index [armather.files.wordpress.com] · came from a fairly prosperous farming family. laplace did...

Beltrami, EugenioBernoulli, DanielBessel, Friedrich WilliamBirkhoff, GarrettCantor, GeorgCauchy, Augustin-LouisCayley, ArthurChebyshev, Pafnuty LvovichCholesky, Andre-LouisCourant, RichardDirichlet, Johann PeterEuler, LeonhardFischer, ErnstFourier, Jean-BaptisteFrobenius, Ferdinand GeorgGauss, Johann Carl FriedrichGivens, J. WallaceGrassmann, HermannHadamard, JacquesHamilton, William RowanHermite, Charles

Hilbert, DavidHölder, Otto LudwigHooke, Robert J.Householder, Alston S.Jacobi, Karl GustavJordan, M. E. C.Kant, ImmanuelKronecker, LeopoldKrylov, A. N.Kummer, Ernst EduardLagrange, Joseph LouisLanczos, CorneliusLaplace, Pierre-SimonLebesgue, HenriLegendre, Adrien-MarieLeibniz, G. W. vonLeontief, WassilyLeverrier, U. J. J.Markov, AndreiMinkowski, HermannMises, Richard von

Neumann, John Louis vonOhm, GeorgPeano, GiuseppePenrose, RogerPerron, OskarPiazzi, GiuseppePoisson, Siméon DenisSchrödinger, ErwinSchur, IssaiSchwarz, Herman AmandusSeki Kowa, TakakazuSylvester, James J.Taussky-Todd, OlgaTodd, JohnToeplitz, OttoTukey, John WilderWeierstrass, K. T. W.Weyl, HermannWielandt, HelmutYoung, David M.

Index

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©2000, Society for Industrial and Applied Mathematics

Arzu ErdemRectangle

Nikolai IvanovichLobachevsky1792 - 1856

János Bolyai1802 - 1860

Eugenio Beltrami was born November 16, 1835 in Cremona, Lombardy,Austrian Empire (now Italy). He studied at Pavia from 1853 to 1856 and then inMilan before being appointed to the University of Bologna in 1862 as a visitingprofessor of algebra and analytic geometry. In 1866 he was appointed professorof rational mechanics. He also worked in universities in Pisa, Rome, and Pavia.

Beltrami is best known for pioneering modern non-Euclidean geometry. Hiswork ranged over almost the whole field of pure and applied mathematics, but heespecially focused on theories of surfaces and space of constant curvature. Hepublished his most famous paper, Essay on an Interpretation of Non-EuclideanGeometry, in 1868. It gives a concrete realization of the non-Euclidean geometryof Nikolai Lobachevsky and János Bolyai and connects it with George Riemann'sgeometry. The concrete realization uses the surface generated by the revolutionof the tractrix about its asymptote.

Beltrami developed what has become known as the Klein-Beltrami disc model ofhyperbolic geometry. The geodiscs are chords in the disc and the isometries areprojective isometries of the plane that map the unit to the disc itself.

Beltrami also worked in optics, thermodynamics, elasticity, and magnetism. Hiscontributions to these topics appeared posthumously in the four-volume work,Opere Matematiche (1902-1920).

He died June 4, 1899 in Rome, Italy.

Eugenio Beltrami

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Jacob Bernoulli1654-1705

Johann Bernoulli1667-1748

Nicolaus Bernoulli1695-1726

Johann Bernoulli III1744-1807

Daniel Bernoulli was born into a family that included several prestigious mathematicians. His father, Johann, and his uncle,Jacob, were both involved in the early development of calculus, and his two brothers each made their mark in the mathematicalcommunity.

Johann's father wanted his son to be a merchant, and Johann wanted the same for his middle son: he even tried to force Danielinto a business career. However, Daniel proved as stubborn as Johann himself, and he did end up in academia. He decided tostudy medicine, but he still found a way to work on the subject he loved. Daniel used his father's theories on energy to develophis doctoral dissertation on the mechanics of breathing.

There were more negative feelings between Daniel and his father than just Daniel's choice of career. Daniel published hismasterpiece, Hydrodynamica, in 1738. Johann studied Daniel's book and used his son's developments to create his own work,which he called Hydraulica. In an attempt to take credit for his son's work, he listed the publication date of Hydraulica as 1732,although its actual date is closer to 1739.

Daniel Bernoulli

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Stamp of Bessel issued byGermany on June 19,

1984, his 200th birthday.

About 300mathematicians have

lunar craters named afterthem. Bessel is one of

them.

Being a less-than-stellar student, Bessel left school and was apprenticed to aBremen merchant house. It was during the course of his bookkeeping workthat he acquired an interest in mathematics. The firm of Kulenkamp dealt inthe import/export business, and the young Friedrich developed an interest ingeography and navigation from working with them. These interests led himto compute the orbit of Halley's Comet from observations made by T.Harriott in 1607.

In 1809 he was appointed director of the Königsberg Observatory andprofessor of astronomy. To hold this post he needed the title of doctor. Itwas on the recommendation of Gauss that a doctorate was awarded to him.During his 30 years at the observatory he completed a catalog of veryaccurate positions for 75,000 stars.

Bessel became the outstanding astronomer of the 19th century. His majorcontribution to applied mathematics was his systematization of the functionsthat now bear his name.

Although he had a happy marriage, his two sons died at an early age. Healso had three daughters. His health began to deteriorate in 1840, and hedied two years later from cancer.

Friedich William Bessel

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'

Issued April 14, 1977 byGermany to

commemorate the 200thanniversary of Gauss'

birth.

Issued February 2, 1955 tocommemorate the

centenary of Gauss' death.

Issued by the GermanDemocratic Republic on

April 19, 1977 to

Johann Carl Friedrich Gauss was born on April 30, 1777 in Braunsweig, Germany. His father worked as a gardener, canaltender, and bricklayer. He was harsh with his sons and tried to thwart every opportunity for advancement that came their way.On the other hand, his mother, Dorothea Benz, expected great things of Carl and used her own sharp intellect and humorousgood sense to help him realize his dreams. She lived with her son for the last 22 years of her life, and he would allow no oneother than himself to wait on her after she went blind. She lived to be 97.

It would be difficult to find a child more precocious than Gauss. He began showing signs of his genius before he was three. Heamazed his early teachers when they learned he could sum the integers from 1 to 100 instantly by seeing that the sum was 50pairs of numbers, each pair adding up to 101. He quickly went beyond the scope of his teacher. It was the schoolmaster'sassistant, Johann Martin Bartels, who developed a friendship with Gauss and led him into the mysteries of algebra. When Gausswas 14, Bartels introduced him to Carl Wilhelm Ferdinand, the Duke of Brunswick. The Duke was so taken with this shy,awkward boy that he agreed to pay for his education. At the age of 18, Gauss entered the University of Göttingen and could notdecide whether to pursue his love of languages or mathematics. His discovery of the polygon of 17 sides was the impetus thatpushed him into mathematics.

The second great stage in his career was the rediscovery of Ceres, which led to Gauss' being proclaimed as the greatestmathematician in the world by Pierre-Simon Laplace. His work on calculating the orbit of Ceres with accuracy led him to spendthe next twenty years of his life working on astronomical calculations. Although Gauss was heavily criticized for spending histime on trivialities such as plotting a minor planet's orbit, he enjoyed the publicity and the many honors he received.

Gauss married in 1805 but his extreme happiness was brief. After only four years his wife died and left him with three smallchildren. He married again the following year and soon had two more sons and a daughter. It is written that Gauss got alongwell with his daughters but had difficulty with his sons. The elder son, Joseph, had his father's gift for mental calculation andwas never a problem, but his other sons ran away to the United States to farm.

In 1806, Gauss' benefactor, Duke Ferdinand, died, and it became necessary for Gauss to find a way to support his large family.He accepted a position as director of the Göttingen Observatory. Although his position brought with it the privilege of teaching,this was not his major interest and he often found his students tiresome.

Johann Carl Friedrich Gauss

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commemorate the 200thanniversary of Gauss'

birth.

Gauss depicted onGerman currency.


lunar craters named afterthem. Gauss is one of

them.

Although Gauss' later years were full of honors, he was not as happy as one might suppose. He worried about dying, and a nearbrush with death made him more conservative than usual. He was viewing a railroad under construction when his horses bolted,throwing him from his carriage. Although he was badly shocked, he was unharmed and lived to be 78.

Johann Carl Friedrich Gauss

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Issued by France onJune 11, 1995 as partof a series of Famous

Frenchmen. Thestamp was issued to

benefit the RedCross.

Napolean Bonaparte1769-1821

Jean-Baptiste JosephFourier

1768-1830

Siméon DenisPoisson

1781-1840

Pierre-Simon Laplace was born on March 23, 1749 in Beaumont-en-Auge, Normandy, France. By some accounts, he was theson of poor peasant farmers. It appears, however, that his father was comfortably well off in the cider trade and that his mothercame from a fairly prosperous farming family. Laplace did not often speak of his roots. He may have been embarrassed that hisfamily aspired to little academic achievement.

Wealthy neighbors became impressed with the young Laplace and offered him an education at a Benedictine priory school as aday pupil. He attended from age seven through sixteen. He intended to enter the Church and enrolled in theology at CaenUniversity. Under the instruction of two mathematics teachers, C. Gadbled and P. Le Canu, of whom little is known, hediscovered his own mathematical talents.

He soon left Caen without taking his degree and traveled to Paris. He used his wealthy contacts to request an audience with Jeand'Alembert, who was not impressed and refused to see him. Laplace then wrote d'Alembert a wonderful letter on the generalprinciples of mechanics. In his reply inviting Laplace to call, d'Alembert wrote, "Sir, you see that I paid little enough attentionsto your recommendations; you don't need any. You have introduced yourself better. That is enough for me; your support is mydue." A few days later, Laplace was appointed professor of mathematics at the Military School of Paris, thanks to d'Alembert'sassistance. He threw himself into his life work--the detailed application of the Newtonian law of gravitation to the entire solarsystem. Later, much of Laplace's work made much of d'Alembert's work obsolete, which strained their relationship.

In 1784 he was appointed an examiner at the Royal Artillery Corps, and in 1785, he examined and passed the sixteen-year-oldNapoleon Bonaparte.

Laplace married Marie-Charlotte de Courty de Romanges, twenty years his junior. They had a son and a daughter. During the1973 Reign of Terror Laplace moved his family out of Paris. Both Laplace and Lagrange escaped the guillotine only becausethey were requisitioned to calculate trajectories for the artillery and to help in directing the manufacture of saltpeter forgunpowder. After the Revolution, Laplace became a versatile politician, changing his party each time power was changed. Heseemed to secure a better job each time the government flopped and it cost him nothing to switch his political loyalties.

However, he did not abandon his moral courage when his true convictions were questioned. In an exchange with NapoleonBonaparte, who asked why Laplace had written a huge book on the system of the world (Celestial Mechanics) without ever oncementioning the author of the universe, Laplace replied, "Sire, I had no need of that hypothesis." When Napoleon repeated this toLagrange, the latter replied, "Ah, but that is a fine hypothesis. It explains so many things."

As a mathematical astronomer Laplace has sometimes been called the Newton of France; as a mathematician he may beregarded as the founder of the modern phase of the theory of probability. He became a full member of the Academy of Sciencesin 1785 at the age of 36. Laplace enjoyed enormous influence after taking a leading role in the study of physics, becoming afounding member of Societe de Arcueil in 1805. Other members included mathematicians Biot and Poisson. After eight years,members began to support the work of other scientists and gradually Laplace's influence diminished. Laplace's corpusculartheory was challenged by Fresnel's wave theory of light. His caloric theory of heat was at odds with the work of Petit and ofFourier. Laplace never conceded that his theories were wrong, writing papers on these topics into his seventies.

Laplace died on March 5, 1827. His last words were, "What we know is not much; what we do not know is immense."

Pierre-Simon Marquis de Laplace

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There is a specialplaque in honor of

Laplace on the facadeof the Eiffel Tower.

Pierre-Simon Marquis de Laplace

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Issued by France onFebruary 15, 1958, as partof a series to honor French

scientists.

Edmond Halley1656-1742

Leonhard Euler1707-1783

Joseph Louis Lagrange was born in Turin on January 25, 1736. He may havebeen the only one of Giuseppe Francesco Lodovico Lagrangia and TeresaGrosso's eleven children to survive after birth. His father was the Treasurerfor the Office of Public Works and Fortifications in Turin, and his motherwas the only daughter of a doctor. Both parents were wealthy. Sadly, thiswealth was squandered by Lagrange's father on unsuccessful financialspeculation. With no wealth to inherit, Lagrange later claimed, "If I had beenrich, I probably would not have devoted myself to mathematics."

Lagrange's family had French connections on his father's side, and Lagrangealways leaned toward his French ancestry. He even used the French form ofhis family name, despite being born an Italian. Lagrange's father had plannedfor his son to be a lawyer, so he sent him to the College of Turin. ClassicalLatin became Lagrange's favorite subject. He became interested inmathematics upon reading Halley's work on the use of algebra in optics. Inmathematics, he was largely self-taught. He had no opportunity to work withleading mathematicians in his youth.

There is some disagreement regarding the age Lagrange was when he wasappointed professor of mathematics at the Royal Artillery School in Turin.According to one account he was sixteen, but another claims he was actuallynineteen. Regardless, he was quite young---younger than most of thestudents he taught.

In 1756, Lagrange sent Euler his results on applying the calculus ofvariations to mechanics. Euler was sufficiently impressed and sought agreater position for Lagrange in Prussia. However, Lagrange turned the offerdown, preferring to devote his time to mathematics rather than prestigiouspositions.

When Frederick II invited him to become a member of the Berlin Academy,he again refused the offer, this time because he thought he could contributenothing more than could Euler, who was then Director of Mathematics of the

Joseph Louis Lagrange

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There is a special plaquein honor of Lagrange onthe facade of the Eiffel

Tower.


lunar craters named afterthem. Lagrange is one of

them.

Academy. Later, upon learning that Euler would be stepping down from thisposition, Lagrange accepted a very generous offer from Frederick to join theAcademy. He succeeded Euler as Director of Mathematics at the young ageof 30. Many Academy members were not pleased to see such a young manin this prestigious position. Since he disliked disputes, Lagrange kept tohimself. Soon most members warmed to him.

Shortly after his arrival in Berlin, Lagrange married his cousin, VittoriaConti. It was a happy marriage, but they produced no children. Lagrangenursed his wife when she became ill and was heartbroken when she died. Hehimself suffered from poor health, due mostly to overwork and not takingcare of himself. After Frederick II died, his position at the Academy becameunpleasant. He entertained offers to return to Italy but instead chose tobecame a member of the Academy of Sciences in Paris, since the offerincluded a clause that Lagrange did not have to teach. This appealed toLagrange since he could devote more time to his mathematics.

His greatest work, Mecanique analytique, which he had written in Berlin,was published in 1788 soon after he moved to Paris. It summarized all thework done in the field of mechanics since the time of Newton and is notablefor its use of the theory of differential equations.

Lagrange appeared to have no fear for his own life, but was deeply dismayedby the cruelties he witnessed during the Revolution. This dismay left himwith little faith in human nature and common sense. His depression waslasting, yet he continued to work. His most important contribution tomathematics during this period was his leading role in perfecting the metricsystem of weights and measures.

Still lonely and despondent despite all of his interesting work, Lagrangeattracted the attention of a young woman forty years his junior. She wastouched by his unhappiness and insisted upon marrying him. She was thedaughter of his astronomer friend Lemonnier. They married when Lagrangewas 56, and the union was ideal for both. His wife made it her life to drawher husband out and reawaken his desire to live. She succeeded, and he wasso taken with her that he gladly went out of his way to please her. He evenaccompanied her to balls. He still desired no children, and they producednone. Of all his successes, the one he prized most highly was "having foundso tender and devoted a companion as his young wife."

Before his death at the age of 76, he wished he had had a wife less good, lesseager to revive his strength, one who would let him end gently. He knew hisdeath would devastate her, yet he was very, very ill and rather lookingforward to death. He died early in the morning on April 10, 1813.



Issued by the SovietUnion on April 17, 1957

to commemorate the250th anniversary of

Euler's birth.

Issued by the GermanDemocratic Republic onSeptember 6, 1983 on the200th anniversary of his

death.

Issued by the GermanDemocratic Republic onJune 7, 1957 as part of afamous scientist series.

Leonhard Euler was born on April 15, 1707 in Basel, Switzerland to PaulEuler and Marguerite Brucker. His father, a former theology student at theUniversity of Basel, counted Jacob and Johann Bernoulli as his friends. PaulEuler had some mathematical training and was able to teach his sonelementary mathematics. The first school Euler attended was in Basel, but itwas rather poor. Euler learned no mathematics during his tenure at theschool, but he read mathematics texts on his own to continue his learning.

Euler's father wanted his son to join him in the religious life and sent him tothe University of Basel to prepare for the ministry. Leonhard entered theUniversity in 1720, at the age of 14. He was to obtain a general educationbefore going on to more advanced studies. Leonhard was still interested inmathematics, and he sought out and was tutored by Johann Bernoulli. Eulercompleted his Master's degree in philosophy in 1723 and began studyingtheology later that year. Euler's enthusiasm for mathematics far outweighedhis interest in theology, and with the help of Johann Bernoulli he eventuallypersuaded his father to let him pursue mathematics.

Euler completed his studies at the University of Basel of 1726. His next taskwas to find an academic position. One became available at the St. PetersburgAcademy of Science upon the death of Nicholaus Bernoulli, and it wasoffered to Euler. Euler accepted the position and joined the Academy twoyears after it had been founded by Catherine I, wife of Peter the Great.Euler's original appointment was to the physiology division, but he wastransferred to the mathematical-physical division through the requests ofDaniel Bernoulli and Jakob Hermann.

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730.During this time he lived with Daniel Bernoulli, who held the senior chair inmathematics. Euler became a professor of physics at the academy in 1730.This allowed him to become a full member of the Academy, so he was ableto give up his navy post. Bernoulli left St. Petersburg in 1733, and Euler was

Leonhard Euler

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Euler on a Swiss banknote.


lunar craters named afterthem. Euler is one of

them.

appointed to the vacant senior chair. The higher salary allowed Euler tomarry, and on January 7, 1734 he wed Katharina Gsell, a painter's daughter.They had 13 children, although only five survived past infancy.

Euler suffered from a severe fever in 1735; he almost lost his life. He beganto have problems with his vision around this time. He lost sight in his righteye shortly after this fever, and eventually a cataract dimmed the light in hisleft eye as well.

Euler shared the Grand Prize of the Paris Academy in 1738 and 1740. Theseawards strengthened his reputation and helped earn him a position in Berlin.Initially Euler intended to remain in St. Petersburg, but political turmoileventually changed his mind. Euler accepted an improved offer fromFrederick the Great and went to join the new Academy of Science.

Euler spent 25 years in Berlin, and during this time he wrote approximately380 articles along with books on the calculus of variations, planetary orbits,artillery and ballistics, and shipbuilding and navigation. In 1759 Eulerassumed the leadership of the Berlin Academy, after the previous president(Maupertuis) died. He did not receive the title of "President," however,because he was no longer on good terms with Frederick. Euler knew it wastime to move on when Frederick offered the Presidency of the Academy toJean d'Alembert in 1763.

Euler returned to St. Petersburg in 1766, greatly angering Frederick. Shortlyafter his return, Euler became almost entirely blind after an illness. A fire in1771 destroyed his home. A brave servant carried Euler through the flames,saving his life. His mathematical manuscripts were also rescued. Despite hishandicap, he was able to continue his work on optics, algebra, and lunarmotion because of his remarkable memory. He received help from his sons,Johann and Christoph, and two other Academy members: W. L. Krafft andA. J. Lexell. Almost half of his total works were produced when he wascompletely blind.

Euler was the most prolific mathematical writer of all time. He published886 books and papers in his lifetime, on various subjects. He madecontributions to geometry, calculus, and number theory and introduced betaand gamma functions and integrating factors for differential equations. Heintegrated Leibniz's differential calculus and Newton's method of fluxionsinto mathematical analysis. He studied continuum mechanics, lunar theory,the three-body problem, elasticity, acoustics, the wave theory of light,hydraulics, and music. He laid the foundation of analytical mechanics.

Euler died on September 18, 1783 after suffering a brain hemorrhage. He leftso much unpublished work that it took the St. Petersburg Academy almost50 years after his death to finally publish it all.

Leonhard Euler


Leonhard Euler


There is a special plaquein honor of Poisson on thefacade of the Eiffel Tower


lunar craters named afterthem. Poisson is one of

them.

Poisson's father was a private soldier who was later given an administrativeposition in his village of Pithviers. When the revolution broke out, he hadlittle time for his son and left him in the care of a nurse. It is recorded thatshe often left him alone suspended by a small cord tied to a nail driven intothe wall. She was concerned that the animals running wild might attack himotherwise. Poisson liked to tell his friends that his swinging back and forthfrom this nail led to his later interest in studying the pendulum.

Poisson was educated by his father. When it was time for Poisson to decideon a career, his uncle offered to teach the young Poisson the art of becominga doctor. Poisson's father readily agreed. Poisson began this first career bylearning to prick the veins of cabbage leaves with a lancet. Not finding this avery agreeable profession, he entered the École Polytechnique at the age of17. His abilities quickly caught the interests of two of histeachers---Lagrange and Laplace, who became his life-long friends. At theage of 18, he wrote a paper on finite differences that so impressed Legendrethat it was published in the Recueil des savants étrangers. Upon graduationhe stayed on as a lecturer. He wrote between 300 and 400 papers during hiscareer.

Toward the end of his career, he discovered what is now called the Poissondistribution. It is believed that the first application of this distributionshowed that the variance in jury decision affected the inferences that couldbe made about the probability of conviction in the French courts. It was laterused to describe the number of deaths in the Prussian army due to horsekicks.

He had always intended to write a book that would cover all of his work inmathematical physics, but he died before he could accomplish this.

Simeon Dennis Poisson

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Simeon Dennis Poisson

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Jean Le Rond d'Alembert1717-1783

This stamp was issued byFrance on June 13, 1959

as part of a seriesdedicated to famous

Frenchmen.

Carl Friedrich Gauss1777-1855

There is a special plaquein honor of Legendre onthe facade of the Eiffel

Tower.

Adrien-Marie Legendre was born on September 18, 1752 in Paris, France.Very little is known about his early life. We know that his family was verywealthy and gave him a top quality education in mathematics and physics atthe College Mazarin in Paris, where he defended his thesis at the age of 18.

Legendre had no need for employment and concentrated on research whileliving in Paris. From 1775 to 1780 he taught with Laplace at École Militaire.Jean d'Alembert had helped him secure this appointment. He won his firstprize after writing an essay on projectiles in response to a task offered by theBerlin Academy. This brought him some fame and launched his researchcareer.

He filled Laplace's vacancy at the Academy of Science in 1783. There hestudied the attraction of ellipsoids, developing the Legendre functions, whichare used to determine the attraction of an ellipsoid at any exterior point. Healso worked on celestial mechanics, number theory, and the theory of ellipticfunctions.

He lost his wealth during the French Revolution, as well as his job at theAcademy. He then married and later praised his wife for helping him to puthis affairs in order and for providing him with the tranquility he needed tocontinue his research and writings.

In 1794, Legendre published Elements de geometrie, which was the leadingelementary text on the topic for around 100 years. The Academy wasreopened and renamed (Institut National des Sciences et des Arts) in 1795.In 1803 Napoleon reorganized the Institut to include a geometry section, thesection Legendre was appointed to.

A dispute with Gauss over who discovered the least squares method leftLegendre bitter, and he fought for many years to have his priority of thework recognized. Gauss, while acknowledging that the least squares methodappeared first in Legendre's book, continued to claim priority for himself,

Adrien-Marie Legendre

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lunar craters named afterthem. Legendre is one of

them.

prompting Legendre to later write, "This excessive impudence isunbelievable in a man who has sufficient personal merit not to have need ofappropriating the discoveries of others."

Legendre's attempt to prove the parallel postulate extended over 30 years.All attempts failed due to his reliability on the Euclidean point of view.Much of his work became obsolete upon publication due to the work ofJacobi and Abel.

His unfortunate choice to refuse to vote for the government's candidate in1824 prompted the suspension of his pension. He died in poverty on January10, 1833. He was 81.

Adrien-Marie Legendre

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lunar craters named afterthem. Jacobi is one of

them.

Karl Gustav Jacob Jacobi was born on December 10, 1804 in Potsdam, Prussia, the second sonof a prosperous banker. Due to his family's wealth, he received a good education at theUniversity of Berlin, but his knowledge of mathematics was a result of self-study since theuniversity had no programs to offer the ambitious student.

After earning his degree, he lectured at the University of Berlin and soon became one of themost inspiring math teachers of his time. His teaching talents soon secured him a position at theUniversity of Königsberg. One year later, his research in the theory of numbers caught theattention of Gauss, who was not an easy man to excite. The Ministry of Education promptlypromoted Jacobi over the heads of his colleagues to an assistant professorship. This was quiteadmirable for a man of twenty-three.

Eight years after his father's death, Jaocbi's prosperity ended when the family fortune was lostin 1840. He had to support his mother, wife, and seven children, and for the first time in his lifefound employment necessary.

The loss of wealth apparently had no effect on his mathematics; he continued to work asassiduously as ever. In 1842, Jacobi met William Hamilton during the British Association atManchester meeting. One of his greatest glories was to continue Hamilton's work in dynamics.Soon after completing this task, he suffered a complete breakdown due to overwork. Thegenerous King of Prussia, Jacobi's benefactor, encouraged him to vacation for several months.The king fully appreciated the honor that Jacobi's research conferred on the kingdom.

Jacobi, on the foolish advice of a physician, began to dabble in politics in an effort to benefit hisnervous system. It was a huge mistake. He ran for political office, became a laughingstock inthe process, and failed abysmally in the election. The king terminated his allowance, and Jacobiwas left penniless. A friend took in his wife and children while Jacobi retired to a dingy hotelroom to continue his research. Once his situation came to the attention of friends, they assistedin procuring him a position at the University of Vienna, in addition to coaxing the king intobecoming his benefactor once again.

His great discovery in Abelian functions is by far his most original contribution to mathematics.This discovery was to nineteenth century analysts what Columbus' discovery of America was tofifteenth century geography.

Jacobi succumbed to small pox on February 18, 1851, in his 47th year.

Karl Gustav Jacob Jacobi

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Karl Gustav Jacob Jacobi

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Issued by Ireland onNovember 15, 1943 to

commemorate thediscovery of quaternions

by Hamilton.

William Wordsworth1770 -1850


lunar craters named after

William Rowan Hamilton is known as one of Ireland's greatest and most eloquent mathematicians. He was born on August 3,1805 in Dublin, Ireland, the youngest of three boys and one girl. His father was employed as a solicitor. From his father,Hamilton inherited exuberant eloquence, religious zeal, and conviviality. His extraordinary intellectual brilliance was probablyinherited from his mother, Sarah Hutton, who came from a family well known for its intelligence.

His parents had little to do with his upbringing. At the age of three he was under the tutelage of his Uncle James, an expertlinguist. His mother died when he was twelve, his father two years later. By thirteen William was able to brag that he hadmastered one language for each year he had lived.

He learned calculating skills from an American child genius, Zorah Colburn, who frankly exposed all of his tricks to William.William, in turn, improved upon what he had been shown. At seventeen, he discovered an error in Laplace's Mechaniqueceleste, and as a result of this, he came to the attention of John Brinkley, the Astronomer Royal of Ireland, who said: "Thisyoung man, I do not say will be, but is, the first mathematician of his age." At eighteen, he enrolled in Trinity College, his firstformal schooling. One year later he fell madly in love with Catherine Disney. Since he was not in a position to marry, havingthree years of study left, she instead married a clergyman fifteen years her senior. This was a decision they both regretted untiltheir deaths. Hamilton entered a deep depression and turned to writing poetry. This new interest later led to a meeting withWilliam Wordsworth, who gently but firmly informed him that his gift was in science, not poetry.

Hamilton accepted the post of Astronomer Royal at the Dunsink observatory. This was a poor choice since he soon lost interestin astronomy and spent all his time on mathematics. Aside from Catherine, he seemed quite fickle with women. He finallysettled on Helen Maria Bayly for a wife when he was 28. Their bland marriage produced two sons and one daughter. Helenbecame an invalid soon after they married, leaving the household to fall into disrepair. The Hamilton family lived in squalorthroughout the remainder of William's life.

Hamilton's discovery of quaternions in 1843 proceeded his succumbing to alcoholism. The disease became worse through theyears since brief interludes with Catherine left Hamilton to despair even further over losing the love of his life to another.

He died September 2, 1865 after a severe attack of gout, shortly after receiving the news that he had been elected the firstforeign member of the National Academy of Sciences of the United States of America.

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them. Hamilton is one ofthem.

William Rowan Hamilton

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There is a special plaquein honor of Fourier on the

facade of the EiffelTower.


lunar craters named afterthem. Fourier is one of

them.

Joseph Fourier was born the son of a tailor in Auxerre, France. He was the ninth of twelve children from his father's secondwife. Fourier's parents died within one year of each other, leaving Joseph an orphan before his tenth birthday. With the influenceof caring neighbors, he was sent to the Benedictine-run École Royale Militaire. It was here that young Joseph first demonstratedhis genius. At the age of twelve, Fourier wrote sermons for the church dignitaries of Paris that they passed off as their own.

Fourier was involved in local politics throughout his life. He joined Napoleon's army in 1798 and was a member of the Legionof Culture that attempted to civilize Egypt. Fourier returned to France in 1801, two years after Napoleon abandoned his army inCairo. He was appointed the Prefect of the Department of Isére by Napoleon and traveled to Grenoble to undertake his newduties.

His responsibilities as Prefect were varied. The two tasks for which he is most remembered include draining the swamps ofBourgoin and managing the construction of a highway from Turin to Grenoble. It was while he lived in Grenoble that Fourierdeveloped his work on the theory of heat. Fourier ended his career as Secretary of the Académie des Sciences.

Jean-Baptiste Joseph Fourier

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George DavidBirkhoff

1884 - 1944

John von Neumann1903-1957

SIAM President1967-1968

Garrett Birkhoff had a special introduction to mathematics through his father,George David Birkhoff (1884-1944), a Harvard mathematician of enormousinternational reputation. Although it is often hard to be recognized in one's ownright when born of a famous parent, Garrett made his way quickly in the world ofmathematics. By the time he was 29, his Lattice Theory was published in theColloquium Series of the American Mathematical Society. His popularundergraduate textbook, A Survey of Modern Algebra, written with SaundersMac Lane, is still available today.

Birkhoff was pleased by the fact that he had no Ph.D. He was one of the firstJunior Fellows, an elite society of young scholars founded by the Harvardpresident in the early 1930s as a meta-Ph.D. During the war, he worked forAberdeen Proving Ground and the Navy and specialized in shaped charges andunderwater ballistics. After the war he focused mostly on applied problems,which raised many eyebrows. His research covered a wide area of pure andapplied mathematics including modern algebra, fluid mechanics, numericalanalysis, and nuclear reactor theory. He tied his work closely to that of John vonNeumann.

Many people remember him as a snappy dresser. He usually dressed in tweeds,dark flannels, shined loafers, and the ever-present bow tie. He carried his notes ina leather attaché and often looked askance at colleagues who lectured without ajacket.

Garrett Birkhoff served the Society for Industrial and Applied Mathematics(SIAM) as its president from 1967 to 1968 and gave the prestigious vonNeumann lecture in 1981. In the words of Werner Rheinboldt, a member of thevon Neumann lecture committee, this was "a long overdue tribute to a mostdistinguished mathematician and firm friend of SIAM."

Garrett Birkoff

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Eugene Paul Wigner1902-1995

Theodore von Kármán1881-1963

David Hilbert1862-1943

John Louis von Neumann was born in Budapest on December 28, 1903. Although his birth name was János, he was calledJancsi as a child. He became Johnny when he went to the United States. His father, Max Neumann, worked as a banker anddespite his Jewish heritage raised his children with a mixture of Jewish and Christian traditions. Max Neumann purchased a titlein 1913 that permitted him to add "von" to his name, but it was his son who first added the article.

Von Neumann demonstrated his mathematical skills even as a child. When he was six years old, he could divide eight-digitnumbers in his head. He was also able to memorize pages of telephone numbers, a trick his parents demonstrated at parties. Hismathematical prowess was noticed at his first school, the Lutheran Gymnasium. He was given special tuition along with hisschoolmate, Eugene Wigner.

In 1921 von Neumann completed his education at the Lutheran Gymnasium, and he published his first paper (written jointlywith Fekete, his tutor at the University of Budapest) in 1922. Max Neumann was concerned that mathematics would not supplyhis son with much money, so he encouraged Theodore von Kármán to speak to John and convince him to pursue a businesscareer. As a compromise, John agreed to study chemistry instead, despite earning entrance to the University of Budapest tostudy mathematics.

Von Neumann entered the University of Berlin in 1921 and studied chemistry there until 1923. He then went to Zürich, wherehe took the examinations given in mathematics at the University of Budapest. His results were outstanding, despite the fact thathe had not attended a single lecture. Von Neumann earned a degree in chemical engineering from the Technische Hochschule inZürich and a doctorate in mathematics from the University of Budapest, both in 1926. His mathematical thesis was on settheory. The definition of ordinal numbers that he published when he was 20 is still in use today. From 1926 to 1929, vonNeumann lectured at Berlin. He lectured at Hamburg from 1929 to 1930. His Rockefeller fellowship allowed him to pursuepostdoctoral studies at the University of Göttingen; it was there that he studied under David Hilbert from 1926 to 1927.

Oswald Veblen invited von Neumann to lecture on quantum theory in Princeton in 1929. Before he went to the United States,von Neumann traveled to Budapest to marry Marietta Kovesi. The two went to Princeton University in 1930, where vonNeumann became a full professor in 1931. He was one of the original six mathematics professor at the newly founded Institutefor Advanced Study, a position he retained for the rest of his life.

Von Neumann's marriage produced a daughter, Marina, in 1936 but ended in divorce the following year. In 1938 von Neumann

John Louis von Neumann

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lunar craters named afterthem. von Neumann is

one of them.

married Klára Dán, whom he met during visits to Europe. She was also from Budapest.

During and after World War II, von Neumann served as a consultant to the armed forces. He was a member of the ScientificAdvisory Committee at the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland in 1940, a member ofthe Navy Bureau of Ordnance from 1941 to 1955, and a consultant to the Los Alamos Scientific Laboratory from 1943 to 1955.He was also a member of the Armed Forces Special Weapons Project in Washington, D.C. President Eisenhower appointed himto the Atomic Energy Commission in 1955.

Von Neumann built a solid framework for quantum mechanics. He also worked in game theory and was one of the pioneers ofcomputer science. His awards are too numerous to list but include two Presidential Awards, the Bôcher Prize, the Medal forMerit in 1947, and the Medal of Freedom in 1956. He also received the Albert Einstein Commemorative Award and the EnricoFermi Award in 1956.

John von Neumann died on February 8, 1957 from an incurable cancer.

John Louis von Neumann

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Johann Peter GustavLejeune Dirichlet

1805-1859

Karl Theodor WilhelmWeierstrass1815-1897


lunar craters named afterthem. Hilbert is one of

David Hilbert was born January 23, 1862 in Königsberg, Prussia. It is believed that his inclination for mathematics came fromhis mother. He attended the University of Königsberg from 1880-1884 and received his Ph.D. in 1885.

Hilbert's first work was on invariant theory. In 1888, he proved his famous Basis Theorem. Hilbert's work in geometry had thegreatest influence in that area after Euclid.

Hilbert's famous 23 Paris problems continue to challenge mathematicians of today. These problems include the continuumhypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, theRiemann hypothesis, the extension of Dirichlet's principle, and many more. It was a major event in mathematics each time aproblem was solved. They were delivered in Hilbert's famous speech before the International Congress of Mathematicians atParis in 1900.

Dirichlet's principle, which was used in boundary value problems, had been discredited by Weierstrass's criticism. Hilbertsalvaged Dirichlet's principle by proving it in 1904.

He died on February 14, 1943.

David Hilbert

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them.

David Hilbert

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Augustin-Louis Cauchy1789-1857


lunar craters named afterthem. Dirichlet is one of

them.

Peter Dirichlet was born in Düren, part of the old French Empire that is nowGermany. He taught at the University of Breslau in 1827. From 1828 to1855, he taught at the University of Berlin. He then attained Gauss's chair atGöttingen.

Dirichlet's most famous works are his papers on the conditions for theconvergence of trigonometric series and the use of the series to representarbitrary functions. Fourier had previously used these series to solvedifferential equations. Earlier work by Poisson on the convergence ofFourier series was shown to be nonrigorous by Cauchy. Dirichlet provedCauchy's work to be erroneous. Dirichlet is considered the founder of thetheory of Fourier series for this work.

Dirichlet was extremely absentminded. He reportedly was so preoccupiedthat when his first child was born, he forgot to tell his in-laws. Hisfather-in-law, when he finally learned the news, complained that Dirichletcould have at least written to them and said that "2 + 1 = 3."

Dirichlet died in 1859 at Göttingen.

Johann Peter Gustav Lejeune Dirichlet

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Issued by France onNovember 10, 1989, to

celebrate Cauchy's 200thbirthday.

There is a special plaquein honor of Cauchy on the

facade of the EiffelTower.


lunar craters named afterthem. Cauchy is one of

them.

Augustin-Louis Cauchy was born in Paris on August 21, 1789. He was the oldest of two sons and four daughters. When Cauchywas four year old, his father moved the family to his country home in Arcueil after the French Revolution in an effort to evadethe guillotine. There wasn't much food available during their stay in Arcueil, and as a result Cauchy was undernourished. Heremained sickly until he reached his early twenties.

Cauchy was educated at home until he was thirteen. He began to win academic prizes as soon as he entered school and evenwon the national prize in humanities. By the age of 21 he received a degree in civil engineering; his first commission was to be amilitary engineer for Napoleon at Cherbourg. Cauchy brought four books with him to Cherbourg: Laplace's Mécanique céleste,Lagrange's Traité des fonctions analytiques, Imitation of Christ by Thomas à Kempis, and a collection of Virgil's Latin works.

Religion was very important to Cauchy, and his staunch Catholic beliefs occasionally got in the way of his mathematics.Twenty-one year-old William Thomson (Lord Kelvin) visited Cauchy and planned to discuss mathematics, but Cauchy spent thetime trying to persuade his young visitor to join the Catholic church.

At the age of sixty-seven, Cauchy developed bronchial trouble. He went to the country to recuperate, but while there hecontracted a fever from which he never recovered. He died on May 23, 1857. He published an astounding 789 papers during hiscareer.

Augustin-Louis Cauchy

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Sofia Kovalevskaya1850-1891


Leopold Kronecker1823-1891

Karl Theodor Wilhelm Weierstrass has been called the father of modernanalysis. He was born on October 31, 1815 to Wilhelm Weierstrass andTheodora Forst in the district of Muenster, Germany. He was the oldest offour children. His father was a tax inspector employed by the French.

The Weierstrass family were devout Catholics. Karl's mother died when hewas eleven, and his father remarried a year later. There is speculation thatKarl's mother felt restrained aversion toward her husband and was quitedisgusted with her marriage. Other possible causes of the discord in thenatural sociability of the children were their father's uncompromisingrighteousness, domineering authority, and Prussian pigheadedness. Heexerted his control over his children even after they had become adults.None of the children ever married.

Weierstrass unknowingly rebelled by failing to earn the degree his father hadinsisted upon. The elder Weierstrass sent Karl to the University of Bonn,where he was to master law and finance. Bored, he spent most of his daysfencing. He reserved his evenings for drinking true German beer. During thisperiod he researched the work of Laplace and Jacobi for his owngratification.

Upon returning home after four years without a degree, his father wasfurious. Karl was looked upon as a failure. A family friend convinced hisfather to send him away to earn a degree in secondary education, animportant stepping stone to his later mathematical eminence, but at the timeKarl seemed totally defeated.

Under the instruction of Christof Gudermann, Weierstrass finally blossomedmathematically. He made the theory of power series---Gudermann'sinspiration---the center of all his work in analysis. During his probationaryyear as a teacher at the Gymnasium in Muenster, Weierstrass wrote amemoir on analytic functions. It was in this memoir that he arrivedindependently at Cauchy's integral theorem---the so-called fundamental

Karl Theodor Wilhelm Weierstrass

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lunar craters named afterthem. Weierstrass is one

of them.

theorem of analysis. Weierstrass did not claim priority on this and many ofhis other discoveries, but he used them as a foundation for his life's work onAbelian functions.

One particular item of interest to note about Weierstrass is his aversion tomusic. It is well known that many mathematicians have a natural affinity formusic, but Weierstrass could not tolerate music in any form. He attemptedmusic lessons at the urging of his sisters, but quickly lost interest. Concertsbored him and he fell asleep during opera performances.

Around 1850 Weierstrass began to suffer from severe dizzy spells. Frequentattacks over the next twelve years made it difficult for him to work. He oncefainted while lecturing and never again trusted himself to write on theblackboard. He enlisted the aid of his students to write for him while hedictated his formulas. These attacks may have been caused by anxiety; theexact cause was never determined. His lectures attracted students from allover the world and his classes were packed---sometimes with fifty studentsin a room designed for thirty.

He enjoyed a professional rivalry with Kronecker, who taught along withhim at the University of Berlin. Their rivalry came to an uncomfortable headin 1877 when Kronecker opposed the work of Cantor, causing a serious riftbetween the two men. It was so serious that Weierstrass considered leavingBerlin for Switzerland, although he did remain in Berlin. Despite theiroccasional disagreements, Weierstrass and Kronecker did remain cordial toeach other.

Of the many students who benefited from Weierstrass' teaching, one inparticular stands out: Sofia Kovalevskaya. She had a gift for mathematics,but she was refused entrance to the university, even at Weierstrass'recommendation. He therefore taught her on his own time, meeting with herevery Sunday afternoon in his home. It was through his efforts that shereceived an honorary doctorate from Göttingen and a post in Stockholm.They corresponded for over twenty years. Upon learning of her prematuredeath, Weierstrass burned all of her letters.

Weierstrass deserves his title of "the father of modern analysis." He devisedtests for the convergence of series and contributed to the theory of periodicfunctions, functions of real variables, elliptic functions, Abelian functions,converging infinite products, and the calculus of variations. He alsoadvanced the theory of bilinear and quadratic forms.

During the last three years of his life, Weierstrass was confined to awheelchair, immobile and dependent. He died of pneumonia in 1897. Hewas eighty-two.



Ernst Eduard Kummer1810-1893

Felix J. Mendelssohn1809-1847

Johann Peter GustavLejeune Dirichlet

1805-1859

Leopold Kronecker, the son of prosperous Jewish parents, was born on December 7, 1823. His father owned a flourishingmercantile business and had an unquenchable thirst for philosophy, which he passed on to his son. Leopold's brother, Hugo, wasborn seventeen years later. His upbringing became the loving responsibility of Leopold, and Hugo later became a distinguishedphysiologist and professor at Berne.

Leopold was a genius at friendships early on, forming lasting bonds with men who had risen in the world or were to rise andwould be helpful to him in either business or mathematics. He was uniformly brilliant at school in the classics, and hismathematical talent appeared early under the expert guidance of Ernst Eduard Kummer (from whom he received specialinstruction). He did not overly concentrate on mathematics, preferring a well-rounded education. In addition to his formalstudies, he took music lessons and became an accomplished pianist and vocalist. Music, he declared when he was an old man, isthe finest of all the fine arts, with the possible exception of mathematics, which he likened to poetry. His home in Berlin laterbecame a meeting place for musicians, among them Felix Mendelssohn.

He entered the University of Berlin in the spring of 1841 and was taught by Dirichlet, Jacobi, and Steiner. Dirichlet's influencebrought about Kronecker's talent in applying analysis to the theory of numbers. Jacobi gave him a taste for elliptic functionswhich he was to cultivate with striking originality and brilliant success, chiefly in novel applications of magical beauty to thetheory of numbers. It appears that Steiner had no influence on him at all.

Kronecker was blessed with a rich uncle in the banking business who also controlled extensive farming enterprises. All of thisbecame Kronecker's inheritance upon his uncle's death. For the next eight years Kronecker managed these properties with greatthoroughness and financial success. To manage the land efficiently he even mastered the principles of agriculture.

During his eight years in business, Kronecker produced no mathematics. He did dabble in it as a hobby so as not to stagnateduring this period. He married the daughter of his deceased uncle in 1848. They had six children and a very happy marriage. Heis the rare mathematician who could properly be called a businessman. He did so well for himself by the time he was thirty thathe could thereafter devote himself to mathematics in considerably greater comfort than most mathematicians can afford.

Leopold Kronecker

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Karl Gustav JacobJacobi

1804-1851

The climax of Kronecker's career in mathematics was his prolonged mathematical war with Karl Wilhelm Theodor Weierstrass.Physically they were opposites: Weierstrass a large imposing figure while Kronecker was quite short and compact. The former'swork was in geometry and analysis; the latter was a born algebraist. The two were well known to be gentlemanly, however, andremained friends throughout their scientific battles.

Kronecker never recovered from his wife's death. A few months after she passed away, he died of a bronchial illness in Berlinon December 29, 1891. He was sixty-nine.

Leopold Kronecker

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Pierre de Fermat1601-1665

Leopold Kronecker1823-1891

William RowenHamilton

1805-1865

Ernst Eduard Kummer was born on January 29, 1810 in Sorau, Brandenburg,Prussia, which is now Germany. His father was a physician who died whenEduard was only three, leaving his mother to raise Eduard and his older brother.Eduard was sent to the University of Halle to study Protestant theology andreceived mathematics teaching as part of his degree. This was supposed toprovide a firm foundation to the study of philosophy. His lecturer, H. F. Scherk,was so inspirational that Kummer was soon studying mathematics as his mainsubject.

Kummer was awarded a doctorate on the strength of one prize-winning essay. Hewas appointed a teaching post at Liegnitz, a position he held for 10 years. Hetaught mathematics and physics with great ability to inspire, as his two mostfamous students (Kronecker and Joachimsthal) would attest to. While teaching,Kummer himself was undertaking his own researches and published a paper onhypergeometric series (a continuation of Gauss's work), a copy of which he sentto Jacobi. Soon he caught the attention of Dirichlet, who corresponded withKummer on mathematical topics. On Dirichlet's recommendation, Kummer waselected to the Berlin Academy of Sciences in 1939, although he was still aschoolteacher. At this point Jacobi began actively seeking a universityprofessorship for Kummer.

In 1840, Kummer married a cousin of Dirichlet's wife. The marriage lasted onlyeight years and ended with his wife's death in 1848. With the support of Jacobiand Dirichlet, he secured a full professorship at the University of Breslau. Hequickly established himself as an outstanding teacher and began his research innumber theory.

Kummer was appointed to the chair left vacant by Dirichlet at the University ofBerlin in 1855. Kronecker was already established there, and Weierstrass wassoon appointed to Berlin. The three soon established the university as one of theleading mathematical centers in the world.

Ernst Eduard Kummer

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Karl TheodorWilhelm Weierstrass

1815-1897

In 1843 Kummer attempted to restore the uniqueness of factorization of integersby introducing ideal numbers to restore efforts to prove Fermat's Last Theorem.This important contribution allowed ring theory and abstract algebra to develop.During his geometric period, he devoted himself to studying the same raysystems as Hamilton, but he treated the problems algebraically. He alsodiscovered the fourth-order surface, now named after him, based on the singularsurface of the quadratic line equation.

He received numerous honors in his long career, chief among them membershipto the Paris Academy of Sciences and fellowship of the Royal Society ofLondon.

Kummer died on May 14, 1893 in Berlin, Germany.

Ernst Eduard Kummer

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Leopold Kronecker1823 - 1891

William Shakespeare1564 - 1616


lunar craters named afterthem. Cantor is one of

them.

Georg Cantor was born in St. Petersburg, Russia. He spent the first eleven years of his life there, before his father moved thefamily to Germany. Both of his parents loved music and the arts and passed this love to their son; Georg was an excellentviolinist.

Although his father wanted him to become an engineer, Georg was determined to study mathematics. He began his studies at theEidgenossische Polytechnikum Zürich but transferred to the University of Berlin after his father died in 1863.

Cantor did marry and have six children, but his personal life was not entirely happy. He suffered bouts of depression; his firstdocumented attack occurred in May of 1884. At the time his peers felt his depression was brought on by resistance to hismathematical theories, but this is no longer felt to be true. Today it is believed that his professional worries were increasedbecause of his illness but were not the cause of it.

The work that his colleagues resisted dealt with infinite sets. Cantor's ideas questioned the validity of modern mathematics,which was what mathematicians like Kronecker were working on. The uproar over Cantor's controversial work kept him fromobtaining a position at the University of Berlin, which he longed for.

When in his depressed state, Cantor turned from mathematics and focused his energy on philosophy and literature. In particular,he was convinced that Francis Bacon was the true author of Shakespeare's plays, and he published several pamphlets stating hisbeliefs in 1896 and 1897.

Georg Ferdinand Ludwig Philipp Cantor

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Georg Ferdinand Ludwig Philipp Cantor

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Trinity College


lunar craters named afterthem. Cayley is one of

them.

Arthur Cayley's gift for advanced mathematics first became apparent when he was about fourteen years old. His mathematicsteacher encouraged his father to allow Arthur to pursue mathematics rather than join the family business as a merchant.

In 1842 Arthur graduated as Senior Wrangler from Trinity College. He won a Cambridge Fellowship and taught there for fouryears, all the while contributing to the Cambridge Mathematical Journal. When the fellowship ended, Arthur turned to law inorder to make a living. Arthur worked as a lawyer for fourteen years while continuing to practice mathematics at night.

In 1863 Arthur became the Sadlerian professor of Pure Mathematics at Cambridge. Although retiring from the law meant adrastic reduction of his finances, Arthur was happy to devote all of his time to mathematics. He went on to publish more than900 papers, and in 1881 he gave a course of lectures at Johns Hopkins University.

Arthur Cayley

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Example of ChebyshevFractal

Andrei AndreyevichMarkov

1856-1922

Chebyshev was interested in and had an impact on many different areas ofmathematics. Although he is mainly remembered for his work in numbertheory, he also worked with prime numbers and integrals. He was interestedin mechanics.

He wrote on many topics including quadratic forms, probability theory,orthogonal functions, construction of maps, and the calculation of geometricvolumes.

He had appointments at the University of St. Petersburg and the Institut deFrance and was a member of the Royal Society. One of his most famousstudents was Andrei Andreyevich Markov.

Pafnuty Lvovish Chebyshev

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Pafnuty LvovichChebyshev1821-1894


lunar craters named afterthem. Markov is one of

them.

Andrei Markov was born on June 14, 1856 in Ryazan, Russia. He attendedschool at St. Petersburg, where he had difficulty with all subjects exceptmathematics. Markov attended the Petersburg University in 1874, where hestudied under Pafnuty Chebyshev. Upon completing his studies in 1878,Markov received a gold medal from the university and was offered aprofessorship.

In 1886, Markov was elected to be a member of the St. Petersburg Academyof Science (founded by Chebyshev). By 1896 he was a full member. Heretired from the Petersburg University in 1905, although he continued toteach.

Markov focused on number theory early in his career, although he is bestremembered for his study of Markov chains. This work on Markov chainsled to the development of stochastic processes.

Poetry interested Markov, and he studied poetic style. He was alsosomewhat of a rebel, which caused friction with his government and peers.In 1907, he renounced his membership of the electorate when therepresentative parliament was dissolved.

Markov had one son who also became a renowned mathematician. Markovdied on July 20, 1922 in Petrograd (now St. Petersburg).

Andrei Adnreyevich Markov

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André-Louis Cholesky was born on October 15, 1875. At the age of 20 he entered theÉcole Polytechnique. He joined the artillery branch upon graduation, and by June1905 he was a member of the Geodesic Section of the Geographic Service.

Beginning in November 1907, Cholesky and two other officers spent three months inGreece doing preliminary surveying of the island of Crete. Cholesky remained on theisland at the end of those three months to execute the triangulation. He completed hiswork on June 15, 1908.

From September 1909 to September 1911 Cholesky was obligated to carry out a tourof duty as a Battery Commander; he returned to the Geodesic Section uponcompletion of the tour. He then went to Algiers to take measurements on behalf of theGovernor General of Algeria and the Regency of Tunis.

Cholesky was assigned to the Ministry of Foreign Affairs in May 1913. He was put incharge of the Topographical Service of the Regency of Tunis. He did not have muchtime to demonstrate his abilities in this position, however, since war broke out soonafter his appointment.

Cholesky understood the importance of geodesy and topography in the organization ofartillery firing, and his technical prowess resulted in his being sent on a mission withthe Geographical Service of the Romanian Army in October 1916. He returned inFebruary 1918.

Cholesky died on August 31, 1918 in battle.

Andre-Louis Cholesky

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Courant Institute ofMathematical

Sciences

Kurt Friedrichs1901-1982

David Hilbert1862-1943

Richard Courant was born on January 8, 1888 in Lublinitz, Prussia. He obtained his doctorate from Göttingen in 1910 underDavid Hilbert's supervision. He taught mathematics at Göttingen until the start of World War I. A few years later, he foundedthe university's Mathematics Institute, where he served as director from 1920 until 1933.

Courant was expelled from Göttingen when the Nazis came to power in 1933. He left for England and then America where hebuilt an applied mathematics research center at New York University, based on the Göttingen style. He served as director of theNew York institute until 1958.

In the years before World War II, numerous mathematicians who were forced to leave Germany were given help by Courant toobtain positions in the U.S. During WWII, Courant's research group, consisting of Kurt O. Friedrichs, James J. Stoker, and afew faculty members, became the nucleus of an expanded group that undertook mathematically challenging problems arisingfrom various war projects, under the sponsorship of the office of Scientific Research and Development. After the war, supportfrom the Office of Naval Research and other government agencies maintained the group and encouraged its growth.

Courant's most important work was in mathematical physics. He published papers in variational problems, finite differencemethods, minimal surfaces, and partial differential equations. Kurt O. Friedrichs said of his longtime friend and colleague, "Onecannot appreciate Courant's scientific achievements simply by enumerating his published work. To be sure, his work wasoriginal, significant, beautiful; but it had a very particular flavor: it never stood alone; it was always connected with problemsand methods of other science, drawing inspiration from them and in turn inspiring them."

Courant is perhaps best known for his scientific organizing and leadership talents, which culminated in renaming the Institutefor him in 1965. He died on January 27, 1972 in New Rochelle, New York.

Richard Courant

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Paul Albert Gordan1837-1912

Emmy AmalieNoether

1882-1935

Ernst Sigismund Fischer was born July 12, 1875 in Vienna, Austria. He is bestknown for the Riesz-Fischer Theorem in which the space of all square-integrablefunctions is complete, in the sense that Hilbert space is complete, and the twospaces are isomorphic by means of a mapping based on a complete orthonormalsystem. This theorem is one of the greatest achievements of the Lebesque theoryof integration.

Beginning in 1894 Fischer studied in Vienna under Franz Mertens and then atZurich and Göttingen with Hermann Minkowski. He became a professor at theUniversity of Brunn in the early 1900s. From 1911 until 1920 he was a professorat Erlangen, replacing the retiring Paul Gordan, known then as the "invariantking." Emmy Noether had been studying under Gordan but continued workingunder Fischer's supervision. He influenced her away from Gordan's constructiviststyle, dominated by forms and formulas, toward Hilbert's more axiomatic andabstract style, characterized by existence proofs. She subsequently became aworld-class algebraist.

From 1920 he was a professor at Cologne. He died on November 14, 1954.

Ernst Fischer

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The characteristic featureof the

Einstein-MinkowskiSpacetime is the Light

Cone.


lunar craters named afterthem. Minkowski is one

of them.

Although he was born in Russia, Minkowski attended and later taught at theUniversities of Berlin and Königsberg. He eventually accepted a chairedposition at the University of Göttingen.

Hermann Minkowski accomplished a great deal in a very short lifetime. Tothe three dimensions of space, he added the concept of a fourth: time. Hisconcept developed from the 1905 theory of relativity developed by AlbertEinstein. Minkowski's work in turn became the framework of Einstein'stheory of general relativity (1916).

He was also interested in investigating quadratic forms. His most originalachievement is believed to be his "geometry of numbers."

Hermann Minkowski died suddenly at the age of 44 from a rupturedappendix.

Hermann Minkowski

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Julius WilhelmRichard Dedekind

1831-1916

William Burnside1852-1927

Ferdinand Georg Frobenius was born on October 26, 1849, in Berlin, Prussia(now Germany). He received his doctorate from the University of Berlin in 1870after working under Karl Weierstrass. His first teaching position was atEidgenössische Polytechnikum Zürich. In 1892, he returned to Berlin to becomeprofessor of mathematics.

Frobenius is best known for his work in group theory. He combined results fromthe theory of algebraic equations, geometry, and number theory, which led himto the study of abstract groups. He collaborated with Issai Schur in representationtheory and character theory of groups.

In 1896, he presented one of his most important papers on group characters to theBerlin Academy, having derived many of his ideas through correspondence withRichard Dedekind. Frobenius was able to construct a complete set ofrepresentations by complex numbers.

Frobenius learned of Theodor Molien's work in 1897 and subsequentlyreformulated his work in terms of matrices. He then showed that his charactersare the traces of irreducible representations. William Burnside later used thischaracter theory with great effect.

Frobenius's representation theory for finite groups later found importantapplications in quantum mechanics.

He died on August 3, 1817.

Ferdinand Georg Frobenius

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Ferdinand GeorgFrobenius1849-1917

William Burnside1852-1927

Issai Schur was born on January 10, 1875 in Mogilyov, Belarus. At age thirteenhe went to Latvia, where he attended the Gymnasium in Libau (now calledLiepaja).

He entered the University of Berlin in 1894 to study math and physics. One ofhis teachers was Frobenius, who had a great influence over Schur and would laterdirect his doctoral studies. Schur learned the foundations of the theory ofrepresentations of groups as groups of matrices, of which Frobenius was afounder along with William Burnside.

In 1901 Schur obtained his doctorate with his thesis on rational representations ofthe general linear group over the complex field. Functions that he introduced inhis thesis are today called S-functions, where the S stands for Schur. In 1903 hebecame a lecturer at the University of Berlin, and from 1911-1916 he wasProfessor of Mathematics at the University of Bonn. He returned to theUniversity of Berlin in 1916 and built his famous school. In 1919 he waspromoted to full professor in Berlin. He was elected to the Prussian Academy in1922.

Schur is most famous for his work on the representation theory of groups, but healso worked in number theory, analysis, reducibility, location of roots, and theconstruction of the Galois group of classes of polynomials.

By the early thirties his life had become miserable. April 1, 1933 was theso-called Boycott Day where Germans carried signs with the message, "Germansdefend yourselves against Jewish atrocity propaganda: buy only at Germanshops." On this day, Jewish professors were banned from the university. Oneweek later, the Nazis passed a law stating that civil servants of non-Aryandescent must retire.

Schur saw himself as a German and not a Jew and could not comprehend thepersecution and humiliation he suffered under the Nazis. Somehow, his dismissal(retirement) was revoked and he was able to carry out some of his duties for a

Issai Schur

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while. He declined all invitations to the United States and Britain, stubbornlyrefusing to leave his native land. Finally, the Nazis officially dismissed him fromhis chair at Berlin in 1935. Incredibly, Schur still continued to work there,suffering great hardships and difficulties. He was not even allowed simple accessto the library; friends had to get the information for him.

After being pressured to resign from the Prussian Academy, he finally made thedecision to go to Palestine in 1939, completely broken in mind and body. Thefinal humiliation was to find a sponsor to pay the "Reichs flight tax" to allow himto leave Germany. He was also forced to sell his beloved academic books inorder to have sufficient funds to live in Palestine. He died two years later on his66th birthday.

Issai Schur

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Oswald Veblen1880-1960

James Hardy Wilkinson1919-1986

SIAM President1969-1970

Givens developed an interest in and ability for mathematics early in life.Born in Alberene, Virginia, he graduated from high school at the age of 14and from Lynchburg College cum laude at the age of 17.

He completed his graduate work at the Universities of Kentucky andVirginia, and completed his Ph.D. at Princeton University in 1936. AtPrinceton, he spent three years assisting Oswald Veblen in the Institute forAdvanced Study.

He began his lifelong teaching career in 1937 at Cornell University, wherehe was appointed Instructor of Mathematics. He then became professor atNorthwestern University. He also taught at the University of Tennessee andWayne State University. Givens served as Director of the AppliedMathematics Division at Argonne National Laboratory beginning in 1964.

Before the term "mathematical software" was invented, Givens advocatedimplementing state-of-the-art algorithms and making them readily availablefor use by scientists and engineers. He and Wilkinson initiated a project fortranslating these algorithms into Fortran programs. Thus, he wasinstrumental in creating the environment for the first of the ANLmathematical software PACKs.

The name of Givens is known to numerical analysts mainly because of theGivens rotations---plane rotation matrices that arise in eigenvaluecomputations. His method was the first roundoff error analysis of matrixcomputations that was deliberately made in the "backward" mode. Althoughnever published in an archival journal, his seminal paper did land in the righthands. James Wilkinson went on to show that floating-point computationwas easier to analyze than fixed-point computation.

Dr. Givens was President of SIAM from 1969 to 1970. He will beremembered as one of the pioneers who created the field of matrixcomputations, as a creative administrator who advocated support of basic

J. Wallace Givens

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research, and as a friend who helped many individuals launch their careers.

J. Wallace Givens

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Giuseppe Peano1858-1932

Richard Dedekind1831-1916

Hermann was one of 12 children in the Grassmann family. Although he did notmarry until he was 40, he had 11 children of his own.

He spent three years in Berlin studying philology and theology, but he never hadany university training in mathematics. He often sought a university position butspent his life as a schoolteacher. It is written that he anticipated much of the workperfected by Giuseppe Peano and Richard Dedekind. However, although hiswork is acknowledged, his name is not linked to these accomplishments.

He wrote many papers that were important contributions to physics andmathematics, but his mathematical achievements were not recognized until acentury later. He essentially prophesized this in the 1862 preface to hisAusdehnungslehre (Theory of Extension).

I remain completely confident that the labour I have expended onthe science presented here and which has demanded a significantpart of my life as well as the most strenuous application of mypowers, will not be lost. But I know and feel obliged to state (thoughI run the risk of seeming arrogant) that even if this work shouldagain remain unused for another seventeen years or even longer,without entering into the actual development of science, still thattime will come when it will be brought forth from the dust ofoblivion and when ideas now dormant will bring forth fruit. I knowthat if I also fail to gather around me (as I have until now desired invain) a circle of scholars, whom I could fructify with these ideas,and whom I could stimulate to develop and enrich them further, yetthere will come a time when these ideas, perhaps in a new form, willarise anew and will enter into a living communication withcontemporary developments.

Grassmann is remembered primarily for his development of a general calculusfor vectors. He also wrote a Sanskrit dictionary that is still used today.

Hermann Grassmann

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'

Hermann Grassmann1808-1887

Gottfried Wilhelmvon Leibniz1646-1716

Giuseppe Peano was born to a poor farming family in Spinetta, Italy. He and hisbrother had to walk five kilometers each way to attend school in Cuneo. He wasan excellent student and moved to Turin to stay with his uncle and finish hisprimary schooling.

Although he initially entered the University of Turin to study engineering, heswitched to mathematics. He joined the staff at the University of Turin andpublished his first paper at the age of 22. He discovered an error in a standarddefinition two years later. In 1888 Peano published the book GeometricalCalculus; this explained with great clarity the ideas of Hermann Grassmann andcontained the first definition of a vector space with modern notation and style.

Peano was very skilled in seeing that theorems were incorrect by spottingexceptions. He pointed out such errors on many occasions, which did not endearhim to his colleagues. Having suffered from Peano's mathematical rigor, CorradoSegre commented that the moment of discovery was more important than arigorous formulation. Peano is said to have countered with "I believe it new inthe history of mathematics that authors knowingly use in their researchpropositions for which exceptions are known, or for which they have no proof."

One of Peano's greatest interests was in finding an artificial language based onLatin but stripped of all grammar. His ideas were based on Leibniz's suggestionof a universal language a century earlier. Because of Peano's work in this area,his mathematical work almost stopped and his career declined. Professorsobjected to his insistence that he teach all his students mathematics and that hegive no exams. His students objected to learning the universal language and all ofits symbols, which they would never use in real life. He was forced to resign in1901.

In spite of this, Peano had a happy life. He was married in 1887 but had nochildren. He became active in politics in his later life and supported a cottonworkers' strike. He died of a heart attack at the age of 74.

Giuseppe Peano

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Charles ÉmilePicard

1856-1941

Jules Tannery1848-1910

Jacques Hadamard was born on December 8, 1865 in Versailles, France. He was good at all subjects except mathematicsthrough the seventh grade, when a good mathematics professor set him on a successful path to math and science. He placedfirst in his entrance exams to the École Polytechnique and École Normale Supérieure. He chose the latter and studied underJules Tannery and Émile Picard.

Hadamard taught school while studying for his doctorate. He wrote his thesis on functions defined by Taylor series andreceived his doctorate in 1892. In that same year he also received the Grand Prix des Sciences Mathematique for his work inentire functions. Hadamard's major contribution to mathematics occurred in 1896, when he proved the prime-number theorem.This theorem states that as n approaches infinity, the limit of the ratio of (n) and n/ln n is 1, where (n) is the number of positiveprime numbers not greater than n. The theory was conjectured in the eighteenth century, a time when the available tools wereinsufficient to prove the theorem. Years later, Hadamard proved it (independent of Charles de la Vallée Poussin) usingcomplex analysis. Hadamard also contributed to the theory of integral functions and singularities of functions represented byTaylor series, and he introduced the word "functional."

He served as a professor at the Collège de France (1897-1935), the École Polytechnique (1912-1935), and the École Centralesdes Arts et Manufactures (1920-1935), all in Paris.

Hadamard was active in politics, moving markedly to the left in between WWI and WWII in response to the Nazi rise topower. He suffered a great tragedy when two of his sons were killed in WWI. He himself escaped France when it fell in 1940and went to the United States. He returned to Paris in 1944 and campaigned actively for peace. As a result he had to rely onthe strong support of mathematicians in the United States to allow him to enter the country for the International Congress inCambridge, Massachusetts in 1950. He was made honorary president of the Congress.

He died October 17, 1963 in Paris.

Jacques Hadamard

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Karl Gustav Jacob Jacobi1804-1851


Charles Hermite was born in Lorraine, France, on December 24, 1822. His mathematical ability was most likely inheritedfrom his father, who had studied engineering. Hermite's father did not particularly like engineering, so he became a clothmerchant instead and married his employer's daughter, Madeleine Lallemand. She was a very domineering woman but ran thebusiness well, aided by her husband.

Charles was the sixth of seven children. He was born with a deformity in his right leg, which saved him from any careerassociated with the army. He was forced to walk with a cane during all of his life. Because the family business absorbed all ofhis parents' time, he was packed off to boarding school at the age of six. Moving on to university at age 18, Hermite alwaysstruggled with examinations, but his professor never gave up on him. Professor Richard recognized his genius for mathematicsdespite Hermite's poor test taking skills. In his private studies, Hermite read Gauss, Euler, Lagrange, and Laplace, masteringall of their work. As an algebraist, he was brilliant, but he struggled with elementary mathematics.

Hermite prepared for several years to enter the École Polytechnique. He passed the entrance exams, but only as 68th in orderof merit. This quite humiliated him. After a year of study, he was thrown out because of his lameness. According to the rulingauthorities, his deformity barred him from any of the positions open to successful students.

He began corresponding with Jacobi on Abelian functions while at the same time seeking a teaching career. Influential friendshelped him pass the certification exams---one of these friends was Joseph Bertrand. Hermite later married Bertrand's sister,Louise, in 1848.

Ironically, one of Hermite's first academic successes was his appointment in 1848 as examiner for admissions to the veryPolytechnique that almost failed to admit him and, in fact, kicked him out. A few months later he was appointed quizmaster atthe same institution. Having finally placed himself in a niche where no examiner could get at him, he settled down to becomea great mathematician. His life was peaceful and uneventful.

Up to the age of forty-three, he was an agnostic like many French scientists of his time. When he fell seriously ill in 1856, hisevangelistic friend, Cauchy, convinced him to convert to Roman Catholicism. From that point on he was a devout Catholicand the practice of his religion gave him much satisfaction.

Despite his reputation as a creative mathematician, he was 47 before he was appointed professor at the École Normale.

Charles Hermite

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Stamp of Henri Poincaré(1854 - 1912) issued byFrance on October 18,1952 honoring famous

French people of the 19thcentury.


lunar craters named afterthem. Hermite is one of

them.

Finally, one year later in 1870, he was appointed professor at the Sorbonne, a position he held until retirement. During histenure he trained a whole generation of French mathem

index [armather.files.wordpress.com] · came from a fairly prosperous farming family. laplace did...

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