Alexandria, Euclid of
is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt.
Cauchy, Augustin Louis
was born on August 21, 1789, in Paris, the eldest of six children. By the time he was 11, both Laplace and Lagrange had recognized Cauchy's extraordinary talent for mathematics. In school he won prizes for Greek, Latin, and the humanities. At the age of 21, he was given a
commission in Napoleon's army as a civil engineer. For the next few years, Cauchy attended to his
engineering duties while carrying out brilliant mathematical research on the side.
In 1815, at the age of 26, Cauchy was made Professor of Mathematics at the Ecole Polytechnique and was recognized as the leading mathematician in France. Cauchy and his contemporary Gauss were the last men to know the whole of mathematics as known at their time, and both made important contributions to nearly every branch, both pure and applied, as well as to physics and astronomy.
Cauchy introduced a new level of rigor into mathematical analysis. We owe our contemporary notions of limit and continuity to him. He gave the first proof of the Fundamental Theorem of Calculus. Cauchy was the founder of complex function theory and a pioneer in the theory or permutation groups and determinants. His total output of mathematics fills 24 large quarto volumes and is second only to that of Euler. He wrote over 500 research papers after the age of 50. Cauchy died at the age of 68 on May 23, 1857.
Descartes, Réné
(1596-1650) is one of the most important Western philosophers of the past few centuries. During his lifetime, Descartes was just as famous as an original physicist, physiologist and mathematician. But it is as a highly original philosopher that he is most frequently read today. He attempted to restart philosophy in a fresh direction. For example, his philosophy refused to accept the Aristotelian and Scholastic traditions that had dominated philosophical thought throughout the Medieval period; it attempted to fully integrate philosophy with the 'new' sciences; and Descartes changed the relationship between philosophy and theology. Such new directions for philosophy made Descartes into a revolutionary figure.
The two most widely known of Descartes' philosophical ideas are those of a method of hyperbolic doubt, and the argument that, though he may doubt, he cannot doubt that he exists. The first of these comprises a key aspect of Descartes' philosophical method. As noted above, he refused to accept the authority of previous philosophers - but he also refused to accept the obviousness of his own senses. In the search for a foundation for philosophy, whatever could be doubted must be rejected. He resolves to trust only that which is clearly and distinctly seen to be beyond any doubt. In this manner, Descartes peels away the layers of beliefs and opinions that clouded his view of the truth. But, very little remains, only the simple fact of doubting itself, and the inescapable inference that something exists doubting, namely Descartes himself.
Einstein, Albert
German-American physicist who, in 1905, published three papers, each of which had a profound effect on the development of physics. In one paper, he proposed the theory of special relativity, which provides a correct description for particles traveling at high speeds. The two postulates of the special theory of relativity were that the speed of light in a vacuum is constant and that the laws of physics are the same for all inertial reference frames. Einstein did know about the Michelson-Morley experiment null result, but was not familiar with Lorentz's work after 1895, so he reinvented the Lorentz transformation for himself (Pais 1982, p. 133).
While special relativity required a modification of the laws of mechanics, the Maxwell equations were found to already satisfy the requirements of special relativity. Using special relativity, Einstein derived the equivalence of rest mass m0 and energy E, expressible as E2-p2c2 = m02c4, where c is the speed of light and p is the (relativistic) momentum . When relativistic mass is used instead (where ), the equation reduces to the famous E = mc2.
Euler, Leonhard
(1707 - 1783) Swiss mathematician who was tutored by Johann Bernoulli. He worked at the Petersburg Academy and Berlin Academy of Science. He had a phenomenal memory, and once did a calculation in his head to settle an argument between students whose computations differed in the fiftieth decimal place. Euler lost sight in his right eye in 1735, and in his left eye in 1766. Nevertheless, aided by his phenomenal memory (and having practiced writing on a large slate when his sight was failing him), he continued to publish his results by dictating them. Euler was the most prolific mathematical writer of all times finding time (even with his 13 children) to publish over 800 papers in his lifetime. He won the Paris Academy Prize 12 times. When asked for an explanation why his memoirs flowed so easily in such huge quantities, Euler is reported to have replied that his pencil seemed to surpass him in intelligence. François Arago said of him "He calculated just as men breathe, as eagles sustain themselves in the air" (Beckmann 1971, p. 143; Boyer 1968, p. 482).
Fermat, Pierre de
(1601-1665) is often called the
"Prince of Amateurs." He was the son of a prosperous
leather merchant, and became a lawyer and magistrate.
Fermat enjoyed the pleasure of discovery more than any
reputation it might gain him and published only one
important manuscript during his lifetime using the
concealing initials M.P.E.A.S. When Roberval offered to
edit and publish some of his works, Fermat replied
"whatever of my works is judged worthy of publication,
I do not want my name to appear there."
However, he carried on voluminous correspondence with
mathematicians, often stating his results piecemeal or
as challenges. Fermat was one of the founders of analytic
geometry, establish probability theory with Pascal,
and helped lay the foundation for calculus. Yet,
his true love was number theory.
In 1640, while studying perfect numbers, Fermat wrote
to Mersenne that if p is prime, then 2p
divides 2p-2. Shortly thereafter he
expanded this into what is now called Fermat's Little
Theorem. As usual, Fermat stated "I would send you a
proof, if I did not fear its being too long." Perhaps
his most famous statement of this form was attached to
(the so-called) Fermat's Last Theorem: about which he
had written in the margin of his copy of Diophantus'
Arithmetica "For this, I have found a truly
wonderful proof, but the margin is too small to contain
it." Few believe Fermat had such a proof, and Wiles
found the first accepted proof in 1995, some 350 years
later.
Fermat developed a method of solving equations of the
form x2-ay2 = 1,
now incorrectly called a Pell equation. He incorrectly
stated that all numbers 22n+1 were
prime. (These are now known as theFermat numbers.)
Fermat developed a method of factoring based on expressing
a number as a difference of squares and was known for his
love of the method of "infinite descent" to solve problems.
Fibonacci, Leonardo
was born in Pisa, Italy around 1170, the son of Guilielmo Bonacci, a secretary of the Republic of Pisa and responsible, beginning around 1192, for directing the Pisan trading colony in Bugia, Algeria.
Some time after 1192, Bonacci brought his son with him to Bugia. The father intended for Leonardo to become a merchant and so arranged for his instruction in calculational techniques, especially those involving the Hindu-Arabic numerals which had not yet been introduced into Europe. Eventually, Bonacci enlisted his son's help in carrying out business for the Pisan republic and sent him on trips to Egypt, Syria, Greece, Sicily, and Provence. Leonardo took the opportunity offered by his travel abroad to study and learn the mathematical techniques employed in these various regions.
Around 1200, Fibonacci returned to Pisa where, for at least the next twenty-five years, he worked on his own mathematical compositions. The five works from this period which have come down to us are: the Liber abbaci (1202, 1228); the Practica geometriae (1220/1221); an undated letter to Theodorus, the imperial philosopher to the court of the Hohenstaufen emperor Frederick II; Flos (1225), a collection of solutions to problems posed in the presence of Frederick II; and the Liber quadratorum (1225), a number-theoretic book concerned with the simultaneous solution of equations quadratic in two or more variables. So great was Leonardo's reputation as a mathematician as a result of these works that Frederick summoned him for an audience when he was in Pisa around 1225.
After 1228, virtually nothing is known of Leonardo's life, except that by decree the Republic of Pisa awarded the "'serious and learned Master Leonardo Bigollo' (discretus et sapiens) a yearly salarium of 'libre XX denariorem' in addition to the usual allowances" (DSB). This stipend rewarded Fibonacci for his pro bono advising to the Republic on matters involving accounting and related mathematical matters.
Fibonacci died some time after 1240, presumably in Pisa.
Fourier, Jean Baptiste Joseph
Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions.
Fourier trained for the priesthood but did not take his vows. Instead took up mathematics studying (1794) and later teaching mathematics at the new École Normale.
In 1798 he joined Napoleon's army in its invasion of Egypt as scientific advisor. He helped establish educational facilities in Egypt and carried out archaeological explorations. He returned to France in 1801 and was appointed prefect of the department of Isere by Napoleon.
He published "Theacuteorie analytique de la chaleur" in 1822 devoted to the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions. In this he introduced the representation of a function as a series of sines or cosines now known as Fourier series.
Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.
Galerkin, Boris Grigorievich
came from a poor family and this was to mean that he had a harder time through his years of education than would otherwise have been the case. He attended secondary school in Minsk, then in 1893 he entered the Petersburg Technological Institute. Here he studied mathematics and engineering but he needed to make money to survive so at first he took on private tutoring, then from 1896 he worked as a designer.
After graduating from the Technological Institute in 1899 he got a job at the Kharkov Locomotive Plant. In 1903 Galerkin went to St Petersburg and there he became engineering manager at the Northern Mechanical and Boiler Plant.
From 1909 Galerkin began to study building sites and construction works throughout Europe. In the same year he began teaching at the Petersburg Technological Institute. His first publication on longitudinal curvature also appeared in 1909, work which carried on from beginnings which had been laid by Euler. This paper was highly relevant to his study of construction sites since the results were applied to the construction of bridges and frames for buildings.
His visits around European construction sites ended around 1914 but his academic work then turned to the area for which he is today best known, namely the method of approximate integration of differential equations known as the Galerkin method. He published his finite element method in 1915.
In 1920 Galerkin was promoted to Head of Structural Mechanics at the Petersburg Technological Institute. By this time he also held two chairs, one in elasticity at the Leningrad Institute of Communications Engineers and one in structural mechanics at Leningrad University.
In 1921 the St Petersburg Mathematical Society was reopened (it had closed in 1917 due to the Russian Revolution) as the Petrograd Physical and Mathematical Society. Galerkin played a major role in the Society along with Steklov, Sergi Bernstein, Friedmann and others.
Other work for which Galerkin is famous is his work on thin elastic plates. His major monograph on this topic Thin Elastic Plates was published in 1937. From 1940 until his death, Galerkin was head of the Institute of Mechanics of the Soviet Academy of Sciences.
Gauss, Carl Friedrich
considered by many to be the greatest mathematician who has ever lived, was born in Brunswick, Germany, on April 30, 1777. By the age of three, he was able to perform long computations in his head; at 10, he studied algebra and analysis. While still a teenager, he made many fundamental discoveries. Among these were the method of "least squares" for handling statistical data, a proof that a 17-sided regular polygon can be constructed with a straight-edge and compass (this result was the first of its kind since discoveries by the Greeks 2000 years earlier), and his quadratic reciprocity theorem. Gauss obatined his Ph.D in 1799 from the University of Helmstedt, under the supervision of Pfaff. In his dissertation, he proved the Fundamental Theorem of Algebra.
In 1801, Gauss published his monumental book on number theory. Disquisitiones Arithmeticae, summarizing previous work in a systematic way and introducing many fundamental ideas of his own, including the notion of modular arithmetic. This book won Gauss great fame among mathematicians.
In 1801, Ceres (an asteroid) was observed by astronomers on three occasions before they lost track of it. In what seemed to be an almost superhuman feat, Gauss used these three observations to calculate the orbit of Ceres. In carrying out this work, he showed that the variation inherent in experimentally derived data follows a bell-shaped curve, now called the Gaussian distribution. Gauss also used the method of least squares in this problem. This achievement established Gauss's reputation as a scientific genius before he was 25 years old.
In 1807, Gauss became professor of astronomy and director of the new observatory at the University of Göttingen. During the decades to come, Gauss continued to make important contributions not only in nearly all branches of mathematics, but also in astronomy, mechanics, optics, geodesy, and magnetism. Gauss also invented, with the physicist Wilhelm Weber, the first practical telegraph.
The acceptance of complex numbers among mathematicians was brought about by Gauss's use of them. Gauss coined the term complex number and popularized the notation i for the square root of negative one. He proved that the ring Z[i] is a unique factorization domain and a Euclidean domain.
Throughout his life, Gauss largely ignored the work of his contemporaries and, in fact, made enemies of many of them. Young mathematicians who sought encouragement from him were usually rebuffed. Despite this fact, Gauss had many outstanding students, including Eisenstein, Riemann, Kummer, Dirichlet, and Dedekind.
Gauss died in Göttingen at the age of 78 on February 23, 1855. At Brunswick, there is a statue of him. Appropriately, the base is in the shape of a 17-point star. In 1989, Germany issued a bank note depicting Gauss and the Gaussian distribution.
Hilbert, David
received his Ph.D. from the University of Königsberg and was a member of staff there from 1886 to 1895 In 1895 he was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his life.
Hilbert's first work was on invariant theory, in 1888 he proved his famous Basis Theorem. First he gave an existence proof but, after Cayley, Gordan, Lindemann and others were baffled, in 1892 Hilbert produced a constructive proof which satisfied everyone.
In 1893 while still at Königsberg he began a work "Zahlbericht" on algebraic number theory. The "Zahlbericht" (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work.
Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance.
He published "Grundlagen der Geometrie" in 1899 putting geometry on a formal axiomatic setting. His famous 23 Paris problems challenged (and still today challenges) mathematicians to solve fundamental questions.
In 1915 Hilbert discovered the correct field equation for general relativity before Einstein but never claimed priority.
In 1934 and 1939 two volumes of "Grundlagen der Mathematik" were published which were intended to lead to a 'proof theory' a direct check for the consistency of mathematics. Göde's paper of 1931 showed that this aim is impossible.
Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.
Jacobi, Karl Gustav Jacob
Jacobi founded the theory of elliptic functions.
Jacobi's father was a banker and his family were prosperous so he received a good education at the University of Berlin. He obtained his Ph.D. in 1825 and taught mathematics at the University of Königsberg from 1826 until his death, being appointed to a chair in 1832.
He founded the theory of elliptic functions based on four theta functions. His "Fundamenta nova theoria functionum ellipticarum" in 1829 and its later supplements made basic contributions to the theory of elliptic functions.
In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is imaginary. This result prompted much further work in this area, in particular by Liouville and Cauchy.
Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.
He also worked on determinants and studied the functional determinant now called the Jacobian. Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy. However Jacobi wrote a long memoir "De determinantibus functionalibus" in 1841 devoted to the this determinant. He proves, among many other things, that if a set of n functions in n variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.
Jacobi's reputation as an excellent teacher attracted many students. He introduced the seminar method to teach students the latest advances in mathematics
Lagrange, Joseph Louis
was born in Italy of French ancestry on January 25, 1736. He became captivated by mathematics at an early age when he read an essay by Halley on Newton's calculus. At the age of 19 he became a professor of mathematics at the Royal Artillery School in Turin. Lagrange made significant contributions to many branches of mathematics and physics, among them the theory of numbers, the theory of equations, ordinary and partial differential equations, the calculus of variations, analytic geometry, fluid dynamics, and celestial mechanics. His methods for solving third and fourth-degree polynomial equations by radicals laid the groundwork for the group-theoretic approach to solving polynomials taken by Galois. Lagrange was a very careful writer with a clear and elegant style.
At the age of 40, Lagrange was appointed Head of the Berlin Academy, succeeding Euler. In offering this appointment, Frederick The Great proclaimed that the "greatest king in Europe" ought to have the "greatest mathematician in Europe" at his court. In 1787, Lagrange was invited to Paris by Louis XVI and became a good friend of the king and his wife, Marie Antionette. In 1793, Lagrange headed a commision, which included Laplace and Lavoisier, to devise a new system of weights and measures. Out of this came the metric system. Late in his life he was made a count by Napoleon. Lagrange died on April 10, 1813.
Laplace, Pierre-Simon
proved the stability of the solar system. In analysis Laplace introduced the potential function and Laplace coefficients. He also put the theory of mathematical probability on a sound footing.
Laplace attended a Benedictine priory school in Beaumont between the ages of 7 and 16. At the age of 16 he entered Caen University intending to study theology. Laplace wrote his first mathematics paper while at Caen.
At the age of 19, mainly through the influence of d'Alembert, Laplace was appointed to a chair of mathematics at the École Militaire in Paris on the recommendation of d'Alembert. In 1773 he became a member of the Paris Academy of Sciences. In 1785, as examiner at the Royal Artillery Corps, he examined and passed the 16 year old Napoleon Bonaparte.
During the French Revolution he helped to establish the metric system. He taught calculus at the École Normale and became a member of the French Institute in 1795. Under Napoleon he was a member, then chancellor, of the Senate, received the Legion of Honour in 1805. However Napoleon, in his memoires written on St Hélčne, says he removed Laplace from office after only six weeks
because he brought the spirit of the infinitely small into the government
Laplace became Count of the Empire in 1806 and he was named a marquis in 1817 after the restoration of the Bourbons. In his later years he lived in Arcueil, where he helped to found the Societe d'Arcueil and encouraged the research of young scientists.
Laplace presented his famous nebular hypothesis in "Exposition du systeme du monde" (1796), which viewed the solar system as originating from the contracting and cooling of a large, flattened, and slowly rotating cloud of incandescent gas.
Laplace discovered the invariability of planetary mean motions. In 1786 he proved that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and self-correcting. These results appear in his greatest work, "Traité du Mécanique Céleste" published in 5 volumes over 26 years (1799-1825).
Laplace also worked on probability and in particular derived the least squares rule. His "Théorie Analytique des Probabilités" was published in 1812.
He also worked on differential equations and geodesy. In analysis Laplace introduced the potential function and Laplace coefficients. He also put the theory of mathematical probability on a sound footing. With Antoine Lavoisier he conducted experiments on capillary action and specific heat. He also contributed to the foundations of the mathematical science of electricity and magnetism.
Maclaurin, Colin
Scottish mathematician who became a disciple of Newton. He published the first systematic formulation of Newton's methods in A Treatise of Fluxions (1742). In this work, he developed a method for expanding functions about the origin in terms of series now known as Maclaurin series. This method was generalized to expansion about an arbitrary point by Brook Taylor. Maclaurin also invented several devices, made astronomical observations, and improved maps of the Scottish isles. Maclaurin knew Cramer's rule probably as early as 1729.
Newton, Sir Isaac
(1642-1727), mathematician and physicist, one of the foremost scientific intellects of all time. Born at Woolsthorpe, near Grantham in Lincolnshire, where he attended school, he entered Cambridge University in 1661; he was elected a Fellow of Trinity College in 1667, and Lucasian Professor of Mathematics in 1669. He remained at the university, lecturing in most years, until 1696. Of these Cambridge years, in which Newton was at the height of his creative power, he singled out 1665-1666 (spent largely in Lincolnshire because of plague in Cambridge) as "the prime of my age for invention". During two to three years of intense mental effort he prepared Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) commonly known as the Principia, although this was not published until 1687.
As a firm opponent of the attempt by King James II to make the universities into Catholic institutions, Newton was elected Member of Parliament for the University of Cambridge to the Convention Parliament of 1689, and sat again in 1701-1702. Meanwhile, in 1696 he had moved to London as Warden of the Royal Mint. He became Master of the Mint in 1699, an office he retained to his death. He was elected a Fellow of the Royal Society of London in 1671, and in 1703 he became President, being annually re-elected for the rest of his life. His major work, Opticks, appeared the next year; he was knighted in Cambridge in 1705.
As Newtonian science became increasingly accepted on the Continent, and especially after a general peace was restored in 1714, following the War of the Spanish Succession, Newton became the most highly esteemed natural philosopher in Europe. His last decades were passed in revising his major works, polishing his studies of ancient history, and defending himself against critics, as well as carrying out his official duties. Newton was modest, diffident, and a man of simple tastes. He was angered by criticism or opposition, and harboured resentment; he was harsh towards enemies but generous to friends. In government, and at the Royal Society, he proved an able administrator. He never married and lived modestly, but was buried with great pomp in Westminster Abbey.
Newton has been regarded for almost 300 years as the founding examplar of modern physical science, his achievements in experimental investigation being as innovative as those in mathematical research. With equal, if not greater, energy and originality he also plunged into chemistry, the early history of Western civilization, and theology; among his special studies was an investigation of the form and dimensions, as described in the Bible, of Solomon's Temple in Jerusalem.
Pascal, Blaise
Pascal's father (Pascal, Étienne) had unorthodox educational views and decided to teach his son himself. He decided that Pascal was not to study mathematics before the age of 15 and all mathematics texts were removed from their house. Pascal however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are 2 right angles and, when his father found out he relented and allowed Pascal a copy of Euclid.
At the age of 14 Pascal started to attend Mersenne's meetings. Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for Fermat, Pascal, Gassendi, and others. At the age of 16 Pascal presented a single piece of paper to one of Mersenne's meetings. It contained a number of projective geometry theorems, including Pascal's mystic hexagon.
Pascal invented the first digital calculator
(1642) to help his father. The device, called the Pascaline, resembled a mechanical calculator of the 1940's.
Further studies in geometry, hydrodynamics, and hydrostatic and atmospheric pressure led him to invent the syringe and hydraulic press and to discover Pascal's law of pressure.
He worked on conic sections and produced important theorems in projective geometry. In correspondence with Fermat he laid the foundation for the theory of probability.
His most famous work in philosophy is "Pensées", a collection of personal thoughts on human suffering and faith in God. 'Pascal's wager' claims to prove that belief in God is rational with the following argument.
"If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing."
His last work was on the cycloid, the curve traced by a point on the circumference of a rolling circle.
Pascal died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain.
Ramanujan, Srinivasa
Indian mathematician who was self-taught and had an uncanny mathematical manipulative ability. Ramanujan was unable to pass his school examinations in India, and could only obtain a clerk's position in the city of Madras. However, he continued to pursue his own mathematics, and sent letters to three mathematicians in England (which arrived in January of 1913) containing some of his results. While two of the three returned the letters unopened, G. H. Hardy recognized Ramanujan's intrinsic mathematical ability and arranged for him to come to Cambridge. Because of his lack of formal training, Ramanujan sometimes did not differentiate between formal proof and apparent truth based on intuitive or numerical evidence. Although his intuition and computational ability allowed him to determine and state highly original and unconventional results which continued to defy formal proof until recently (Berndt 1985-1997), Ramanujan did occasionally state incorrect results.
Ramanujan had an intimate familiarity with numbers, and excelled especially in number theory and modular function theory. His familiarity with numbers were demonstrated by the following incident. During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=13+123=93+103.
Unfortunately, Ramanujan's health deteriorated rapidly in England, due perhaps to the unfamiliar climate, food, and to the isolation which Ramanujan felt as the sole Indian in a culture which was largely foreign to him. Ramanujan was sent home to recuperate in 1919, but tragically died the next year at the very young age of 32.
Ramanujan published some of his results in journals, and many are beautiful indeed. However, his working notebooks contained much additional unorganized material which remained uninvestigated until the sustained efforts of Berndt and his coworkers who systematically examined and proved Ramanujan's sometimes vague or ambiguous statements. For anyone with a little knowledge of number theory, Ramanujan's notebooks make absolutely fascinating reading. It is therefore a great pity that their publisher, Springer-Verlag, has chosen to price these slim volumes at the ridiculous price of about $100 apiece.
Runge, Carle David Tolm
worked on a procedure for the numerical solution of algebraic equations and later studied the wavelengths of the spectral lines of elements.
At the age of 19, after leaving school, Runge spent 6 months with his mother visiting the cultural centres of Italy. On his return to Germany he enrolled at the University of Munich to study literature. However after 6 weeks of the course he changed to mathematics and physics.
Runge attended courses with Max Planck and they became close friends. In 1877 both went to Berlin but Runge turned to pure mathematics after attending Weierstrass' lectures. His doctoral dissertation (1880) dealt with differential geometry.
After taking his secondary school teachers examinations he returned to Berlin where he was influenced by Kronecker. Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
Runge published little at that stage but after visiting Mittag-Leffler in Stockholm in September 1884 he produced a large number of papers in Mittag-Leffler's journal "Acta mathematica" (1885).
Runge obtained a chair at Hanover in 1886 and remained there for 18 years. Within a year Runge had moved away from pure mathematics to study the wavelengths of the spectral lines of elements other than hydrogen (J J Balmer had constructed a formula for the spectral lines of hydrogen.)
Runge did a great deal of experimental work and published a great quantity of results. He succeeded in arranging the spectral lines of helium in two spectral series and, until 1897, this was thought to be evidence that hydrogen was a mixture of two elements.
In 1904 Klein persuaded Göttingen to offer Runge a chair of Applied Mathematics, a post which Runge held until he retired in 1925.
Runge was always a fit and active man and on his 70 th birthday he entertained his grandchildren by doing handstands. However a few months later he had a heart attack and died.