Bio Index

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

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


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


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 mathematicians. Henri Poincaré was to become the most famous among hisstudents.

Some of his most noteworthy contributions to mathematics include his work in the theory of functions, in particular theapplication of elliptic functions to the general equation of the fifth degree, the quintic equation. In 1873 he published the firstproof that e is a transcendental number. He is known for a number of mathematical entities that bear his name: Hermitepolynomials, Hermite's differential equation, Hermite's formula of interpolation, and Hermitian matrices.

He died on January 14, 1901.

Charles Hermite

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1815-1897


Ludwig Otto Hölder was born December 22, 1859 in Stuttgart, Germany. Hestudied engineering at the polytechnic in Stuttgart for a year and then went on tothe University of Berlin. He attended the lectures of Weierstrass, Kronecker, andKummer. His interest in algebra came partly through the influence of Kronecker.He presented his thesis at the University of Tübingen in 1882 on analyticfunctions and summation procedures by arithmetic means.

Hölder became a lecturer at Göttingen in 1884 and began his work on theconvergence of the Fourier series. He soon discovered the inequality named afterhim. While at Göttingen, he became interested in group theory.

In 1889 he was offered a post at Tübingen, but he suffered a mental collapse. Thefaculty kept their confidence in him and Hölder made a steady recovery. He gavehis inaugural lecture a year later.

His contributions to group theory include clarification of the notion of a factorgroup and the Jordan-Hölder theorem, which proves the uniqueness of the factorgroups in a composition series. He introduced the concepts of inner and outerautomorphisms.

Hölder died August 29, 1937 in Leipzig, Germany.

Ludwig Otto Holder

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Gregorian ReflectingTelescope

Cover of Micrographia1665

Hooke's drawing ofcork

Robert Hooke is most well known for the law of elasticity that bears his name,but he was influential in many other areas as well. His interests includedchemistry, astronomy, biology, physics, and geology. He was also a notedarchitect who helped rebuild London after the Great Fire of 1666.

Hooke had an antagonistic relationship with one of his contemporaries, SirIsaac Newton. Hooke wrote to Newton in 1679 about a possible inverse squarelaw of gravitation, which Newton later proved. Hooke claimed the theory ashis. The argument grew bitter, and as a result Newton removed all references toHooke in his Principia.

Some of Hooke's inventions include the microscope, conical pendulum,Gregorian reflecting telescope, balance spring watch, and marine barometer.The invention of the microscope led to Hooke's 1665 book Micrographia. Thisbook contains pictures of objects Hooke studied through his microscope, mostnotably slices of cork.

Robert J. Hooke

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Alston Householder, OlgaTaussky-Todd, and John

Todd at a GatlinburgSymposium.

James Hardy Wilkinson1919-1986

Magnus R. Hestenes

SIAM President1964.

Alston Scott Householder was born on May 5, 1904 in Rockford, Illinois.The family moved to Alabama shortly after he was born; Householderspent his childhood there.

Householder earned a BA in Philosophy from Northwestern University ofEvanston, Illinois in 1925. From there he went to Cornell University inIthaca, New York, where he received his MA (also in philosophy) in 1927.

Householder then taught mathematics at many different places. Thisexperience interested him so much that he went back to school at theUniversity of Chicago. He earned a Ph.D. in mathematics in 1937 for histhesis on the calculus of variations. His interests moved towardmathematical applications, particularly applying mathematics to biology.After receiving his degree he joined the Committee for MathematicalBiology at the University of Chicago. He worked with the committee until1944, when he became involved in the war effort. Householder joined OakRidge National Laboratory in 1946 after the war ended. He started theORNL Division of Mathematics and served as the first director, a post heheld until 1969.

It was during his time at Oak Ridge that Householder trained himself innumerical analysis. Householder published The Theory of Matrices inNumerical Analysis, one of his most important books, in 1964. This book,still in print today, was one of the first ones to organize this field.

Householder will be remembered for more than his research. During histime at Oak Ridge, many distinguished mathematicians visited him. Theyincluded James H. Wilkinson, Alexander Ostrowski, Wallace Givens,Magnus Hestenes, Olga Taussky-Todd, and John Todd. Together they didmuch to advance numerical analysis and machine computation.Householder was concerned with worldwide collaboration amongnumerical analysts. To achieve a unified effort in this direction, heorganized the Gatlinburg Symposium on Numerical Linear Algebra, the

Alston S. Householder

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first of which was held in 1961. After the fourth conference, theHouseholder Prize was established to be awarded to the author of the bestthesis in numerical linear algebra. Although the meetings were eventuallyheld in places other than Gatlinburg, Tennessee, the name "GatlinburgMeeting" was kept for about 20 years. The meetings were later renamed"Householder Symposium" to honor a pioneer of this area of research.

Long active in the Society for Industrial and Applied Mathematics, heserved as its president in 1964.

Householder died on July 4, 1993 of a massive stroke. Just two weeksbefore his death, he had attended the 12th Gatlinburg meeting.

Alston S. Householder

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Olga Taussky-Todd andJohn Todd

John Todd, Olga'shusband

Emmy Amalie Noether1882-1935

Olga Taussky-Todd was born in Czechoslovakia, which was part of theAustro-Hungarian empire. She enjoyed writing and grammar when she wasyounger, and even wrote poetry and music. Science first drew her attentionin high school; it was her interest in astronomy that led her to mathematics.Olga received her Ph.D. in 1930 and spent the next several years at variouspositions.

Shortly after receiving her degree, she accepted a position to work asRichard Courant's assistant at the Mathematisches Institut in Göttingen,helping with the publication of the first volume of the collected works ofDavid Hilbert. Later in her career she worked with Emmy Noether at BrynMawr College.

She met her husband and fellow mathematician, John (Jack) Todd, whileworking at the University of London. According to Olga, they met whenJack asked her for help on a technical mathematical question, which she wasinitially unable to supply. Their 1938 marriage didn't slow them down, andthey continued to travel throughout their careers.

Being a woman, Olga faced additional career challenges. During oneinterview she was asked whether she was the senior or junior author onjointly authored papers, an improper question that another member of theinterviewing committee told her not to answer. Also, the senior womanmathematician at one of Olga's positions would not allow women to work ontheir theses with Olga, since she felt it was detrimental to a juniormathematician's career to have a female supervisor.

Despite her early obstacles, Olga was widely recognized for herachievements. Among the many citations she received are the Ford Prize;the Austrian Gold Cross of Honor, First Class, for Science and Art; and theLos Angeles Times Woman of the Year Award. In addition to being a greatmathematician, she was an inspiring teacher, colleague and friend to many.

Olga Taussky-Todd

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John E. Littlewood1885-1977

Olga Taussky-Todd,John's wife1906-1995

John Todd was born in Northern Ireland to schoolteacher parents. His father wasan expert chess player. John attended Methodist College in Belfast, where heworked in engineering. He completed his undergraduate studies at Queen'sUniversity, which is also in Belfast. His graduate work was completed at St.John's College in Cambridge; his supervisor was J. E. Littlewood.

Todd taught at Queen's University for four years before joining King's College inthe University of London. He stayed in London for twelve years, some of whichincluded military service. His position as the youngest in the department meantthat he got all the chores to do; a fortunate one led to his meeting Olga Taussky,whom he married in 1938.

One result of his war service was a change in his interests from modern realvariable theory to numerical mathematics, table making, and computers. TheTodds were invited to the National Bureau of Standards to help set up NationalApplied Mathematics Laboratories at UCLA and NBS in Washington, DC.

Since teaching was still in their blood, the couple went to Caltech in 1957, whereJohn was a Professor and Olga a Research Associate. Olga conducted seminarsand supervised theses, and in 1971 she was granted the rank of Professor. Johnspent some of his free time at Caltech playing on their cricket team.

John Todd has continued to serve the mathematical community through hisresearch papers, books and services to societies such as the AmericanMathematical Society and the Society for Industrial and Applied Mathematics.

John Todd

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Stamp of Evariste Galois1811-1832

Issued by France onNovember 10, 1984.

Galois died in a duel atthe age of 21.

Camille Jordan was born in Lyon, France on January 5, 1838. He studiedmathematics at the École Polytechnique. In 1870, Jordan published Treatiseon Substitutions and Algebraic Equations, a work which won him thePoncelet Prize of the Académie des Science. Beginning in 1873, he taught atthe École Polytechnique and at the Collège de France.

He was greatly influenced by the work of Evariste Galois and systematicallydeveloped the theory of finite groups and arrived at the concept of infinitegroups.

Jordan's contemporaries thought highly of his work in algebra and grouptheory. He is best remembered today for his work involving curves. Jordanproved that a simple closed curve divides the plane into exactly two parts, asimple theorem requiring a difficult proof.

Some attribute him with being the greatest exponent of algebra in his day.

Marie Ennemond Camille Jordan

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Pierre-Simon Laplace1749-1827

Immanuel was the fourth of nine children born to Johann George (a harness maker) and Anna Regina Kant. The family wasvery pious, and a concern for religion touched all aspects of their lives.

In 1740 Kant began his studies at the University of Königsberg. He was interested in philosophy, mathematics, and the naturalsciences. His father's early death required Kant to work as a private tutor for seven years to earn enough money to help supportthe family. During this time he published several important papers, one of which anticipated Laplace's hypothesis (developedmore than 40 years later).

Kant was barely 5 feet tall. He was very thin and never had very good health. He always attributed his longevity to an unvariedroutine. After he woke up, he drank a cup of tea, smoked a pipe, and meditated for an hour. From 6:00-7:00 he prepared thelectures that he gave from 7:00-9:00. He worked until 1:00 in his study. He invited friends for very long dinners (his one mealeach day) and then took a walk from 4:00-5:00. He was so punctual at this that people began to set their watches by him. Afterhis walk he read until he went to bed at 10:00.

Kant became increasingly antisocial and bitter as he aged. He was saddened at the loss of his memory and his ability to work.He was totally blind when he died in 1804.

Immanuel Kant

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Pafnuty LvovichChebyshev1821-1894

Leonhard Euler1707-1783


lunar craters named afterthem. Krylov is one of

them.

Aleksei Nikolaevich Krylov was born on August 15, 1863 in Visyaga,Simbirskoy (now Ulyanovskaya), Russia. He attended the Maritime HighSchool in St. Petersburg from 1878 to 1884. After he graduated, he joinedthe compass unit of the Main Hydrographic Administration.

In 1880, Krylov joined the St. Petersburg Maritime Academy in thedepartment of ship construction. He learned mathematics from A. N.Korkin, one of Pafnuty Lvovich Chebyshev's students. After graduating in1890, Krylov taught at the Academy for almost 50 years.

Krylov received a doctorate in applied mathematics from MoscowUniversity in 1914. He was the director of the Physics - MathematicsInstitute of the Soviet Academy of Sciences from 1927 to 1932.

Leonhard Euler had earlier described Krylov's field of study as "navalscience." This science studies the theories of buoyancy, stability, rolling andpitching, vibrations, and compass theories. Compass deviation was a topicKrylov worked on throughout his career.

In 1943, Krylov was awarded a state prize for his compass theory work. Hewas made a hero of socialist labour. He died on October 26, 1945 inLeningrad (now St. Petersburg, Russia) at the age of 82.

Aleksei Nikolaevich Krylov

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Albert Einstein1879-1955.

This stamp was issued byNicaragua in 1971 tocelebrate Einstein's

Theory of Relativity.

Richard Courant1888-1972

Cornelius Lanczos was born on February 2, 1893 near Budapest, Hungary, to a Jewish family. He was the oldest of fivechildren. His father was a well-known lawyer and a highly cultured man. The family was fluent in French, Hungarian, andGerman. Education was a priority for all the children. Cornelius attended a Catholic secondary school where he received ahigh-level education. This is where he first began to excel in mathematics. In 1911, he was admitted to the Faculty of Arts atthe University of Budapest where he studied physics, mathematics, and philosophy.

His younger years coincided with the development of the theory of relativity. He chose to write his Ph.D. thesis onquarternionic formulation of the theory of special relativity.

Lanczos left Hungary for Germany in 1920 due to anti-Semitic laws that limited the number of university positions for Jews.His first position in Germany was at the University of Freiburg, where he assisted the university's leading physicist, FranzHimstedt. He then moved on to the University of Frankfurt where he was to work along with Richard Courant, BernhardBaule, and Erich Bessel-Hagen.

He spent one full year as an assistant to Albert Einstein, in Berlin, Germany. He declared many times that this year was themost decisive period of his life. His task was to study the equations of motion of the general theory of relativity.

He married his first wife, Maria Rupp, in the Fall of 1929, but she soon became ill with tuberculosis and spent much of hertime is Swiss sanatoriums. Much of Lanczos' salary was spent on her care.

Lanczos was offered a position in the United States at Purdue University in 1932 as Professor of Physics. His wife could notaccompany him due to her illness. He traveled to Europe every six months to be with her. It was a stressful time for himbecause his new job in America required that he teach. This meant that he had to quickly master English since his successrelied heavily on how good a teacher he was. He also contributed to the cultural scene at Purdue, performing at several pianorecitals and participating in the Cosmopolitan Club, an organization for international students.

Lanczos made his first significant impact in numerical analysis in 1938 with the publication of his paper, TrigonometricInterpolation of Empirical and Analytical Functions. He combined advantages of the power series with the Fourier seriesand derived a very effective approximation method for both empirical and analytical functions. It became widely known in

Cornelius Lanczos

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the literature as the Tau method because of the coefficient in the error term.

In the early 1940s, Lanczos's personal life was in turmoil. After his wife died in Europe, he brought his son Elmar to theUnited States where he raised him on his own. His position at Purdue became uncertain when a new department head tookover. He decided to leave Purdue to work at Boeing in 1946, his first industry position. In 1949 he joined the Institute forNumerical Analysis of the National Bureau of Standards, and he published his first textbook, The Variational Principles ofMechanics, in 1949. The book was reprinted four times and became a standard university text.

During the McCarthy period, many staff members of the INA experienced unjustified security investigations, which affectedthe morale of the entire institution. Thus, Lanczos welcomed an invitation to become Senior Professor of Theoretical Physicsof the Dublin Institute for Advanced Studies in Ireland. This position allowed him to return to the study of the theory ofrelativity and he was quite happy. He had married a highly educated German lady, Ilse Hildebrand, and they were like fosterparents to many scholars who attended the institute.

In the last years of his life he returned to his native Hungary for several visits with leading scientists. During one of thesevisits he suffered a massive heart attack and died on June 25, 1974 at the age of eighty-one. He was buried in Budapest closeto his birthplace.

Cornelius Lanczos

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Jean-Baptiste JosephFourier

1768-1830


lunar craters named afterthem. Lebesgue is one of

them.

Henri Lebesgue was born in Beauvais, Oise, Picardie, France on June 28, 1875. He attended the École Normale Supérieure andtaught in the Lucée at Nancy from 1899 to 1902. He then went to the University of Rennes, where he was a lecture master until1906. From Rennes he went to Poitiers, where he was promoted from an assistant lecturer to a professor.

Lebesgue is best remembered for his definition of the definite integral, which he gave in 1902. The Lebesgue integral greatlyexpands the scope of Fourier analysis and is one of the great achievements of modern real analysis.

Lebesgue passed away on July 26, 1941, in Paris. His two major books were Lessons on Integration and Analysis of PrimitiveFunctions (1904) and Lessons on the Trigonometric Series (1906). He was awarded the Prix Saintour in 1917.

Henri Lebesgue

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Blaise Pascal invented theCalculating Machine to free

his father, a tax collector,from a tedious task. Thereare about 50 machines still

in existence today.

Sir Isaac Newton1642-1727

Christian Huygens1629-1695

Gottfried Wilhelm von Leibniz was born at Leipzig on July 1, 1646. His father was a professor of moral philosophy and diedwhen Leibniz was only six. His mother, Catherine Schmuck, the daughter of a lawyer, raised him. Although formally schooled,he taught himself to be fluent in the classical languages of Greek and Latin, which was probably motivated by his desire to readhis father's books.

He became a master of all trades; mathematics was only one area in which he excelled. The brilliant young Leibniz entereduniversity at fourteen and received his law degree by age twenty. He was to become a diplomat, historian, philosopher, andmathematician, performing enough work in each field to fill one ordinary working life. Gauss is known to have said that Leibnizsquandered his splendid talent for mathematics on a diversity of subjects in which no human being could hope to be supreme,whereas (according to Gauss) Leibniz had in him supremacy in mathematics.

After receiving his doctorate in law, he accepted a position to revise some of the local legal codes and serve on severalcommissions. He became secretary to the Nuremberg alchemical society and took up residence in the courts of Mainzo in aneffort to improve Roman civil law code.

One of his more ambitious, yet futile projects as a diplomat involved reuniting the Catholic and Protestant churches. When thatfailed, he attempted to unite the two Protestant factions of the day. That too failed.

One of his earlier mathematical achievements was inventing an improvement to Pascal's calculating machine. Leibniz's versionwent beyond simple addition and subtraction; it could multiply, divide, and extract roots as well.

While visiting Paris as a diplomat, he met Christian Huygens, who was primarily a physicist with a lot of mathematicsknowledge. He agreed to teach Leibniz the newer mathematics, and after several years Leibniz discovered many of the formulasof calculus as well as the Fundamental Theorem of Calculus. This led to a long dispute with Newton, who failed to publish hisown results years prior to Leibniz's discovery. Thus he, and many of his followers, accused Leibniz of plagiarism.

Leibniz's other great achievements in mathematics included his development of the binary system of arithmetic and his work ondeterminants, which arose from his developing methods to solve systems of linear equations.

The remaining forty years of his life were spent as a librarian for the Brunswick family, cataloging the exact history of thefamily's existence, an extremely complex task that he was unable to complete.

Gottfried William von Leibniz

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About 300 mathematicianshave lunar craters namedafter them. Leibniz is one

of them.

Diplomacy was Leibniz's main livelihood, and it provided him with a venue to pursue his zeal for mathematics. When he died atthe age of seventy it is said that only his secretary attended his funeral. This is rather perplexing for one who had secured anominal amount of fame and a secure position in life, but the controversy about who discovered calculus was still very fresh.After careful study it appears that Newton made the first great strides in calculus, while Leibniz developed the notation we usetoday, due in part to Newton being quite unwilling to share his own material.

Gottfried William von Leibniz

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Leo Tolstoy1828-1910

Leontief won theNobel Prize in

Economic Science in1973.

Wassily Leontief was born in St. Petersburg, Russia on August 5, 1906. Hisfather was a professor of economics. Young Wassily spent his youth in St.Petersburg (now Leningrad) and remembered his country mourning the death ofLeo Tolstoy, the whistling bullets of the February Revolution, and VladimirLenin addressing a mass meeting in front of the Winter Palace.

Leontief entered the University of Leningrad in 1921, where he studiedphilosophy, sociology, and economics and received his degree of LearnedEconomist in 1925. He continued his studies at the University of Berlin,receiving his Ph.D. in 1928. From 1927 to 1930 he served as member of the staffof the Institute for World Economics at the University of Kiel. In 1929 he wentChina to serve as advisor to the Ministry of Railroads. In 1931, Leontief movedto New York and served at the National Bureau of Economic Research, movingon to Harvard's Department of Economics in 1932, where he was appointedProfessor of Economics in 1946. While there, he organized the HarvardEconomic Research Project (1948) and served as its Director until 1973. From1975 until his death on February 5, 1999, he was a Professor of Economics atNew York University and was named Director of the school's Institute forEconomic Analysis in 1978.

He married Estelle Marks, a poet, in 1932. They had one daughter.

Leontief is best known for his input-output analysis theory. In 1973, he wasawarded the Nobel Prize in Economics for his work in input-output tables. Thisgrid-like table shows what individual industries buy from and sell to oneanother. With the addition of government, consumers, foreign countries, andother elements, a general outline of the goods and services circulating in anational economy emerges. This system is used in various forms by a largenumber of industrialized countries for both planning and forecasting. One of themore recent developments of this method is its extension to include residuals ofthe production system---smoke, water pollution, scrap, etc, and the furtherprocessing of these. In this way the effects of the production on the environmentcan be studied.

Wassily Leontief

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Leontief is also known for having developed linear programming, amathematical technique for solving complex problems of economic operations.The phenomenon called Leontief Paradox succeeded economists' prior theory,which held that a country's exports reflect the commodity most abundant in thatcountry, such as labor or capital. However, Leontief pointed out that althoughthe United States has more capital than most other nations, the majority of itsexports were of labor-intensive goods; conversely, the majority of U.S. importswere of capital-intensive goods.

Leontief's major publications include The Structure of the American Economy1919-1929: An Empirical Application of Equilibrium Analysis (1941) andInput-Output Economics, 2nd ed. (1986).

Wassily Leontief

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John Couch Adams1819 - 1892

Stamp issued byNicaragua on May 15,

1971 to celebrateEinstein's general theory

of relativity.

Urbain Jean Joseph Leverrier was born on March 11, 1811 at Saint-Lô inNormandy. He attended the École Polytechnique to prepare for his career asa scientist. Leverrier's early interest was chemistry, but he took a teachingpost in astronomy when it became vacant.

Leverrier dealt with celestial mechanics, the mathematical analysis of theplanetary motions. According to the rules of this science, planets movearound the sun in orbits that are basically elliptical. Deviations to the orbittend to be due to attractions by the rest of the planets. Although thecomputations were very complicated, at this time scientists had sufficientlyexplained the orbits of all planets except Uranus.

In 1845 Leverrier began to study the orbit of Uranus. He and John CouchAdams independently concluded that the irregular orbit was probably causedan unknown planet. Through a process of detailed calculations, he estimatedthe location of this unknown planet. Neptune was discovered on September23, 1846, less than one degree from the spot suggested by Leverrier. As oneof his colleagues noted, Leverrier "...discovered a star with the tip of his pen,without any instruments other than the strength of his calculations alone."

The work Leverrier did that led to this discovery was hailed as one of theoutstanding scientific achievements of all time. He received muchrecognition for his work. He was awarded the Copley Medal of the RoyalSociety of London, and in France he became an officer in the Legion ofHonour.

Leverrier became director of the Paris Observatory in 1854. He provedunpopular and was removed from the post in 1870. His successor died in1873, and the post was again offered to Leverrier, but this time a councilsupervised him and greatly restricted his authority.

Leverrier discovered a slight anomaly in Mercury's orbit in 1855 andpostulated either an asteroid belt so close to the sun as to be invisible or a

Urbain Jean Joseph Leverrier

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planet (which he called "Vulcan") closer to the sun than Mercury. He wasnot able to prove the existence of either, and the mystery wasn't solved until1915, when Einstein's general theory of relativity explained the orbit ofMercury without the need for perturbing bodies.

Leverrier died in Paris on September 23, 1877.

Urbain Jean Joseph Leverrier

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lunar craters named afterthem. Ohm is one of them.

Born in Erlangen, Bavaria, Georg Ohm was raised by a locksmith fatherwho cared about the sciences and philosophy. Although poor, his fathermanaged to get Georg and his brother, Martin, into a preparatory school.There they studied and excelled for five years, Georg in the area of scienceand Martin in mathematics.

After their education at the preparatory school ended, they got into theUniversity of Erlangen because of their abilities. Georg's education wasinterrupted when he was exiled to Switzerland over a family disagreement.He eventually returned to receive a degree in 1811 and teach mathematics.He later went to the Jesuit College in Cologne for almost ten years.

Ohm was intensely interested in electricity and its effects and relationshipsbetween different properties that acted upon it. With his interest in the area,his skill with mechanics that he learned from locksmithing, and his intellectin mathematics and science, he eventually was responsible for what is nowcalled Ohm's Law. His work was initially received with very littleenthusiasm, and he resigned his position at Cologne.

It was not until 1841 that his work was recognized by the Royal Society. Hewas awarded the Copley Medal that same year.

George Ohm

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An example of thePenrose tile.

Stephen Hawking

Maurits Cornelius Escher1898-1972

Roger Penrose was born August 8, 1931 in Colchester, Essex, England. Hewas raised in a family with strong mathematical interests. His mother was adoctor and his father a medical geneticist. One brother became amathematician, the other a chess champion. Roger was originally alsointerested in medicine, but he decided to pursue mathematics whenscheduling conflicts at school forced him to choose between the two. Hestudied at University College in London. After obtaining his doctorate inalgebraic geometry from Cambridge, he was professor at Birkbeck Collegein London from 1966 to 1973, and then moved to Oxford University.

Roger and his father are the creators of the famous Penrose staircase and theimpossible triangle known as the tribar. Both creations have providedinspiration for the famous graphic artist, M.C. Escher.

Penrose is very well known for formulating some of the fundamentaltheorems that describe black holes, including the singularity theorems,which he developed jointly with English physicist Stephen Hawking. Thesetheorems state that once the gravitational collapse of a star has proceeded toa certain degree, singularities (which form the center of black holes) areinevitable. He accomplished most of this work in the 1960s.

Penrose has also proposed a new model for the universe. He holds that allcalculations about both the macroscopic and the microscopic worlds shoulduse complex numbers, requiring reformulation of the major laws of physicsand of space-time. His proposed universe model uses basic building blockscalled "twistors."

In 1994, Penrose was knighted for his outstanding contributions tomathematics. His passion, however, is recreational mathematics. He isfascinated with a field of geometry known as tessellation, the covering of asurface with tiles of a prescribed shape. With only notebook and pencil,Penrose set about developing sets of tiles that produce quasi-periodicpatterns; at first glance the patterns seem to repeat regularly, but upon closer

Roger Penrose

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One of Penrose's populartitles.

examination this is not so. After years of research and careful study, hefound that two shapes successfully cover a surface, one of a large bird andone of a small bird. While this all sounds somewhat trivial, it soon becameobvious that certain chemical substances (crystals) form in a quasi-periodicmanner. This finding has a useful application: such crystals make anexcellent nonscratch coating for frying pans.

Roger Penrose is currently the Emeritus Rouse Ball Professor ofMathematics at the University of Oxford, the Gresham Professor ofGeometry, Gresham College, London, and he has a part time appointment asFrancis and Helen Pentz Distinguished Professor of Physics andMathematics at Penn State University. Penrose has received a number ofawards including the 1988 Wolf Prize which he shared with StephenHawking for their understanding of the universe, the Dirac Medal, and theAlbert Einstein Prize. His 1989 book, The Emperor's New Mind, became abest seller and won the 1990 Rhone-Poulenc Science Book Prize.

Roger Penrose

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Carl LouisFerdinand von

Lindemann1852-1939

Oskar Perron was born on May 7, 1880 in Frankenthal, Pfalz, Germany. He had aclassic early education and studied mathematics in his spare time. Perron's fatherwanted him to continue the family business, but Perron wanted to continue hiseducation. He prevailed and entered the University of Munich in 1898. He also studiedat universities in Berlin, Tübingen, and Göttingen. His doctoral thesis on geometry wasdirected by Carl Lindemann.

Perron was appointed a lecturer at Munich in 1906; he then accepted a post asextraordinary professor at Tübingen in 1910. By 1914 Perron was a professor atHeidelberg. Perron's career at Heidelberg was interrupted by World War I in 1915. Hewon the Iron Cross for his work during the war. When the fighting was over, hereturned to Heidelberg, where he remained until 1922. He then accepted a chair atMunich.

Perron retired in 1951, although he continued to teach some courses at Munich until1960. Even after he stopped teaching he still continued to do research; he published 18papers between 1964 and 1973. In addition to his love of mathematics, Perron enjoyedmountain climbing. He climbed one 2200-meter peak more than 20 times, the last timewhen he was 74. He died on February 22, 1975 in Munich.

Oskar Perron

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Carl Friedrich Gauss1777-1855

Giuseppe Piazzi was first and foremost a Theatine monk, secondly a professor ofmathematics at the Academy of Palermo, and thirdly a great astronomer. He setup an observatory at Palermo in 1789 from which he published a catalog of 7646stars.

In 1801 he was observing the sky for a new catalog when he noticed a small,star-like object that was not on any of his star maps. He recorded the movementover the next six weeks and noticed that it was moving relative to the backgroundstars. Its rapid movement indicated it was not a star but an object in the solarsystem. The first asteroid had been discovered!

Piazzi had discovered Ceres because Gauss had recently developed themathematical techniques that allowed the orbit to be calculated. The 1000thasteroid to be discovered was named Piazzia in Giuseppe's honor. Asteroiddiscoveries were rare from that point until 1891, when photographic methodswere introduced. More than 3000 asteroids have been discovered to date, andastronomers estimate that there are another 100,000 still to be found.

Giuseppe Piazzi

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Max Karl Ernst LudwigPlanck

1858-1947

Albert Einstein1879-1955

Paul Adrien MauriceDirac

1902-1984

Erwin Schrödinger was born on August 12, 1887 in Vienna, Austria. Hisfather, Rudolf, ran a small linoleum factory. His mother, Emily Bauer, washalf-English, which led to Schrödinger's fluency in English as well asGerman.

Schrödinger had a private tutor at his home until he was 10 years old. Hethen entered the Akademisches Gymnasium in the autumn of 1898, where heremained until he graduated in 1906. Schrödinger entered the University ofVienna that same year, where he studied theoretical physics. He received hisdoctorate on May 20, 1910; his dissertation was titled On the conduction ofelectricity on the surface of insulators in moist air.

At the beginning of World War I, Schrödinger was summoned to duty at theItalian border. He was able to continue his theoretical research during histime in Italy, and he even submitted a paper for publication from the front.He was transferred to Hungary in 1915, where he submitted another paper.When he was sent back to the Italian front, Schrödinger received a citationfor outstanding service for commanding a battery during battle.

The spring of 1917 found Schrödinger back in Vienna, where he taught acourse in meteorology. He remained in Vienna after the war, and from 1918to 1920 he made substantial contributions to color theory. He also worked onradioactivity in Vienna; he proved the statistical nature of radioactive decay.He was offered an associate professorship at Vienna in January 1920 on thebasis of his work. At this time Schrödinger was looking to marry, so hedeclined the associate professorship since it would not support him and anunemployed wife.

Schrödinger married Annamaria Bertel on March 24, 1920, around the timehe accepted an assistantship in Jena. The newlyweds were in Jena for a shorttime before Schrödinger accepted a chair in Stuttgart. Another rapid movefound the couple in Breslau, where Schrödinger had accepted another chair.It was their third move in eighteen months. Their rapid travel was not over

Erwin Schrodinger

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Schrödinger shared the1933 Nobel Prize in

Physics with Paul Dirac.


lunar craters named afterthem. Schrödinger is one

of them.

yet, as Schrödinger accepted the chair of theoretical physics at Zürich in late1921. His years at Zürich (1921-1927) were the most productive of hiscareer.

In 1926 Schrödinger published his revolutionary work relating to wavemechanics and the general theory of relativity in a series of six papers. Heproposed wave mechanics in these papers, which are the second formulationof quantum theory. These papers were highly regarded, and Schrödingerreceived praise from colleagues such as Max Planck and Albert Einstein.

Schrödinger was offered Planck's chair of theoretical physics in Berlin whenit became vacant, a position he accepted. He assumed the post on October 1,1927. Schrödinger was a Catholic, but he disagreed with the persecution ofJews in Berlin, so he decided to leave Germany in 1933. After spending thesummer in South Tyrol, he went to Oxford. It was here where Schrödingerheard he was to share the 1933 Nobel Prize in Physics with Paul Dirac.

He was offered posts at Princeton University and the University ofEdinburgh, but in 1936 he accepted a position at the University of Graz inAustria, where the Nazi regime again entered his life. After the Anschluss,the Nazis controlled Graz. Schrödinger was advised to write a letter to theUniversity Senate apologizing for his earlier desertion of Germany. Heregretted writing this letter for the rest of his life and explained to Einstein "Iwanted to remain free --- and could not do so without great duplicity." Theletter did not soothe the Nazis, and in August 1938 he was dismissed fromhis post for "political unreliability."

Schrödinger and Anny left Germany and fled to Rome, and then to Oxford.They spent a year in Gent after Schrödinger was offered a one-year visitingprofessorship at the University of Gent. When that position endedSchrödinger went to Dublin and joined the Institute for Advanced Studies asDirector of the School for Theoretical Physics. This was a position he helduntil his retirement. He published a new unified field theory in January 1947without the benefit of critical analysis; this new theory proved to be nothingof merit. Schrödinger was devastated when he read Einstein's opinion of hislatest work.

Other topics besides a unified field theory interested Schrödinger during histime in Dublin. He published two works during these years, What is life(1944) and Nature and the Greeks (1954). He retired to a position of honorin Vienna in 1956 and published his last book, Meine Weltansicht, in 1961.

Schrödinger died in Vienna on January 4, 1961 after a long illness. Hefathered three daughters with three different women, none of whom was hiswife.

Erwin Schrodinger


Erwin Schrodinger


Eduard Kummer1810-1893


1815-1897

Herman Schwarz was born on January 25, 1843 in Hermsdorf, Poland (nowGermany). He originally studied Chemistry at Berlin, but he switched tomathematics due to the influence of Eduard Kummer and Karl Weierstrass.Weierstrass supervised Schwarz's doctorate, which he received from theUniversity of Berlin in 1864. Schwarz was appointed to chairs at Zürich (1869)and Göttingen (1875). He taught at Berlin from 1892 to 1917, succeedingWeierstrass.

Not all of Schwarz's life was devoted to mathematics. He was the captain of thelocal Voluntary Fire Brigade, and he aided the stationmaster at the local railwaystation.

Schwarz married Kummer's daughter. He died on November 30, 1921 in Berlin.

Herman Amandus Schwarz

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Although he was born into a samurai warrior family, Takakazu was later adopted by anoble family whose name he bears. A servant introduced him to mathematics when hewas nine and Seki Kowa continued to teach himself. He collected a library ofmathematics books and soon became known as an expert in the field.

No specific accomplishments are attributed to him due to the secrecy surrounding theschools in Japan at the time but it is known that he worked on determinants prior toLeibniz and what came to be called Bernoulli numbers before Jacob Bernoulli.

He was a rigorous mathematician who was well liked by his students.

Takakazu Seki Kowa

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Arthur Cayley1821-1895

Florence Nightengale1821-1910


lunar craters named afterthem. Sylvester is one of

them.

James Joseph Sylvester was born to Jewish parents on September 3, 1814. He was the youngest of several children. Between theages of six and fourteen, he attended private schools, where his mathematical genius began to show itself. The last half of hisfourteenth year was spent at the University of London, where he studied under De Morgan.

In 1829 Sylvester entered the Royal Institution at Liverpool. At the end of his first year, he was so far ahead of his fellowstudents in mathematics that he was placed in a special class by himself. He won two prizes during his two years at this school.Sylvester's days at the Royal Institution were not happy ones. Because he did not hide his religion, he suffered persecution fromsome of his Christian schoolmates.

Sylvester entered St. John's College in Cambridge in 1831 when he was 17. Sylvester did not graduate from this institutebecause he would not sign a religious oath to the Church of England. Because of this refusal he was eligible for neither a Smith'sprize nor a Fellowship. Eventually the college stopped requiring its students to sign this oath. As soon as this change occurred in1871, Sylvester promptly received his degrees.

Beginning in 1838, Sylvester taught physics at the University of London. He spent three years at this school, which was one ofthe few places that did not ban him due to his religion. De Morgan was one of his colleagues.

At the age of 27 Sylvester was appointed to a chair in the University of Virginia. During his few months in the United States, heused a stick to strike a student who had insulted him. The student collapsed, causing Sylvester to (incorrectly) believe he hadkilled the young man. Sylvester fled to New York and boarded the first available ship to England.

Once back in England, Sylvester worked as an actuary and a lawyer. He also tutored in mathematics; Florence Nightingale wasone of his students. During his time at the London courts he encountered Arthur Cayley, who was also a lawyer with an interestin mathematics. The two discussed their shared interest in their spare time and became life-long friends. It was during thesediscussions that Sylvester first became interested in matrix theory.

In 1854, when he was 40, Sylvester applied for the professorship of mathematics at the Royal Military Academy in Woolwich.He did not receive it until the following year, when the original recipient passed away. Sylvester held the position at Woolwichfor 16 years. His mandatory retirement came when Sylvester was 55. It was thought that Sylvester would retire frommathematics with his retirement from Woolwich, and this belief was reinforced by the 1870 publication of Sylvester's only

James J. Sylvester

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book, The Laws of Verse. It was on poetry.

Sylvester rejoined his mathematical life in 1876, when he accepted a chair at Johns Hopkins University. He founded theAmerican Journal of Mathematics, the first mathematical journal in the United States, the following year.

In 1883, the 68-year-old Sylvester was appointed to the Savilian chair of Geometry at Oxford. He remained in this position untilhis eyesight and memory began to fail. He continued to work on mathematics until he suffered a paralytic stroke early in Marchof 1897. He died on March 15, 1897 at the age of 83. He never married.

James J. Sylvester

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The Enjoyment ofMath is still available

today.

Hans Rademacher1892-1969

Otto Toeplitz was born on August 1, 1881 in Breslau, Germany (now Wroclaw,Poland). He was appointed professor at Bonn, but the Nazis dismissed him in1935 because of his Jewish background.

He began his academic career after studying under Hilbert at Göttingen. Hisscientific work centered on the theory of integral equations and the theory offunctions of infinitely many variables. He made lasting contributions to thesefields.

In the 1930s he developed a general theory of infinite dimensional spaces.

Toeplitz thoroughly enjoyed the history of mathematics. He wrote an excellentbook on the history of calculus, The Calculus: A Genetic Approach, whichpublished in 1949 in German. In 1963, the English version appeared; bothversions were published posthumously.

His other very famous book was a joint effort with Hans Rademacher, TheEnjoyment of Math, which is still available today. This was originally publishedin German in the early 1930s. The English version was published in themid-1950s, also posthumously.

Toeplitz moved to Jerusalem in the spring of 1939 and died less than a year lateron February 19, 1940.

Otto Toeplitz

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James W. Cooley

John Tukey's parents were both teachers. They realized when he was quiteyoung that he had great potential and decided to keep him under their wingand home-school him. His formal education began at Brown Universitywhere he studied chemistry and mathematics. He later went on to PrincetonUniversity where he received his Ph.D. in mathematics.

Tukey stayed on at Princeton as an instructor and later joined the FireControl Research office where he began his work in statistics. After the war,he returned to Princeton as a professor of statistician the mathematicsdepartment. He had so much energy that he joined AT&T Bell Labs at thesame time.

In 1965 Tukey and James W. Cooley laid out a scheme that sped up one ofthe most common activities in scientific and engineering practice-thecomputation of Fourier transforms. Their algorithm, which soon came to becalled the fast Fourier transform (FFT) is widely credited with making manyinnovations in modern technology feasible. Its impact extends frombiomedical engineering to the design of aerodynamically efficient aircraft.

Tukey has made substantial contributions to the analysis of variance and theproblem of making simultaneous inferences about a set of parameter valuedfrom a single experiment. His colleague and co-author Fred Mosteller hassaid of him, "John loves to work with others, and many have had thepleasure in participating in his genius. Variety and breadth mark hisaccomplishments. He works successfully on both large- and small-scaleproblems and on both practical and theoretical problems. . . . He is alwayseager to respond to new questions, and he gives generously of his time andideas."

John Wilder Tukey

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Probability,Statistics andTruth, written

by von Mises, isstill in print.

John Venn1834-1923

Richard von Mises was born April 19, 1883 in Lemberg, Austria (now Lviv, Ukraine).He studied at Vienna and became professor at the University of Strassburg in 1909. Hemoved to Germany in 1920 when he was appointed to direct the Institute for AppliedMathematics at Berlin, a position he kept until the Nazis forced him out in 1933. VonMises first went to Istanbul and in 1939 moved to the United States, where he taughtat Harvard University.

His scientific and mathematical contributions are diverse. His first interest was fluidmechanics, particularly in relation to aerodynamics and aeronautics. Here he madefundamental advances in boundary-layer-flow theory and airfoil design. In 1915 hebuilt an aeroplane for the Austrian military and became a pilot for the Austrian Armyin World War I.

His primary work in statistics involved the theory of measure and appliedmathematics. He also made considerable progress in the area of frequency analysisstarted by John Venn. His association with the Viennese school logical positivismdrew him to probability theory. Eventually he came to the conclusion that aprobability cannot be simply the limiting value of a relative frequency, and added theproviso that any event should be irregularly or randomly distributed in the series ofoccasions in which its probability is measured.

In 1951 von Mises wrote a philosophical book, Positivism: A Study in HumanUnderstanding, where he summarized his views on science and life.

He died on July 14, 1953 in Boston.

Richard von Mises

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Albert Einstein1879-1955


lunar craters named afterthem. Weyl is one of

them.

Hermann Weyl (known as Peter to his close friends) was born on November 9, 1885 in Elmshorn, Germany. He was educated atthe universities of Munich and Göttingen before winning his doctorate from the latter in 1908. David Hilbert served as Weyl'ssupervisor at Göttingen, and it was here that Weyl began his teaching career.

From 1913 to 1930 Weyl held the chair of mathematics at Zürich Technische Hochschule, where he counted Albert Einstein asone of his colleagues. Weyl then moved to Göttingen, where he held the chair of mathematics from 1930 to 1933. After leavingGöttingen in 1933 to escape the Nazi regime, Weyl moved to the Institute for Advanced Study at Princeton, where he remaineduntil his retirement in 1952.

One of Weyl's outstanding qualities was his ability to join previously unrelated subjects. In Die Idee der Riemannschen Fläche(1913), Weyl united analysis, geometry, and topology, thereby creating a new branch of mathematics.

Weyl also published works on philosophy, logic, and the history of mathematics. He passed away on December 8, 1955, inZürich, Switzerland.

Hermann Weyl

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Hermann Weyl

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Issai Schur1875-1941

Erhard Schmidt1876-1959

Helmut Wielandt was born in Niedereggenen, Lörrach, Germany, on December 19,1910. In 1929 he entered the University of Berlin, where he studied mathematics,physics, and philosophy. He was awarded a doctorate in 1935; his dissertation wason permutation groups. During his time in Berlin, he was greatly influenced byErhard Schmidt and Issai Schur.

Wielandt worked on the editorial staff of Jahrbuch über die Fortschritte derMathematik in Berlin from 1934 to 1938. He then went to be an assistant atTübingen; it was here that he submitted his habilitation thesis in 1939. World WarII interrupted Wielandt's career at Tübingen in 1939, although he was formally stillpart of the staff until 1946.

The end of World War II brought Wielandt to the University of Mainz, where hewas appointed an associate professor. He stayed at Mainz until 1951; he then wentto the University of Tübingen as an Ordinary Professor. Wielandt remained atTübingen until he retired in 1977. He held a number of visiting positions atdifferent universities, including the University of Wisconsin, Madison; theUniversity of Warwick, England; and the University of Brazil. He was the editor ofMathematische Zeitschrift from 1952 to 1972.

Helmut Wielandt

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Garrett Birkhoff1911-1996

David Kincaid

David M. Young, Jr. is a leader in the field of computational mathematics. Hebecame involved in this field when he was a graduate student at Harvard in thelate 1940s. He obtained his Ph.D. in 1950 under the direction of Garrett Birkhoff.He is best known for the development of theoretical analysis associated with thesuccessive overrelaxation (SOR) method. This material is now consideredclassical analysis and appears in most textbooks on numerical analysis.

Young is former director of the Computation Center at the University of Texas atAustin. While serving in this post he acquired two large computer systems thathelped establish the university as the leader in scientific computing---the ControlData 1604 computer in 1961 and the Control Data 6600 in 1966. He left theComputation Center in 1970 to become director of the Center for NumericalAnalysis.

Young has worked extensively on the numerical solution of partial differentialequations, emphasizing the use of iterative methods for solving large sparsesystems of linear algebraic equations. He and David Kincaid are providing theinfrastructure for scientists in a wide variety of disciplines to use high performancecomputers in their research. Young and Kincaid specialize in developingalgorithms for using supercomputers in solving scientific problems. "Large sparsesystems arise in many applications-reservoir simulation in petroleum engineering,heat and groundwater flow in fluid mechanics, computer and telecommunicationsnetworks in computer science, reactor modeling in nuclear engineering; tomention a few," explains Young.

Young has been at the University of Texas at Austin since 1958. He is currently anAshbel Smith Profesor of Mathematics and Computer Sciences. He is the authorof several books, including Iterative Solution of Large Linear Systems, A Survey ofMathematics with R. Gregory, and Applied Iterative Method with L. Hageman. Heis still active in both research and teaching and he plays tennis regularly!

David M. Young

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