not approved; accordingly, he was not a formal member of the Bureau, did not receive payment, could not participate in meetings, and could not submit papers. Still Cauchy refused to take any oaths; however, he did feel loyal enough to direct his research to celestial mechanics. In 1840, he presented a dozen papers on this topic to the Academy. He described and illustrated the signed-digit representation of numbers, an innovation presented in England in 1727 by John Colson. The confounded membership of the Bureau lasted until the end of 1843, when Cauchy was replaced by Poinsot.

Throughout the nineteenth century the French educational system struggled over the separation of church and state. After losing control of the public education system, the Catholic Church sought to establish its own branch of education and found in Cauchy a staunch and illustrious ally. He lent his prestige and knowledge to the

In 1848 King Louis-Philippe 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 astronomy. After political turmoil all through 1848, France chose to become a Republic, under the Presidency of

His name is one of the 72 names inscribed on the Eiffel Tower.

Work

Early work

The genius of Cauchy was illustrated in his simple solution of the

Other significant contributions include being the first to prove the Fermat polygonal number theorem.

Complex functions

Cauchy is most famous for his single-handed development of

where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. The contour integral is taken along the contour C. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form the theorem was given in 1825.^[17]

In 1826 Cauchy gave a formal definition of a

${\displaystyle f(z)=\varphi (z)+{\frac {B_{1}}{z-a}}+{\frac {B_{2}}{(z-a)^{2}}}+\cdots +{\frac {B_{n}}{(z-a)^{n}}},\quad B_{i},z,a\in \mathbb {C} ,}$

where φ(z) is analytic (i.e., well-behaved without singularities), then f is said to have a pole of order n in the point a. If n = 1, the pole is called simple. The coefficient B₁ is called by Cauchy the residue of function f at a. If f is non-singular at a then the residue of f is zero at a. Clearly, the residue is in the case of a simple pole equal to

${\displaystyle {\underset {z=a}{\mathrm {Res} }}f(z)=\lim _{z\rightarrow a}(z-a)f(z),}$

where we replaced B₁ by the modern notation of the residue.

In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first^[19] he proposed the formula now known as Cauchy's integral formula,

${\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{C}{\frac {f(z)}{z-a}}dz,}$

where f(z) is analytic on C and within the region bounded by the contour C and the complex number a is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at z = a. In the second paper^[20] he presented the residue theorem,

${\displaystyle {\frac {1}{2\pi i}}\oint _{C}f(z)dz=\sum _{k=1}^{n}{\underset {z=a_{k}}{\mathrm {Res} }}f(z),}$

where the sum is over all the n poles of f(z) on and within the contour C. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with Pierre Alphonse Laurent being the first mathematician besides Cauchy to make a substantial contribution (his work on what are now known as Laurent series, published in 1843).

Cours d'Analyse

In his book Cours d'Analyse Cauchy stressed the importance of rigor in analysis. Rigor in this case meant the rejection of the principle of Generality of algebra (of earlier authors such as Euler and Lagrange) and its replacement by geometry and infinitesimals.^[21] Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe".^[22] The book is frequently noted as being the first place that inequalities, and ${\displaystyle \delta -\varepsilon }$ arguments were introduced into calculus. Here Cauchy defined continuity as follows: The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.

M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement.^[23] Gilain notes that when the portion of the curriculum devoted to Analyse Algébrique was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus.^[24] Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853.

Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing that they lead from everything from the usual "epsilontic" definitions or to the notions of

He was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.^[5] He wrote a textbook^[26] (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a limit in the form that is still taught. Also Cauchy's well-known test for absolute convergence stems from this book: Cauchy condensation test. In 1829 he defined for the first time a complex function of a complex variable in another textbook.^[27] In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods;^[28] thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity.

Argument principle, stability

In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "

Cauchy was very productive, in number of papers second only to Leonhard Euler. It took almost a century to collect all his writings into 27 large volumes:

• Oeuvres complètes d'Augustin Cauchy publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique (27 volumes) at the Wayback Machine (archived July 24, 2007)(Paris : Gauthier-Villars et fils, 1882–1974)
• Œuvres complètes d'Augustin Cauchy. Académie des sciences (France). 1882–1938 – via Ministère de l'éducation nationale.

His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises:

• "Analyse Algébrique".
  Bibliothèque du Roi. 1821. online at the Internet Archive
  .
• Le Calcul infinitésimal (1823)
• Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828)^[5]

His other works include:

• Mémoire sur les intégrales définies, prises entre des limites imaginaires [A Memorandum on definite integrals taken between imaginary limits] (in French). submitted to the Académie des Sciences on February 28: Paris, De Bure frères. 1825.{{cite book}}: CS1 maint: location (link)
• Exercices de mathematiques. Paris. 1826.
• Exercices de mathematiques. Vol. Seconde Année. Paris. 1827.
• Leçons sur le calcul différentiel. Paris: De Bure frères. 1829.
• Sur la mecanique celeste et sur un nouveau calcul qui s'applique a un grand nombre de questions diverses etc [On Celestial Mechanics and on a new calculation which is applicable to a large number of diverse questions] (in French). presented to the Academy of Sciences of Turin, October 11. 1831.{{cite book}}: CS1 maint: location (link)
• Exercices d'analyse et de physique mathematique (Volume 1)
• Exercices d'analyse et de physique mathematique (Volume 2)
• Exercices d'analyse et de physique mathematique (Volume 3)
• Exercices d'analyse et de physique mathematique (Volume 4) (Paris: Bachelier, 1840–1847)
• Analyse algèbrique (Imprimerie Royale, 1821)
• Nouveaux exercices de mathématiques (Paris : Gauthier-Villars, 1895)
• Courses of mechanics (for the École Polytechnique)
• Higher algebra (for the Faculté des sciences de Paris [fr])
• Mathematical physics (for the Collège de France).
• Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843) credited as originating the operational calculus.

Politics and religious beliefs

Augustin-Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to Arcueil during the French Revolution. Their life there during that time was apparently hard; Augustin-Louis's father, Louis François, spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis François to his mother in Rouen says:^[29]

We never had more than a one-half pound (230 g) of bread — and sometimes not even that. This we supplement with little supply of hard crackers and rice that we are allotted. Otherwise, we are getting along quite well, which is the important thing and goes to show that human beings can get by with little. I should tell you that for my children's pap I still have a bit of fine flour, made from wheat that I grew on my own land. I had three bushels, and I also have a few pounds of potato starch. It is as white as snow and very good, too, especially for very young children. It, too, was grown on my own land.^[30]

In any event, he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X.

He was an equally staunch Catholic and a member of the

His royalism and religious zeal made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic"^[32] and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when Guglielmo Libri Carucci dalla Sommaja was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by Joseph Liouville rather than Cauchy, which caused a rift between Liouville and Cauchy. Another dispute with political overtones concerned Jean-Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by Jean-Victor Poncelet, to be wrong.

See also

References

Notes

Citations

Sources

• Belhoste, Bruno (1991). Augustin-Louis Cauchy: A Biography. Translated by Frank Ragland. Ann Arbor, Michigan: Springer. p. 134.
  ISBN 3-540-97220-X
  .
• ISBN 9780671628185
  .
• S2CID 119320059
  .
• Bradley, Robert E.; Sandifer, Charles Edward (2010). Buchwald, J. Z. (ed.). Cauchy's Cours d'analyse: An Annotated Translation. Sources and Studies in the History of Mathematics and Physical Sciences. Cauchy, Augustin-Louis. Springer. pp. 10, 285.
  LCCN 2009932254
  .
• Brock, Henry Matthias (1908). "Augustin-Louis Cauchy" . In Herbermann, Charles (ed.). Catholic Encyclopedia. Vol. 3. New York: Robert Appleton Company.
• Bruno, Leonard C.; Baker, Lawrence W. (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Detroit, Mich.: U X L.
  OCLC 41497065
  .
• Chisholm, Hugh, ed. (1911). "Cauchy, Augustin Louis" . Encyclopædia Britannica. Vol. 5 (11th ed.). Cambridge University Press. pp. 555–556.
• Freudenthal, Hans (2008). "Cauchy, Augustin-Louis". In Gillispie, Charles (ed.).
  ISBN 978-0-684-10114-9
  – via American Council of Learned Societies.
• Kline, Morris (1982). Mathematics: The Loss of Certainty. Oxford University Press.
  ISBN 978-0-19-503085-3
  .
• Valson, Claude-Alphonse (1868). La vie et les travaux du baron Cauchy: membre de l'académie des sciences [The Life and Works of Baron Cauchy: Member of the Academy of Scinces] (in French). Gauthier-Villars.
• This article incorporates material from the
  Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL
  .

Further reading

• Barany, Michael (2013), "Stuck in the Middle: Cauchy's Intermediate Value Theorem and the History of Analytic Rigor", Notices of the American Mathematical Society, 60 (10): 1334–1338,
  doi:10.1090/noti1049
• Barany, Michael (2011), "God, king, and geometry: revisiting the introduction to Cauchy's Cours d'analyse", Historia Mathematica, 38 (3): 368–388,
  doi:10.1016/j.hm.2010.12.001
• Boyer, C.: The concepts of the calculus. Hafner Publishing Company, 1949.
• Benis-Sinaceur, Hourya (1973). "Cauchy et Bolzano" (PDF). Revue d'Histoire des Sciences. 26 (2): 97–112.
  doi:10.3406/rhs.1973.3315. Archived
  (PDF) from the original on 2022-10-09.
• S2CID 120890300
  .
• Gilain, C. (1989), "Cauchy et le Course d'Analyse de l'École Polytechnique", Bulletin de la Société des amis de la Bibliothèque de l'École polytechnique, 5: 3–145
• Grabiner, Judith V. (1981). The Origins of Cauchy's Rigorous Calculus. Cambridge: MIT Press.
  ISBN 0-387-90527-8
  .
• Rassias, Th. M. (1989). Topics in Mathematical Analysis, A Volume Dedicated to the Memory of A. L. Cauchy. Singapore, New Jersey, London: World Scientific Co. Archived from the original on 2012-03-25. Retrieved 2011-01-27.
• Smithies, F. (1986). "Cauchy's Conception of Rigour in Analysis". Archive for History of Exact Sciences. 36 (1): 41–61.
  S2CID 120781880
  .
• "Cauchy, Augustin Louis" .
  New International Encyclopedia
  . 1905.

External links

Wikiquote has quotations related to Augustin Louis Cauchy.

Wikimedia Commons has media related to Augustin Louis Cauchy.

• O'Connor, John J.; Robertson, Edmund F., "Augustin-Louis Cauchy", MacTutor History of Mathematics Archive, University of St Andrews
• Cauchy criterion for convergence at the Wayback Machine (archived June 17, 2005)
• Augustin-Louis Cauchy – Œuvres complètes (in 2 series) Gallica-Math
• Augustin-Louis Cauchy at the Mathematics Genealogy Project
• Augustin-Louis Cauchy – Cauchy's Life by Robin Hartshorne