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- Mathematics In Europe - Pierre de Fermat (1607/1608–December 1, 1665); by Heinz Klaus Strick, Germany

to work out the details of the methods that he had developed Many found his work Methodus ad disquirendam maximam et minimam on determining the tangents to a curve extreme values and surface areas under the curve of a power function the Fermat parabola y x n and the Fermat hyperbola y 1 x n to be extremely opaque However Newton 1643 1727 called it a source of inspiration and Laplace 1749 1827 in the spirit of nationalistic enthusiasm saw in Fermat the true inventor of the calculus The extreme value problems solved by Fermat include the following At what point in the interior of a triangle with internal angles less than 120 is the sum of the distances to the three vertices minimal For this Fermat point the following holds The line segments joining this point to the three vertices always form angles of 120 see the figure above Fermat criticized René Descartes s work on optics as flawed which provoked an angry response from Descartes who in Fermat recognized a rival of at least equal stature Twenty years later Fermat again attacked the problem of the refraction of light and derived a fundamental law of optics that describes the path of a light ray as it passes from one medium to another Light takes the fastest not the shortest route between points A and B figure to the right From 1643 to 1654 Fermat lost contact with his mathematical colleagues in Paris due to civil war and an epidemic of plague Inspired by the Arithmetica of Diophantus ca 250 B C E he immersed himself in a branch of mathematics that had been largely neglected by the mathematicians of his time number theory Five years after Fermat s death his son Clément Samuel discovered in the margin of his father s copy of the Arithmetica a Latin translation with commentaries published by Bachet de Méziriac 1581 1638 an assertion that became known as Fermat s last theorem The Diophantine equation x n y n z n has no nontrivial solutions in integers x y z for n 2 Instead of stating an idea of a proof Fermat wrote down the following famous sentence Cuius rei demonstrationem mirabilem sane detexi Hanc marginis exiguitas non caperet I have found a truly wonderful proof but this margin is too small to contain it One must assume that Fermat was mistaken Over the centuries many mathematicians searched for a proof but it was not until 1995 that Andrew Wiles published a proof which required much of the machinery unavailable to Fermat of modern mathematics Fermat never again referred to this theorem in its full generality which suggests perhaps that he had become aware of his error He did give a proof of the theorem for the special case n 4 using the method of infinite descent descente infinie that he had developed On the assumption that there is a solution x y z to the equation x 4 y 4 z 4

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Open archived version from archive - Mathematics In Europe - Josef Fourier (March 21, 1768–May16, 1830); bei Heiz Klaus Strick, Germany

joined the local revolutionary committee Thanks to his talents as a speaker he soon wielded considerable influence over the development of the revolution in the region His involvement in politics almost cost him his head He became embroiled in an argument with the ROBESPIERRE faction which managed to have him arrested He avoided the guillotine only because the influence of his adversaries waned after the execution of ROBESPIERRE himself In 1794 FOURIER was accepted into the mathematics program of the newly established École Normale in Paris His teachers were JOSEPH LOUIS LAGRANGE PIERRE SIMON LAPLACE and GASPARD MONGE Simultaneously as an outstanding student he obtained a position as an instructor at the Collège de France and then moved on to the newly founded École Polytechnique He was again arrested because of his revolutionary activities but was soon released In 1797 he was appointed to a professorship in the chair for analysis and mechanics vacated by LAGRANGE and his fame as an outstanding teacher spread far and wide Lagrange and Laplace In 1798 he joined the group of scientists that accompanied NAPOLEON on his Egyptian campaign When the French fleet was destroyed by the fleet of ADMIRAL NELSON at the Battle of the Nile the expedition was unable to return to France FOURIER became involved in reform of the educational system in Egypt He was named secretary of the Institut d Égypte and he organized archaeological expeditions NAPOLEON charged him with collecting valuable archaeological finds NAPOLEON himself returned to France to seize power for himself In 1801 FOURIER was able to return to Paris where he would have liked to resume his duties as a professor of mathematics NAPOLEON however named him against his wishes prefect of the Département Isère His new duties included draining the swamps around Lyon and building a road from Grenoble to Turin He carried out these tasks to the great satisfaction of his superiors though during this time he also managed to complete two extensive monographs On the Propagation of Heat in Solid Bodies and Description de l Égypte The book about Egypt could not be published until NAPOLEON was satisfied that he had been sufficiently praised in every passage in later editions of the work FOURIER removed almost all the passages relating to NAPOLEON s merits FOURIER s enthusiasm for the culture of ancient Egypt was infectious When in 1802 he showed the twelve year old JEAN FRANÇOIS CHAMPOLLION a replica of the Rosetta Stone the boy became obsessed with the idea of deciphering the hieroglyphics that appeared on the stone In 1822 CHAMPOLLION presented his research to the Académie des Inscriptions et Belles Lettres in Paris he had deciphered the hieroglyphics FOURIER s work on the propagation of heat in solid bodies was met with resistance and not only because of the unusual physical modelling of how heat propagates Namely the mathematical treatment of the problem was initially rejected by LAGRANGE und LAPLACE since they could not warm up to the idea of representing

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Open archived version from archive - Mathematics In Europe - Évariste Galois (October 25, 1811--May 31, 1832); by Heinz Klaus Strick, Germany

the insurgents he was publicly attacked by Galois in a letter to the Gazette des Écoles which resulted in his expulsion from the École Normale Galois was now destitute and desperate being unable to attend any university He joined the Republican National Guard which however by the end of 1830 was outlawed by the new king Louis Phillipe the so called citizen king Some of the officers were arrested for treason Siméon Denis Poisson 1781 1840 suggested to Galois that he resubmit his article for the third time to the Academy which he did at once At a celebration on the occasion of the release of the imprisoned officers in May 1831 Galois proposed a toast to the king with a dagger in his hand He was arrested and later released On July 14 the anniversary of the storming of the Bastille he was again arrested following his appearance in the streets heavily armed and wearing the uniform of the outlawed National Guard While in prison he received the news that Poisson had judged his article to suffer from an insufficiently clear exposition Galois attempted suicide but was prevented from doing so by his fellow prisoners When the cholera epidemic reached the prison he was transferred to hospital there he fell in love with Stéphanie the daughter of the attending physician It has never been satisfactorily explained how on May 30 1832 he engaged in a duel with a fellow republican During the night before the duel Galois committed to paper everything that he wished to communicate to the world should he perish the following day The name Stéphanie appears repeatedly in a farewell letter Wounded in the duel abandoned by his own seconds he died the following day A friend wrote up his mathematical legacy and sent the papers to several renowned mathematicians among them Carl Friedrich Gauss to no avail Recognition came only eleven years later when Joseph Liouville 1809 1882 understood the significance of the theories developed by Galois and published them in his journal in 1846 In the mid sixteenth century mathematicians finally succeeded in finding mathematical expressions for the solutions of polynomial equations of the third and fourth degree the solutions are expressed in terms of the coefficients of the equation together with the fundamental arithmetic operations and extraction of roots solution by radicals François Viète 1540 1603 pointed out Viète s laws certain relationships between the coefficients of polynomial equations and their solutions For example for degree 2 if we take the equation x 2 x 1 x 2 cdot x x 1 cdot x 2 0 then x 1 and x 2 are the two solutions In degree 3 if x 3 x 1 x 2 x 3 cdot x 2 x 1 cdot x 2 x 1 cdot x 3 x 2 cdot x 3 cdot x x 1 cdot x 2 cdot x 3 0 then x 1 x 2 x 3 are the solutions The search for a solution procedure

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Open archived version from archive - Mathematics In Europe - Carl Friedrich Gauss (April 30, 1777-February 23, 1855); by Heinz Klaus Strick, Germany

had an unfortunate temporal overlap with the Disquisitiones The individual chapters deal with the theory of arithmetic and congruences with the proof of Leonhard Euler s 1707 1783 conjecture on quadratic reciprocity with the theory of quadratic forms the solution of equations of the form ax 2 2bxy cy 2 m with continued fractions and primality tests and with the solution of equations of the form x n 1 n in mathbb N and x m equiv 1 bmod p x 0 1 x 1 frac 12 i cdot frac sqrt 3 2 x 2 frac 12 i cdot frac sqrt 3 2 Solutions of x 3 1 x 0 1 x 1 i x 2 1 x 3 i Solutions of x 4 1 x 0 1 x 1 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 2 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 3 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 4 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 Solutions of x 5 1 x 0 1 x 1 frac 12 i cdot frac sqrt 3 2 x 2 frac 12 i cdot frac sqrt 3 2 x 3 1 x 4 frac 12 i cdot frac sqrt 3 2 x 5 frac 12 i cdot frac sqrt 3 2 Solutions of x 6 1 The law of quadratic reciprocity describes the conditions under which quadratic congruence equations are solvable If p and q are prime numbers then the two congruence equations x 2 equiv p bmod q and x 2 equiv q bmod p are either both solvable or both not solvable unless p and q each have remainder 3 on division by 4 in which case one equation is solvable while the other is not Here are a few examples The equation x 2 equiv5 bmod7 is not solvable that is there are no perfect squares in the sequence 5 12 19 26 33 40 dotsc since the reciprocal equation x 2 equiv 7 bmod 5 equiv 2 bmod 5 has no solution There is no square whose final digit is 2 or 7 The equation x 2 equiv5 bmod11 has a solution since it is clear that perfect squares appear in the sequence of numbers 5 16 27 38 49 60 dotsc But then the reciprocal equation x 2 equiv11 bmod 5 equiv1 bmod 5 must have at least one solution there exist squares with 1 or 6 as final digit The equation x 2 equiv3 bmod 11 is solvable since in the sequence 3 14 25 36 47 58 dotsc there are some perfect squares But then the reciprocal equation x 2 equiv11 bmod 3 equiv2 bmod 3 has no solution that is in the sequence 2 5 8 11 14 17 20 23 dotsc are to be found no perfect squares An equation of the form x n 1 is called a cyclotomic from the Greek kyklos circle temnein to cut equation since one can write down the n solutions in the form x k cos left frac k cdot2 pi n right i cdot sin left frac k cdot2 pi n right with k 0 1 2 dotsc n 1 If one then draws these points as was Gauss s practice after 1820 in the complex plane also known as the Gaussian plane in which the number 1 is drawn to the right of the origin and the number i above it then these points form the vertices of a regular n gon on the unit circle see the figures above Gauss had suddenly become famous As an expression of gratitude to the duke he turned down an invitation to take up residence in Saint Petersburg hoping that his sovereign would build him an observatory in Braunschweig His particular interest in astronomy was strengthened by yet another sensational accomplishment one that made his name known among nonmathematicians as well On January 1 1801 the Italian astronomer Giuseppe Piazzi discovered the asteroid Ceres but then after a couple of days lost all trace of it as the asteroid vanished behind the sun Using Piazzi s data from his sightings Gauss calculated the asteroid s orbit using the method of least squares making possible the rediscovery of Ceres the following year by Heinrich Olbers Gauss had come up with the method of minimizing errors by considering the squares of deviations from a given model when he was only 17 years old He presented the theory behind the method together with a description of its practical application to astronomy in 1809 in the paper Motus corporum coelestium in sectionibus conicis solem ambientium the motion of the celestial bodies that orbit the sun in conic sections Again here there is a priority dispute since Legendre proposed the same method independently of Gauss in 1806 Gauss and Bessel Nicaragua 1994 After the duke of Braunschweig was mortally wounded in 1807 in the Battle of Jena and Auerstedt Gauss had to move to Göttingen to assume a professorship of astronomy in order to provide for himself and his new family He was not pleased that the post included teaching duties and he made every effort to keep them to a minimum His happy marriage to Johanna Osthoff lasted only four years She died giving birth to their third child To provide care for his children he married a friend of Johanna s Minna Waldeck with whom he had an additional three children When his second wife died in 1831 his youngest daughter Therese took care of the household in which Gauss s mother had resided since the death of her husband she died in 1839 at the age of 95 Gaussian bell curve sextant As director of the observatory in Göttingen Gauss worked on improving the design of telescopes and investigated the question of constructing optics with

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Open archived version from archive - Mathematics In Europe - Charles Hermite (December 24, 1822–January 14, 1901); by Heinz Klaus Strick, Germany

transcendental number that is that there is no polynomial equation of any degree to which e is a solution In 1873 CHARLES HERMITE was able to show precisely that He gave a proof which cannot be presented in an elementary way in an article comprising thirty pages It was then nine years later that FERDINAND VON LINDEMANN employed HERMITE s proof method to prove that the number Pi the ratio of a circle s circumference to its diameter is also transcendental This result had as a corollary the solution to the millennia old problem of squaring the circle the question of the constructability using only straightedge and compass of a circle with area equal to that of a given square namely that such a construction is impossible since every construction with straightedge and compass leads to numbers in a limited class of algebraic numbers After LINDEMANN had succeeded in proving the transcendence of p HERMITE s star began to fade though he was not bothered by the fact CHARLES HERMITE s father worked as an engineer in the salt mines of Lorraine After marrying he went to work for his parents in law who were textile merchants Interested more in art than in commercial trade the father then opened a business in Nancy in order to be able to take part in the cultural life of the provincial capital CHARLES was the sixth of his seven children He was born with a congenital defect a malformed right foot His schooling began with attendance at a collège in Nancy but then he moved to Paris where in the years 1840 1841 he received mathematical instruction under LOUIS RICHARD who had taught ÉVARISTE GALOIS at the Lycée Louis le Grand Like his predecessor GALOIS HERMITE was interested less in the material of the standard curriculum than in the writings of EULER GAUSS and LAGRANGE which he read in his free time While still a school pupil he submitted two articles to the recently established journal Nouvelles Annales des Mathématiques one of them a commentary on an article by LAGRANGE on the solvability of equations of the fifth degree though he did so without knowledge of GALOIS s contributions to this subject apparently the editors of the journal were also at the time unaware of the writings of GALOIS Like GALOIS he hoped to study at the École Polytechnique but unlike GALOIS he passed the entrance examination though to be sure with a rather mediocre performance After a year at the École Polytechnique whose main task was to prepare a new generation for the military he was not allowed to continue the reason given that his physical infirmity made him unfit for military service The decision to expel him was countermanded after the intervention of advocates on his behalf but offended HERMITE left that institution in order to complete his education elsewhere He corresponded with JACOBI on the subject of a special differential equation for which he had found a method of solution

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/981-charles-hermite-december-24-1822-january-14-1901 (2013-11-18)

Open archived version from archive - Mathematics In Europe - David Hilbert (January 23, 1862–February 14, 1943); by Heinz Klaus Strick, Germany

the number π a result that had as a corollary a proof that the classical problem of squaring the circle with straightedge and compass alone was impossible After short study journeys to Leipzig Felix Klein and Paris Henri Poincaré Charles Hermite he completed his habilitation in Königsberg where he became a privatdozent lecturer and at age 30 an assistant non tenured professor In 1891 he discovered a curve that today is called the Hilbert curve It is defined by recursion and it has the surprising property that it is continuous and that its range completely fills a square In 1895 he was offered a chair at the University of Göttingen which from the time of Carl Friedrich Gauss had become one of the most important centres of mathematical research in Germany Hilbert made Göttingen the international centre for mathematics and physics Despite numerous offers from other universities Hilbert remained in Göttingen until the end of his life From 1933 he was forced to experience the destruction of his life s work by the racial politics of National Socialism In 1897 the German Mathematical Society asked Hilbert to provide a summary of the current state of research in algebraic number theory As a result of this proposal he wrote in collaboration with his university friend Hermann Minkowski at the time still in Königsberg later in Göttingen the famous Zahlbericht number report in which he systematized the theories of Ernst Eduard Kummer Leopold Kronecker and Richard Dedekind and not only with regard to notation for he also found and mended holes in some of their proofs After completing the Zahlbericht Hilbert set about placing Euclidean geometry on a solid axiomatic foundation He formulated a complete system of twenty axioms from which all the theorems of geometry in three dimensional space could be derived in the strictest logical sense While Hilbert used such descriptive notions as point line and plane he asserted that they could just as well be replaced with terms such as table chair and beer mug These geometrical objects were described by eight axioms of combination or incidence for example I 1 Two distinct points P and Q always determine a line g four axioms of order for example II 1 If B lies between A and C then B also lies between C and A five axioms of congruence III the parallel axiom IV Let g be an arbitrary line and P a point external to g then in the plane determined by g and P there is at most one line g that passes through P and does not intersect g as well as two axioms of continuity for example the Archimedean axiom V 1 If AB and CD are line segments then there exists a number n such that if one lays the segment CD on the ray beginning at A and proceeding in the direction of B then lays the segment CD again on this ray placing the point C where the point D was and

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Open archived version from archive - Mathematics In Europe - Carl Gustav Jacob Jacobi (December 10, 1804–February, 18, 1851); by Heinz Klaus Strick, Germany

and applied mathematics Crelle s journal The new ideas presented in the work of the one stimulated new ideas in the work of the other and within several months the theory of elliptic functions had undergone a substantial development In a letter LEGENDRE gave the following assessment They are making such rapid progress my good sirs in their speculations that it is scarcely possible to follow them in particular for an old man I congratulate myself that I have lived long enough to witness this good natured competition between two young equally strong athletes who are directing all their efforts to the benefit of our science to push back ever farther the boundaries of knowledge This good natured competition came to an abrupt end with the death of ABEL in April 1829 That same year JACOBI published his compendium Fundamenta nova theoriae functionum ellipticarum The positive evaluations of GAUSS and LEGENDRE had the result that shortly after his 23rd birthday JACOBI was named assistant professor and full professor at the age of 27 The enormous reputation that JACOBI s lectures enjoyed led to ever greater numbers of students enrolling at the University of Königsberg He made a habit of including the most recent result of mathematical research in his lectures He was the first to institute mathematical seminars based on the model of the classics In 1842 BESSEL and JACOBI attended as representatives of Prussia a conference in Manchester hosted by the British Association for the Advancement of Science There they met the Irish astronomer and mathematician WILLIAM ROWAN HAMILTON This meeting would result in a series of articles by JACOBI on the mathematics of the solar system On his return trip by way of Paris JACOBI gave a lecture to the Académie des Sciences He had arrived at the peak of his renown When in 1843 JACOBI was diagnosed with diabetes his doctor recommended that he recuperate in a milder climate However JACOBI was unable to afford such a trip the fortune he had inherited had been lost in a bank crash ALEXANDER VON HUMBOLDT with whom JACOBI had maintained a correspondence since 1828 supported his application to the king of Prussia and so JACOBI was able to journey to Italy accompanied by DIRICHLET with whom he had become friendly In the mild climate his health improved noticeably and he began to work again on his mathematical research In Rome he studied DIOPHANTUS s Arithmetica which was housed in the Vatican Library JACOBI for whom the change in climate had obviously done a great deal of good had no desire to return to Königsberg In recognition of his service the king of Prussia agreed to a transfer to Berlin accompanied by an increase in salary justified by the increased cost of living in the capital as well as a supplement to offset the expected costs associated with his illness As a member of the Prussian Academy of Sciences he was entitled but not obliged to give lectures During the

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Open archived version from archive - Mathematics In Europe - Felix Klein (April 25, 1849–June 22, 1925), by Heinz Klaus Strick

geometry Then for every line AB and for every point P external to this line there are arbitrarily many parallel lines namely all lines that do not pass through the triangle ABP In 1875 FELIX KLEIN moved to the Technical University Munich Technische Hochschule München There he instituted a required four semester series of courses called higher mathematics for students of engineering an innovation that was soon adopted by the other technical universities in Germany In 1880 KLEIN was called to a chair in geometry at the University of Leipzig Within a short period of time he implemented the idea of establishing a mathematical seminar that is a separate building in which lectures are held containing its own library including a collection of models depicting aspects of geometry Soon similar facilities were established at other universities Here as well he instituted required courses for beginning students These were enhanced with tutorials He had already promoted the idea of a collaboration between university and industry in his inaugural lecture in Leipzig Über die Beziehungen der neueren Mathematik zu den Anwendungen On the relationships between today s mathematics and its applications in which he also proposed a chair in applied mathematics In 1882 he discovered a geometric object between whose interior and exterior it was impossible to distinguish today it is known as the KLEIN bottle perhaps on account of a mistaken translation into English resulting from a confusion of a single letter between the words Flasche bottle and Fläche or Flaeche surface KLEIN was now at the peak of his creative powers In his paper On RIEMANN s Theory of Algebraic Functions and Their Integrals he pointed out connections between various branches of mathematics In his investigation of the mappings of the regular icosahedron to itself he discovered connections with the problem of the algebraic solution of equations of the fifth degree In a lecture on this subject he mentioned the smallest noncyclic group which today in his honour is known as the KLEIN four group it is the group of four elements that can be interpreted for example as the group of symmetries of a rectangle In fall 1882 however he suffered a physical and emotional breakdown what triggered it appears to have been a professional and personal quarrel with the French mathematician HENRI POINCARÉ POINCARÉ had named a certain class of mappings Fuchsian functions although unbeknownst to him it was KLEIN and not LAZARUS FUCHS who had discovered them The correspondence between these two outstanding mathematicians which had begun on a strictly professional level quickly developed into a bitter conflict between two rivals When POINCARÉ then proceeded to publish new and far reaching results in non Euclidean geometry which was the area of KLEIN s specialization this touched off for KLEIN a creative crisis In one phase of his depression he decided to end all mathematical research activity in order to free himself of the pressure of competition FRIEDRICH ALTHOFF an influential official in the Prussian Ministry of Culture

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Open archived version from archive