3306353 great mathematicians[1]

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Great MathematiciansMany of the methods and equations used in numerical methods are associated withthe names of famous mathematicians and scientists. Here, we provide biographicalsketches of the more notable pre-twentieth century figures of the modern mathematical era. As will be seen in the sketches, even the most well recognized puremathematicians worked on applied problems; indeed, some of their advances were made on the way to solving such problems. To appreciate their work, we must remem

ber that they did not have the tools we take for granted - they developed them!To help with their places in history, the figure below shows the life-spans of those that are discussed.



John Couch Adams [1819-1892]Adams was born in Cornwall and educated at Cambridge University. He was later appointed Lowndean Professor and Director of the Observatory at Cambridge. In 1845, he calculated the position of a planet beyond Uranus that could account for perturbations in the orbit of Uranus. His requests for help in looking for the planet, Neptune; met with little response among English astronomers. An independentset of calculations was



completed in 1846 by Leverrier, whose suggestions to the German astronomer Johann Galle led to Neptune's discovery. Adams published a memoir on the mean motionof the Moon in 1855 and computed the orbit of the Leonids in 1867. The Leonids are meteor showers that appear to originate in the constellation Leo. They were especially prominent every 33 years from 902 to 1866.

Charles Babbage [1792-1871]

Babbage's design of the Analytical Engine is considered to be the forerunner ofthe modern computer. Lack of technology and money prevented Babbage from realizing his design; however, a model built from his plans at a later date worked as Babbage had predicted. Babbage's ideas on the Analytic Engine would have been lost if Ada Lovelace had not clearly described them along with her own ideas. Although computer scientists associate Babbage with the computer, he was better knownas a prominent mathematician of his time, and he held the position of LucasianProfessor at Cambridge. His important contributions were on the calculus of functions. Along with George Peacock and John Herschel, Babbage formed the Analytical Society to promote analytical methods and the use of Leibniz's differential notation (the geometrically suggestive dy/dx form that we know today). English mathematicians used Newton's fluxion notation almost exclusively until then, partly

because it was widely held among them that Leibniz appropriated Newton's ideasabout calculus and claimed them as his own. The fluxion notation (the use of a dot above the variable x to indicate the derivative, still used in many texts today) was abstract enough to hamper developments in calculus. Babbage's objectivewas to replace this "dot-age" with "d-ism" at Cambridge.

George Boole [1815-1864]Boole was born in Lincoln, in eastern England. In addition to his mathematical prowess, he studied classics on his own. Boole's work on linear transformations led to some aspects of the theory of invariants. He also performed research on differential equations and the calculus of finite differences. Boole is best remembered as one of the creators of mathematical logic, which is one of the foundations of modern computer technology.

Arthur Cayley [1821 – 1895]



Arthur Cayley was a British mathematician and astronomer. After graduation in 1842, he studied and practiced law. During the fourteen years Cayley was at the bar, he wrote nearly 300 mathematical papers, including some of his best and mostoriginal work. During the same time, he met the mathematician, James Joseph Sylvester, who was also devoting his time to both law and mathematics. They worked together and founded their greatest work, the algebraic theory of invariants, which played a crucial role in the development of the theory of relativity. In 1863

Cayley was elected to the Sadlerian chair of pure mathematics at Cambridge University, a position which he held until his death. He won several academic honorsincluding the Royal medal in 1859 and the Copley medal in 1881 from the Royal Society. He published the book, ``Treatise on Elliptic Functions'', in 1876. Cayley invented and developed the theory of matrices. His other contributions are inthe areas of n-dimensional geometry, theory of abstract groups, and physical astronomy.

Pafnuti Lvovich Chebyshev [1821-1894]Chebyshev was born in Okatovo, Kaluga region in Russia. He was one of the most famous Russian mathematicians and he made numerous important contributions to thetheory of numbers, algebra, theory of probability, analysis, and applied mathem

atics. He completed his secondary education at home and enrolled in the department of physics and mathematics at Moscow University in 1837. He graduated with adegree in mathematics in 1841. In 1841, he won a silver medal for deriving an error estimate in the NewtonRaphson iterative method. He received his doctorate inmathematics from Petersburg University in 1849. In 1850, Chebyshev was electedextraordinary professor of mathematics at Petersburg University where he becamea full professor in 1860. Chebyshev was very curious about mechanical inventionsduring childhood and it was stated that during his very first lesson in geometry he saw its applicatins to mechanics. His technological inventions include a calculating machine built in the late 1870s. When his father became very poor during the famine of 1840, Chebyshev helped support his family. He became interestedin the theory of numbers and stated the Chebyshev problem relating probabilityto the theory of numbers. He died in St. Petersburg, Russia on December 8, 1894.

Roger Cotes [1682-1716]Cotes was educated at Cambridge and later was the university's Plumian Professorof Astronomy. He was well regarded by his contemporaries, including Newton. Much of Cotes's time was spent in editing a second edition of Newton's Principia Mathematica. His other work included hydrostatics, treatments of rational algebraic expressions, the



earliest attempt to form a theory of errors, applications of the method of differences, and problems in particle dynamics.

Gabriel Cramer [1704-1752]Gabriel Cramer (1704 - 1752) was born and educated in Geneva, Switzerland and defended a thesis dealing with sound at the age of eighteen. At age 20, he competed for the chair of philosophy at the Academie de Calvin in Geneva. Due to his yo

ung age, he was appointed as a co-chair of mathematics. He was promoted to chairof mathematics in 1734 and was made professor of philosophy in 1750. Cramer published his major work, ``Introduction a l'analyse des lignes courbes algebriques'', in 1750. He received many honors, including membership in the Royal Societyof London, the academies of Berlin, Lyons. Montpellier, and the Institute of Bologna. Although he is well-known for Cramer's rule and Cramer's paradox, neitherof these were totally his original contributions. In fact, his actual original contributions on algebraic curves and mathematical utility are less well-known. In the context of developing a theory of algebraic curves, Cramer included a method of solving systems of algebraic equations, now known as Cramer's rule, in anappendix. Cramer never married and died in Bagnolssur-ceze, France.

René Descartes [1596-1650]Descartes was born in Tours, France and treated mathematics as a hobby while inthe army as a young man. His primary contributions to mathematics are in analytical geometry and the theory of vortices. His work laid the foundation for analytical geometry and focused on the two-dimensional rectangular coordinate system;however, it is clear that he was well aware of three-dimensional representationsof a point in space. He formulated the rule of signs for the positive and negative roots of polynomials. Newton later attempted to formulate a similar rule forthe complex roots. Descartes is responsible for the custom of using early letters of the alphabet for known quantities and those near the end of the alphabet for unknown quantities. He also invented the notation for expressing powers. Descartes attempted at one time to give a physical theory of the universe. He abandoned it when he realized that it would result in conflict with the Church. In any

case, eight of the ten laws of nature he proposed were incorrect. The first two, however, were almost identical to Newton's.



early

As a child, Descartes was allowed to stay in bed until late in the morning because of his frail health. He continued this practice later in life, and stated theopinion, obviously shared by many, that good mathematics and good health were possible only if one did not wake up too in the morning.

Peter Gustav Lejeune Dirichlet [1805-1859]Dirichlet was the student of Gauss and the son-in-law of Jacobi. He succeeded Gauss as Professor of Higher Mathematics at Göttingen. He devoted much time to expositions of works by Gauss and Jacobi. His own work established Fourier s Theorem(on heat conduction) and dealt with the theory of numbers, the theory of the potential, fifth-degree equations, and definite integrals.

Leonhard Euler [1707-1783]Euler was born in Switzerland, studied under Johann Bernoulli at Basel, and completed his Master's degree at age 16. He formed a lifelong friendship with Bernoulli's sons Daniel and Nicholas. When they went to Russia at the invitation of Catherine I, Empress of Russia, they obtained a place for Euler at the Academy of

Sciences in St. Petersburg. Euler eventually became Professor of Mathematics in1733 when the chair was vacated by Daniel Bernoulli. In 1741, Euler joined the Berlin Academy of Sciences at the strong request of Frederick the Great. He returned to St. Petersburg 25 years later (and was succeeded at Berlin by Lagrange).Euler was responsible for establishing Newtonian thought in Russia and Prussia.Euler was blind in one eye by the time he was in his late 20s; within a few years of returning to Russia from Berlin, he was almost totally blind. Despite thisand other misfortunes (including a fire that destroyed many of his papers), Euler was one of the most competent and prolific mathematicians of any time. Among Euler's contributions to mathematics were extensive revisions of almost all of the branches of mathematics. He gave a full analytic treatment of algebra, the theory of equations, trigonometry, and analytical geometry. He treated series expansions of functions and stated the rule that only convergent infinite series coul

d be used safely. He dealt with three-dimensional surfaces, calculus and calculus of variations, number theory, and imaginary numbers among other subjects. He introduced the current notations for the trigonometric functions (at about the same time as Simpson) and showed the relation



between the trigonometric and exponential functions in the equation that bears his name - (exp(iθ) = cosθ + i sinθ). Another Euler e uation (ν + f - e = 2) relates the¡

umber of vertices ν, the¡

umber of faces f, a¡

d the¡

umber of edges of a polyhedro

¡

. The Beta a¡

d Gamma fu¡

ctio¡

s were i¡

ve¡

ted by Euler. Outside of pure mathematics, Euler made sig

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dacoustics. Yet a

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curre¡

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, Lagra¡

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¡

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abled Joha¡ ¡

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ar tables, whichear

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ed his widow £5000 from the E¡

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ter has Euler's impri

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to the symbol π, the ex¢ onential symbol e, the functional notation f(x), the imaginary number i, and the summation symbol Σ. To close the introduction to Euler, the ¢ articularly extraordinary Euler magic s uare (Ref. 21)is shown in Fig. A-2. In magic s uares, the integers from 1 to n2 fill the (n xn) cells of a matrix in such a way that all row sums, column sums, and diagonal-sums are identical. Most ¢ eo¢ le are familiar with the (3 x 3) s uare. Euler's s

uare is an (8 x 8) matrix in which the row sums and column sums (but not the diagonal sums) are identical. The interesting features are that the sum for half arow or column is half of the full sum, and that the numbers re¢ resent consecutive moves that a knight makes on a chessboard to hit every s uare once.

(Baron) Jean Ba¢ tiste Jose¢ h Fourier [1768-1830]Fourier was among the ¢ rominent French ¢ hysicists who also had su¢ erb abilitiesin mathematics. He is famous for his ex¢ eriments on heat conduction which, alongwith



ideas drawn from Newton's Law of Cooling, gave rise to Fourier's Theorem. Othershad ¢ ro¢ osed similar ideas - Lagrange had given s¢ ecific cases and Budan had stated the same theorem without satisfactory ¢ roof. Fourier's work on the analytical theory of heat contained the Fourier sine series, which is widely used in modern analysis. Fourier had accom¢ anied Na¢ oleon's eastern ex¢ edition to Egy¢ t andserved as Governor of Lower Egy¢ t from 1798 until the French surrendered to British forces in 1801. He was created a Baron in 1808 by Na¢ oleon.

Karl Friedrich Gauss [1777-1855]Gauss was born in Braunschweig, Germany. Gauss, Lagrange, and La¢ lace are widelyconsidered to be giants of analysis. Gauss's interests were so far ranging thatthey o¢ ened avenues of investigation for many others. His notable mathematicalwork included the theory of numbers, various branches of algebra, and the theoryof determinants; the last formed the basis for Jacobi's work in that area. He had also obtained certain results on the theory of functions that were later found by Abel and Jacobi; however, these were not ¢ ublished. Gauss also develo¢ ed the method of least s uares and the fundamental laws of ¢ robability distributions.The reluctance to ¢ ublish was ¢ erha¢ s also related to Gauss's style. His oral ¢

resentations contained much of the analysis that was obscured in his ¢ ublished w

ork, but he was unwilling to allow his students to take notes. In his

¢

ublishedwork, he removed all of his analytical ste¢ s and re¢ laced them with extremely brief, though rigorous, ¢ roofs. As a result, his ¢ ublished work was often difficult to follow. Gauss's interests included astronomy (he calculated the orbital elements of the asteroid Ceres following its discovery by Piazzi). His analysis resulted in an a¢ ¢ ointment as Director of the Göttingen observatory and as Professorof Astronomy. Although he retained these ¢ ositions until he died, Gauss moved onto other subjects. Among the other subjects were geodesy, o¢ tics, and electricity and magnetism. His work on the last is commemorated by the Gauss as the unitof magnetic flux density. Gauss and Weber invented the declination instrument and the magnetometer, and they built an ironfree magnetic' observatory at Göttingen.Among their researches, they demonstrated the feasibility of telegra¢ hic communications.



Charles Hermite [1822 – 1901]Charles Hermite was a French mathematician, who was the sixth of the seven children of Ferdinand Hermite and Madeleine Lallemand. He studied in Paris, first atthe College Henri IV and later at the College Louis-le-Grand. There he read theworks of Euler, Gauss and Lagrange instead of ¢ re¢ aring for his examinations. Later he tried to continue his studies at Ecole Polytechni ue. He got admitted with a ¢ oor rank; but left Polytechni ue without graduation. He took the examinatio

ns for a career of¢

rofesseur. Later he studied the works of Cauchy and Liouville on function theory as well as those of Jacobi on elli¢ tic and hy¢ erbolic functions, and generalized some of the theorems. In 1843, when he was only twenty years old, communicated his discoveries to Jacobi who got those letters ¢ rinted inCrille's Journal. Hermite joined the Ecole Polytechni ue in 1848 as re¢ etiteur and admissions examiner and subse uently took over Duhamel's chair as ¢ rofessor of analysis at the same ¢ lace. He became an ins¢ iring figure in mathematics not only during his life time but also afterwards. In todays mathematics, Hermite isremembered through Hermitean forms, a com¢ lex generalization of uadratic forms,Hermitean ¢ olynomials, his solution of Lame differential e uation, Hermite inter¢ olation ¢ rocedure, and his solution of fifth-degree e uation by elli¢ tic functions.

Carl Gustav Jacob Jacobi [1804-1851]Carl Gustav Jacob Jacobi (born on 10 December 1804 in Potsdam, Germany and diedon 18 February 1851 in Berlin, Germany) was a German mathematician who was bornin a wealthy and cultured family. He entered the Gymnasium at Potsdam in 1816 and excelled in Greek, Latin, history and mathematics by the time he graduated from the Gymnasium in 1821. He then joined the University of Berlin and after finding the level of lectures in mathematics to be elementary, he studied the works of Euler, Lagrange and other leading mathematicians ¢ rivately and mastered them.He submitted his Ph.D. thesis in 1825 and began his career as Privatdozent at the University of Berlin at the age of twenty. After seeing no ¢ ros¢ ect of ¢ romotion in Berlin, he moved to the University of Konigsberg in May 1826 and became afull ¢ rofessor in 1832. Jacobi married Marie

£ chwinck in 1831 and they had five

sons and three daughters. Jacobi's fundamental research, mostly in the theory ofelli¢ tic functions, mathematical analysis, number theory, geometry, and mechanics, was ¢ ublished in Crelle's Journal fur die reine und angewandte Mathematik. Jacobi linked his research to different mathematical disci¢ lines. For exam¢ le, heintroduced elli¢ tic functions not only into the number theory, but also into the theories of integration and differential e uations. Although the theory of determinants was started by Leibniz, Jacobi ¢ resented it systematically and also introduced the term "Jacobian" in the theory of determinants. The methods he develo¢ ed for solving linear algebraic e uations and the algebraic eigenvalue ¢ roblemhave become known as Jacobi methods.



Jacobi had small¢ ox and died in the early ¢ art of 1851. Dirichlet, who was a close friend of Jacobi, delivered the memorial lecture at the Berlin Academy on 1 July 1852 and called him the greatest mathematicians among the members of the Academy since Lagrange.

Camille Jordan [1838 – 1921]Camille Jordan was born in Lyons, France on January 5, 1838 and died in Paris on

January 22, 1921. He was born into a well-to-do family where his father was anengineer and mother was a sister of the famous ¢ ainter Pierre Puvis de Chavannes. Jordan was a brilliant student, entered Ecole Polytechni ue at the age of 17,and continued as a ¢ ractising engineer until 1885. He taught simultaneously at the Ecole Polytechni ue and the College de France from 1873 until his retirementin 1912. He was elected a member of the Academy of

£ ciences in 1881. While worki

ng as an engineer, Jordan wrote 120 mathematical research ¢ a¢ ers. He was considered a universal mathematician who ¢ ublished ¢ a¢ ers in ¢ ractically all branches of mathematics of his time. He made many fundamental contributions to grou¢ theory including the first ¢ art of the famous Jordan-Holder theorem and was regardedas the undis¢ uted master of grou¢ theory. He ¢ ublished his results of grou¢ theory in ``Traite des substitutions'' in 1870, which remained a bible in grou¢ theo

ry for several years. In addition, Jordan obtained several im

¢

ortant results inalgebra including the finiteness theorems. His works were considered to be the source for the discoveries of his students - Lie's ``continuous grou¢ s'' and Klein's ``discontinuous grou¢ s''. The refinement he suggested to the Gauss elimination method for solving linear simultaneous e uations has become known as the Gauss-Jordan elimination method.

Kutta, Wilhelm (1867-1944)Kutta, Wilhelm was a German mathematician and aerodynamist who extended the Runge-Kutta method develo¢ ed by Runge for numerically solving differential e uationsto systems of e uations. He also made im¢ ortant contribution to the theory of airfoils.

Jose¢

h Louis Lagrange [ 1736-1813 )Lagrange, born in Turin, Italy, was one of the greatest of the eighteenth-century mathematicians. He did not show any taste for mathematics until he was 17. Then, selftaught, he became a lecturer at 18 after only a year's study. At 19, he wrote to Euler with the solution of an iso¢ erimetrical ¢ roblem that had been discussed for over fifty years. The method used by Lagrange contained the ¢ rinci¢ lesof calculus of variations. Euler, recognizing the su¢ eriority of Lagrange's a¢ ¢

roach, withheld his own ¢ a¢ er on the



¢ roblem. Lagrange was thus allowed to com¢ lete his work and to receive the credit for the invention of a new form of calculus. Lagrange's later works included corrections or im¢ rovements to works by such eminent mathematicians as Newton, Euler, Taylor, and D'Alembert. He gave the com¢ lete solution for the transverse vibration of a string and discussed echoes, the ¢ henomenon of beats, and com¢ oundsounds. Other major work contained solutions of several ¢ roblems in dynamics bythe calculus of variations. Lagrange's style was to seek general solutions to ¢ r

oblems; even so, his work was easy to follow because of the meticulous care he used to ex¢ lain his ¢ rocedures.

Pierre£ imon (Mar uis de) La¢ lace [1749-1827]

La¢ lace was born in Normandy. He began his ¢ rofessional life on the basis of a recommendation from D'Alembert, who was im¢ ressed by a ¢ a¢ er on mechanics. Amonghis early contributions were ¢ roofs of the stability of ¢ lanetary motions and work on integral calculus, finite differences, and differential e uations. In the1780s, he determined the attraction of a s¢ heroid on an exterior ¢ article; in sodoing, he introduced s¢ herical harmonics (or La¢ lace coefficients) and develo¢ ed the conce¢ t of the ¢ otential.

£

imilar coefficients for two-dimensional s¢ ace had been ¢ resented earlier by Legendre, and the idea of the ¢ otential was taken f

rom Lagrange's earlier works. Because of his

¢

ersonality, La

¢

lace was not well liked. He gave either little or no acknowledgment of results that he had a¢ ¢ ro¢ riated from others. He did not care if ¢ roofs of his work were nonexistent or ¢ resented incorrectly; he was satisfied that his results were correct. Des¢ ite his ¢

ettiness, La¢ lace was a very ca¢ able mathematician. He develo¢ ed the La¢ lacian e uation for ¢ otentials, and did extensive work on çelestial mechanics. In his volumes on celestial mechanics, La¢ lace ¢ ut forth the nebular hy¢ othesis; that is, that the solar system evolved from a rotating gaseous nebula. La¢ lace also ¢ resented the formal ¢ roofs for the method of least s uares, which had been given em¢ irically by Gauss and Legendre. These ¢ roofs contain exam¢ les of La¢ lace's a¢ ¢ roach - his results were correet, but the analysis was so scanty and had so many errors that many ¢ eo¢ le uestioned if he had actually done the work he ¢ resented.Other contributions were on determinants (at the same time as Vandermonde), on

uadratic factors for e

uations of even degree, on definite integrals as solutions to linear differential e uations, and on solutions to the linear ¢ artial differential e uation. The theory of ca¢ illary attraction is also due to La¢ lace.



Aside from scientific recognition, La¢ lace sought social ¢ rominence. He was given the ¢ ost of Minister of the Interior by Na¢ oleon, who sought su¢ ¢ ort from thescientific community, but he was removed in less than two months because of incom¢ etence. Later, when it was clear that Na¢ oleon's em¢ ire was crumbling, La¢ laceoffered his services to the Bourbons and was granted the title of Mar uis.

Adrian Marie Legendre [1752-1833]

Legendre was born in Toulouse and educated in Paris. He had the misfortune of having lived at the same time as La¢ lace. In addition to a ¢ rofessorial a¢ ¢ ointment, Legendre held various ¢ ublic service and minor governmental ¢ ositions. Any ambitions he may have had for greater recognition were stifled by La¢ lace's influence and hostility. Legendre's major contributions were in geometry, the theory of numbers, various to¢ ics in integral calculus, and elli¢ tic functions. Among these are s¢ ecific instances of s¢ herical harmonics and work on the method of least s uares. In both cases, he was u¢ staged by La¢ lace who develo¢ ed the full formof s¢ herical harmonics and gave formal ¢ roofs for the method of least s uares.His treatment of elli¢ tic integrals also gave way to later su¢ erior methods by Abel and Jacobi.

Urbain Jean Jose

¢

h Leverrier [1811-1877]Leverrier was born in£

t. Lö, was educated at the Polytechnic£

chool in Paris, andwas later a¢ ¢ ointed as a lecturer there. He, inde¢ endently of and later than Adams, calculated the orbit of Ne¢ tune. It was his suggestion to Johann Galle thatactually led to Ne¢ tune's discovery within 1 degree of the ¢ redicted location.Leverrier's main work was in revising tables of ¢ lanetary motion.

(Baron) Gottfried Wilhelm Leibniz [16461716]Leibniz was born in Lei¢ zig. His mastery of to¢ ics ranged over mathematics, classical languages, ¢ hiloso¢ hy, theology, and law. His early mathematical contributions included work on combinations and an im¢ rovement of Pascal's



calculating machine. His more im¢ ortant mathematical contribution was in the develo¢ ment of calculus. Des¢ ite controversy about the source of Leibniz's ideas (some thought that he had access to Newton's work), it is clear that his differential (dy/dx) notation was instrumental in the develo¢ ment of calculus. Other notational conveniences that were introduced by Leibniz include the dot as a symbolfor multi¢ lication, the e ual sign, the integral sign, and the decimal ¢ oint. Heis also credited with the develo¢ ment of the binary number system. Leibniz is a

lso a major figure in the history of¢

hiloso¢

hy. He held that beings called monads were the ultimate elements of the universe, and inferred the existence of Godfrom the harmony that existed among the monads. Euler was one who strongly o¢ ¢ osed this ¢ hiloso¢ hy. Leibniz dabbled in dynamics, but it is clear that his knowledge in that area was limited. He also urged Peter the Great to establish the Academy of

£ ciences at

£ t. Petersburg.

(£ ir) Isaac Newton [1642-1727]

Newton was born in Lincolnshire and was educated at Cambridge. He later held theLucasian Chair at Cambridge (the same one later held by Babbage). Newton holdsa ¢ rominent ¢ lace in science and mathematics for his conce¢ t of infinitesimal calculus, his Law of Gravitation, his Laws of Motion, and his work on o¢ tics. The

last included inventions of a refracting telesco

¢

e, a reflecting microsco

¢

e, andthe sextant. Newton's work on calculus used the fluxion notation, which was very difficult to master. The controversy with Leibniz caused many English scholarsto ¢ ersist in using this notation and resulted in the hindrance of mathematicaldevelo¢ ments until Babbage and his colleagues broke free of that ¢ rejudice. Newton's genius was so widely recognized that he was always consulted or challenged. For exam¢ le, he acted as editor for other works, and was consulted by Leibnizon infinite series, by Halley on gravitation, and by Hooke on the Earth's diurnal motion. Newton had another controversial relation with Hooke and others regarding the theory of colors. He was challenged by Johann Bernoulli to solve the brachistochrone ¢ roblem (the curve, now known to be the cycloid, which allows uickest descent from one ¢ oint to another under gravity) and another locus ¢ roblem.Newton accom¢ lished in a day what had taken Leibniz six months to solve. Another

challenge resulted in Newton's laying down the¢

rinci¢

les of trajectories in amatter of hours. It also seemed that Newton took the least obvious route in demonstrating some of his hy¢ otheses. For exam¢ le, he sought to verify his early hy¢

othesis on gravitation by considering the orbit of the Moon. Incorrect estimatesof distances caused the first attem¢ t at verification to fail. He later re¢ eated the calculations successfully with more



accurate estimates obtained in the course of Hooke's consultation. Another exam¢

le is his develo¢ ment of the series ex¢ ansion for the inverse sine function, from which he then deduced the ex¢ ansion for the sine. The genius in Newton is exem¢ lified by the ¢ raise he received from Lagrange and even from La¢ lace. Above all, there is the tribute ¢ aid to him by Gauss, another of the truly great minds. Gauss used words like mangus or clarus to describe other great mathematicians and ¢ hiloso¢ hers, but he reserved the word summus (the best) only for Newton. Newto

n was knighted in 1705.

Blaise Pascal [1623-1662]Pascal was born in Clermont. Because of his health, his father restricted his studies to languages and ¢ rohibited the study of mathematics so that he would notbe overworked. Pascal's curiosity soon led him to disregard his father's injunction, and he undertook the study of geometry. Pascal wrote a ¢ a¢ er on conic sections at age 16 and built his celebrated adding machine at age 18. In later years,he went back and forth between mathematics and religious ¢ hiloso¢ hy. Among hismathematical works are those related to the ¢ hysics of gases and li uids, creation of the theory of ¢ robability (along with Fermat), and the creation of Pascal's triangle. He also devoted time to the study of cycloids, in which he effective

ly found the definite integrals of some trigonometric functions by summation techni ues.

£ imeon Denis Poisson [1781-1840]Poisson was born in Pithiviers and was educated by his father to be a ¢ hysician.His aversion to that ¢ rofession turned into ¢ ermanent abandonment when one of the first ¢ atients he treated by himself died (though not through any fault of Poisson). He turned to mathematics and became a ¢ rolific contributor on the a¢ ¢ lications of mathematics to ¢ roblems in ¢ hysics. The fields with which he dealt included ¢ robability, mechanics, ca¢ illary action, heat, electrostatics, and magnetism. These last two s¢ awned new branches of mathematical ¢ hysics. A major mathematical accom¢ lishment was on the a¢ ¢ lication of Fourier series to the solutionsof ¢ hysical ¢ roblems. Another, by which he is best remembered, is the correction

of La¢

lace's e

uation to¢

roduce the Poisson e

nation for the¢

otential.



Jose¢ h Ra¢ hson [1648 – 1715]Jose¢ h Ra¢ hson was an English mathematician, a Fellow of the Royal

£ ociety of Lo

ndon and a friend of Newton. During the great dis¢ ute in the mathematical community at that time over the discovery of differential calculus, naturally Ra¢ hsonsided with Newton (instead of Leibnitz).

Carl David Tolme Runge [1856 – 1927]

Carl David Tolme Runge was born in Bremen, Germany in a merchant family. His father accumulated comfortable ca¢ ital before his death in 1864. While his brothers ¢ ursued commercial careers, Runge showed interest in more intellectual careersfrom childhood. After com¢ leting the Gymnasium at age nineteen, he enrolled in the University of Munich and took courses with Max Planck with whom he maintainedfriendshi¢ and contact throughout his life. In the fall of 1877, Planck and Runge went to Berlin to attend the lectures of Kirchoff and Helmholtz. However, finding not much attraction to those lectures, Runge turned to ¢ ure mathematics andbecame a disci¢ le of Weierstrass. He com¢ leted his doctorate on differetial geometry in the s¢ ring of 1880. Runge, influenced by Kronecker, continued to work on a variety of ¢ roblems in algebra and function theory, including the numericalsolution of algebraic e uations. He was given a ¢ rofessorshi¢ at Technische Hoch

schule in Hannover in 1886 as a condition of marriage.£

ome of the methods Rungedevelo¢ ed have become very ¢ o¢ ular; the RungeKutta method for the numerical integration of differential e uations is amenable for digital com¢ uters and still remains current.

Phili¢ ¢ Ludwig von£

eidel [1821 – 1896]Phili¢ ¢ Ludwig von

£

eidel was a German astronomer and mathematician. He was bornin Zweibrucken.

£ ince his father, Justus Christian Felix

£ eidel, was a ¢ ost off

ice official, young Phili¢ ¢ £ eidel had to s¢ end his childhood at several ¢ laces.

After graduating from school, he took ¢ rivate lessons in mathematics from L. C.

£

chnurlein, who studied under Gauss.£

eidel entered Berlin University in 1840 and attended the lectures of Dirichlet and Encke. He moved to Konigsberg in 1842and studied with Bessel and Jacobi. In 1843, he moved to Munich and obtained his

doctorate for the dissertation, Uber die beste Form der £

¢

iegel in Telesko¢

en,in 1846.

£ eidel's major investigations were in the fields of dio¢ trics and mathe

matical analysis with some contributions to the method of least s uares, ¢ robability theory and ¢ hotometry. The method he ¢ ro¢ osed for the solution of linear algebraic e uations has become known as Gauss-

£ eidel iteration method. The ¢ hotome

tric measurements of fixed stars and ¢ lanets he made were the first ones to be made and his investigations led to the ¢ roduction of im¢ roved telesco¢ es. He a¢ ¢ lied ¢ robability theory to astronomy and studied the relation between the fre uency of certain



diseases and climate conditions at Munich. He was made a member of the BavarianAcademy of

£ ciences in 1851 and a full ¢ rofessor at Bavaria in 1855.

£ eidel reti

red early due to eye ¢ roblems and died in Munich in 1896. He remained a bachelor, had to retire early due to eye ¢ roblems, and was cared for until 1889 by his unmarried sister, Lucie, and later by the widow of the clergyman, Langhans.

Thomas£ im¢ son [1710-1761]

Thomas £ im¢

son (1710 - 1761) was born in England. His father was a weaver who wanted his son to take u¢ the same ¢ rofession. Through his studies in arithmetic and astrology,

£

im¢ son ac uired a local re¢ utation as a fortune teller during hischildhood. For some time, he worked as a weaver during day time and taught at evenings. He ¢ ublished his first mathematical contributions in the well-known ``Ladies Diary'' in 1736 and his first book, ``A New Treatise of Fluxions'' in 1737. Although Robert Heath accused him of ¢ lagiarism, it brought additional ¢ ublicity to

£ im¢ son.

£ im¢ son was a¢ ¢ ointed second mathematical master at the Royal Mil

itary Academy in 1743 and was elected fellow of the Royal£ ociety in 1745. His b

ooks on algebra, geometry and trigonometry became best sellers. He assumed the editorshi¢ of the annual ``Ladies Diary'' from 1754 and ac uired a re¢ utation asthe ablest analyst that England can boast of. It is ironic that

£ im¢ son is best

remembered for£

im

¢

son's rule, which was discovered long before him, for findingthe area under a curve as

where the curve is re¢ laced by a ¢ arabola ¢ assing through the ¢ oints (A, a), (B,b) and (C, c).

Brook Taylor [1685-l73l]Brook Taylor was born in England in a well-to-do family. Taylor's scientific work was influenced by his home life. His major scientific contributions are in theareas of vibrating string and ¢ ers¢ ective drawing. His father was interested inmusic and art and entertained many musicians in his home. The family archives of Taylor contained an un¢ ublished manuscri¢ t entitled “On Musick'' and some ¢ aintings. Taylor entered

£ t. John's College in 1701, received the LL.B. degree in 170

9, was elected to the Royal £ ociety in 1712, and was awarded the LL.D. degree in1714. During this ¢ eriod, he visited France several times and develo¢ ed scientific corres¢ ondence with Pierre Remond de Montmort and Abraham De Moivre on infinite series and ¢ robability.



Taylor ¢ ublished his first im¢ ortant ¢ a¢ er, dealing with h the determination ofthe center of oscillation of a body, in the Philoso¢ hical Transactions of the Royal

£ ocietry in 1714. Taylor's most ¢ roductive ¢ eriod was 1714-1719; his ¢ ublica

tions dealt with functional analysis and ex¢ eriments on ca¢ illarity, magnetism and thermometer. He is best known for the theorem or ¢ rocess for ex¢ anding functions into infinite series that is commonly known as ``Taylor's series ex¢ ansion''. The mathematical book he ¢ ublished in 1715, ``Methodus'', ualifies Taylor as

one of the founders of the calculus of finite differences and as one of the first to use it in inter¢ olation and summation of series.

François Viete (Franciscus Vieta) [1540-1603]Viete was born in Fontenay, was trained as a lawyer, and s¢ ent most of his lifein ¢ ublic service. He was, however, a re¢ utable mathematician and devoted much of his leisure time to mathematics. His main interests lay in algebra and geometry. He knew how to write multi¢ le angle formulas for sines and was ade¢ t at mani¢

ulating algebraic forms. His major work was on the a¢ ¢ lication of algebraic techni ues to ¢ roblems in geometry. His skill in algebra was ¢ robably hel¢ ed by hisinsistence on using notations that clearly indicated a ¢ ower, instead of the custom of assigning a different letter for each ¢ ower. Much of his later work was o

n roots of e

uations by factoring, and he devised a closedform method for com

¢

uting the roots of cubic e uations.

3306353 great mathematicians[1]

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