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- Mathematics In Europe - Niels Henrik Abel (August 5, 1802–April 6, 1829); by Heinz Klaus Strick, Germany

in his proof And some years he understood why it is in fact impossible to find a general procedure for arbitrary equations of the fifth order At the age of 19 Abel matriculated at the University of Christiania where however there were no courses in higher mathematics Holmboë convinced his former professor Christopher Hansteen who later became known for his research on the Earth s magnetic field to win this unusual talent for science Hansteen took a personal interest in Abel including financing a trip to Copenhagen so that he could discuss his ideas with the Danish mathematician Ferdinand Degen This gave Abel the imp etus to study the so called elliptic integrals For example to determine the arc length of an ellipse whence the name certain integrals must be calculated the function f defined by f x a cdot int 0 x sqrt 1 varepsilon 2 cos 2 t mathrm d t associates an angle x with the arc length f x see the figure Here a and b are the semiaxes of the ellipse and varepsilon sqrt a 2 b 2 a is the numeric eccentricity Such integrals including other related integrals cannot be calculated in an elementary way that is one cannot express in terms of elementary functions a function whose derivative is the expression inside the integral sign Therefore the values of such integrals can be computed only numerically Abel had the idea to investigate the associated inverse function whereby a particular arc length of the ellipse is related to the associated angle thereby creating the theory of elliptic functions He became convinced that to further his work he would have to make personal contact with the greatest mathematicians of his time namely Carl Friedrich Gauss in Göttingen and Adrien Marie Legendre the world expert on elliptic functions and Augustin Louis Cauchy in Paris Over the following two year s he studied the French and German languages intensively and continued his work on mathematical problems In 1824 he published his Mémoire sur les équations algébriques ou on démontre l impossibilité de la résolution de l équation générale du cinquième degré though it appeared in an abridged form to save on the expense of publication and he sent a copy to Gauss who put it aside unread Because of his limited contact with the scientific world Abel was unaware that Paolo Ruffini 1765 1822 had already in 1799 published a paper expounding similar ideas although his proof was marred by several gaps Finally in 1825 he was granted the necessary funds for a journey Carrying letters of introduction he first visited the engineer August Leopold Crelle in Berlin who at the time was preparing to launch a mathematical journal that would be independent of the universities the Journal für die reine und angewandte Mathematik Journal for Pure and Applied Mathematics The first issue of what came to be called Crelle s journal contained seven papers by Abel including the Re cherches sur les fonctions elliptiques and Beweis

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Open archived version from archive - Mathematics In Europe - Jean Le Rond D’Alembert (November 16, 1717–October 29, 1783); by Heinz Klaus Strick, Germany

F called a D Alembert force For D Alembert mechanics should be seen as a branch of mathematics in contrast to Newton he considered experimentation as an aid in the solution of physical problems to be superfluous which can be seen in his work Réflexions sur la cause générale des vents For this work he received in 1744 a prize from the Prussian Academy of Sciences D Alembert claimed that the tides were the single cause of the emergence of winds nonetheless he was the first to describe physical processes in terms of partial differential equations that is equations of several variables in which partial derivatives in terms of these variables appear In 1747 he published an article on vibrating strings whose motion he characterized in terms of a differential equation This contribution is also original and brilliant from a mathematical point of view Euler recognized the possibilities offered by such ideas and methods and he developed the theory much further In the process he criticized D Alembert s work for its imprecise formulations D Alembert who until this time had been on friendly terms with Euler now came into conflict with him even to the point of accusing him of having stolen his ideas Before he had already accused others of his contemporaries D Alembert was quick to become vigorously embroiled in argument and had great difficulty in acknowledging his own errors After repeated conflicts with members of the French Academy of Sciences he determined no longer to submit his articles for publication in Paris but turned instead to Berlin However in Berlin the Director of the Division of Mathematics at the Prussian Academy of Sciences the person responsible for receiving submissions for publication was none other then Euler The flames of conflict were further fanned when Frederick the Great invited D Alembert to succeed Pierre Louis Moreau de Maupertuis as president of the Prussian Academy of Sciences an appointment that was opposed by Euler D Alembert turned down the offer and stopped submitting articles for publication altogether preferring to collect and publish them later in eight volumes under the title Opuscules mathématiques mathematical booklets These volumes contain extraordinary works on such topics as complex valued functions approaches to making precise the notion of limit and the quotient criterion for infinite series later developed further by Cauchy In connection with the integration of rational functions Recherches sur le calcul intégral D Alembert discovered the method of partial fraction decomposition He realized that every real polynomial of even degree can be decomposed as a product of quadratic factors with real coefficients with complex roots always appearing as complex conjugate pairs Even today the so called fundamental theorem of algebra which states that a polynomial of degree n always has n roots counted with multiplicity in the complex plane is known in France as le théorème de D Alembert although D Alembert s proof was deficient with the first acceptable proof given only in 1799 by Carl Friedrich Gauss as part

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Open archived version from archive - Mathematics In Europe - Stefan Banach (March 30, 1892 – August 8, 1945); by Heinz Klaus Strick, Germany

opérations dans les ensembles abstraits et leur application aux équations intégrales On operations in abstract sets and their application to integral equations he was awarded a doctorate This work is considered to represent the origin of a new field of mathematics functional analysis Beginning with vector spaces whose elements are functions BANACH investigated the properties of spaces in which a norm can be defined by defining it is possible to measure the distance between elements a b BANACH s ground breaking work dealt with the properties of complete spaces X that is spaces in which Cauchy sequences in X have their limit in X A short while thereafter such spaces began to be called BANACH spaces and the vocabulary and terminology used by BANACH in his work were taken up by other mathematicians In quick succession there followed new articles by BANACH on the abstract spaces that he was investigating By 1922 he had completed his habilitation and was named an associate professor After an academic year in Paris he continued his work as a professor in Lwów founding a new mathematical journal with STEINHAUS which was devoted in particular to the further development of functional analysis and wrote a number of successful mathematical textbooks In 1932 he published what became the standard text in functional analysis the Théorie des opérations linéaires BANACH s working method was unusual in that while other mathematicians prefer the quiet of a study or library BANACH preferred the bustling atmosphere of a café where he could concentrate on his thoughts unperturbed by the noise and activity around him When his favourite café the Scottish Café Kawiarnia Szkocka closed he would often go the cafeteria at the railway station which was open at all hours In 1939 BANACH was elected president of the Polish Mathematical Society After the start of World War II Soviet troops occupied Lwów following the signing of the MOLOTOV RIBBENTROP pact Since BANACH had spent considerable time in Moscow during the 1930s and had good connections with Soviet mathematicians he was able to continue his work and was even named a dean of the university When the German troops occupied the city in 1941 the situation became difficult for BANACH It was only by chance that he escaped the mass murder of Polish intellectuals by the SS After the liberation of Lwów in July 1944 the renewal of academic work in Polish universities was supported by Soviet mathematicians BANACH however had only a short time remaining for his work weakened by the privations of the German occupation he had no resistance left and in August 1945 STEFAN BANACH died of lung cancer Among the most important theorems formulated by BANACH is the theorem that today goes under the name BANACH fixed point theorem If for a mapping f of a complete metric space R into itself there exists a q in 0 1 such that for all a b one has d f a f b le q d a b

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Open archived version from archive - Mathematics In Europe - Jacob Bernoulli (January 6, 1655 – August 16, 1705); by Heinz Klaus Strick, Germany

applied the differential calculus with great success and published papers on the calculation of tangents and surface areas In 1690 he succeeded in solving a problem posed by LEIBNIZ using the differential calculus what is the curve along which a body falls with constant speed the so called isochrone In his article he was the first to speak of an integral calculus after which LEIBNIZ adopted the term integral in his writings Equations representing the relationship between one or more quantities and their rates of change so called differential equations arise frequently in physics Some of these can be solved using the method of separation of variables an idea originating with JACOB BERNOULLI For example the relationship y x y between the variables x and y und the derivative of the latter becomes after rearrangement and integration yy x and hence int y dy int x dx that is y 2 2 x 2 2 C or y 2 x 2 2C which is the equation of a hyperbola in the figure can be seen the associated field of tangents of the differential equation an idea originating with JOHANN BERNOULLI At the lattice points of the coordinate system are shown the tangents whose slope can be calculated from the differential equation Together the Bernoulli brothers studied caustics envelopes of reflected rays and in this connection derived a formula for the osculating circle of a curve for a differentiable function its radius r can be calculated as follows r frac 1 f a 2 1 5 f a Picture of a caustic from Wikipedia Other papers prove that JACOB BERNOULLI knew how to apply the new calculus Catenary Lemniscate Brachistochrone What is the curve that takes the form of a chain hanging from two points of equal height The solution is the so called catenary f x a 2 e x a e x a What is the geometric locus of all points such that the product of their distance from two fixed points is constant The solution is the lemniscate x 2 y 2 2 a 2 x 2 y 2 Through what curve must two points at different heights be joined so that a body falling without friction travels from the upper point to the lower point in the least amount of time The brachistochrone curve was found as a solution by NEWTON LEIBNIZ und L HOSPITAL Together the brothers played a significant role in the dissemination and development of the calculus However beginning with little sensitivities and petty jealousies that made working together difficult over the course of years there developed an implacable hatred that did not remain hidden from other scientists The ambitious JOHANN BERNOULLI left Basel to take up a professorship of mathematics in Groningen Only after his brother s death did he return to Basel succeeding him in his chair at the university The year 1713 saw the beginning of the priority dispute between LEIBNIZ and NEWTON over who had invented the differential calculus and

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Open archived version from archive - Mathematics In Europe - János Bolyai (December 15, 1802–January 27, 1860)); by Heinz Klaus Strick, Germany

interior angles on one side together are smaller than two right angles then if the two lines are extended indefinitely they will intersect on the side on which lie the angles that are smaller than two right angles The question of the independence of the fifth postulate had been considered by many mathematicians since antiquity In all of these unsuccessful efforts an important role was played by from today s point of view imprecise terminology and insufficient mathematical rigor The very definition of parallel lines posed a problem EUCLID understood these to be lines that do not intersect an equivalent notion is the property that parallel lines are those that are perpendicular to the same line The formulation parallel lines are those that are everywhere equidistant already uses the parallel postulate Many mathematicians discovered formulations equivalent to the parallel postulate which however were no more amenable to proof than EUCLID s postulate itself For example in 1733 GIROLAMO SACCHERI showed that the formulation the sum of the angles of a triangle is equal to two right angles is equivalent to the parallel postulate and in 1705 JOHN PLAYFAIR proposed the following formulation which was also known in antiquity for every line g and every point S external to g there is exactly one line that is parallel to g and passes through the point S Now it follows already from the first four postulates that there is at least one such point thus an equivalent formulation would be the following for every line g and every point S external to g there is at most one line that is parallel to g and passes through the point S SACCHERI and later 1766 JOHANN HEINRICH LAMBERT who also proved that pi is an irrational number took indirect approaches they assumed that the first through fourth postulates were valid and that the fifth postulate was false and attempted to derive a contradiction They failed to find such a contradiction but they discovered remarkable properties that would exist in such a geometry without realizing that they had discovered a new geometry When FARKAS BOLYAI learned of his son s intention to work on the fifth postulate he wrote to him You must not pursue parallel lines along that path I know that path to its very end I too have lived through that endless night Every ray of light every joy of my life has been extinguished in it I beseech you in God s name Let the theory of parallels alone You should have the same disgust towards it as you would towards keeping debauched company It will consume all your leisure your health your peace and all your joy in life But JÁNOS did not give it up Toward the end of 1823 he informed his father that he had succeeded in creating a new and different world out of nothing He was disappointed when he realized that he was unable to convince his father of the validity of his ideas

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Open archived version from archive - Mathematics In Europe - Bernard Bolzano (October 5, 1781–December 18, 1848); by Heinz Klaus Strick, Germany

his works failed to receive the recognition that they would otherwise have obtained had there not been a prohibition against publication and also had many of his ideas not been so far ahead of their time It thus came to pass that a number of his ideas were rediscovered only decades later Already in his early mathematical writings BOLZANO was concerned with making mathematical proofs and the chain of their argument more rigorous In 1810 his Contributions to a better grounded presentation of mathematics appeared In 1816 there followed The binomial theorem as a consequence of it the polynomial theorem and the series that serve for the calculation of the logarithmic and exponential functions demonstrated more precisely than ever before In this work BOLZANO criticised the brilliant yet insufficiently precise methods of LEONHARD EULER and JOSEPH LOUIS LAGRANGE By the binomial theorem is meant here the binomial series that can be defined not only for natural number exponents but also for integer rational and even arbitrary real exponents n 1 x n n choose 0 n choose 1 x n choose 2 x 2 cdots 1 nx frac n n 1 2 x 2 cdots BOLZANO wrote that the difference between 1 x n and the r th term of the series 1 nx frac n n 1 2 x 2 cdots can be made less than any given quantity if one takes a sufficient number of terms of the series and this makes sense only for x 1 In 1817 he wrote his Pure analytical proof of the mean value theorem today known as Bolzano s theorem If a function f is continuous on a closed interval a b a b in Bbb R and if furthermore f a cdot f b 0 then at least one zero of f lies in the interval a b An important precondition for a proof of Bolzano s theorem is a precise definition of the notion of continuity for BOLZANO continuity in an interval meant that when x is an arbitrary number in the interval the difference f x omega f x can be made less than any given value if one takes omega sufficiently small a formulation that differs little from that used in contemporary mathematics CAUCHY s convergence criterion a sequence a n n in Bbb N converges if and only if for every epsilon 0 there exists a number n 0 such that for all n m with n m ge n 0 one has a n a m le epsilon AUGUSTIN LOUIS CAUCHY published this theorem in 1821 Yet four years earlier BOLZANO in the above mentioned paper had given the same necessary and sufficient conditions for the convergence of an infinite series that is an infinite sum but it went unnoticed When a sequence of values F 1 x F 2 x ldots F n x ldots F n r x is such that the difference between its n th term F n x and every subsequent

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Open archived version from archive - Mathematics In Europe - Georges-Louis Leclerc Buffon (1707–1788); by Heinz Klaus Strick, Germany

multivolume Histoire naturelle générale et particulière A Natural History in General and in Detail planned to comprise fifty volumes dealing with the origin of the Earth minerals and the development of organisms in particular the various species of animals The first volumes appeared in 1749 and three years later it appeared in German translation and it was also translated into many other languages including English Publication led to conflict with the Church because of the work s advocacy of a theory that the Earth arose through collision of a comet with the Sun and that the motion of the planets is to be explained not by divine intervention but by the laws of mechanics Buffon formulated the hypothesis that all life developed from tiny particles and that further development was influenced by climate change On the basis of research in comparative anatomy he conjectured among other things that apes and human beings have the same ancestry His research led him to estimate the age of the Earth at about 75000 years Until that time the Church based on biblical genealogy had declared the age of the Earth to be about 6000 years During Buffon s lifetime 36 volumes of the encyclopaedia were published They were among the most widely disseminated works of the Age of Enlightenment and they brought Buffon international renown For his services he was granted by the king in 1773 the title of Comte He was elected to the Académie française in 1753 in recognition of his cultivated writing style indeed his texts were reprinted in French readers until well into the twentieth century Critics among whom can be counted the mathematician and physicist Jean Baptist le Rond d Alembert 1717 1783 belittled his style calling him a grand phrasemonger However throughout his life he serenely sidestepped criticism on the principle that criticism would redound unfavourably on the critics Buffon married twice His only son on whom he pinned his hopes for a continuation of his work was a disappointment His life ended under the guillotine during the turmoil of the French Revolution In 1777 Buffon s Essai d Arithmétique Morale appeared in a supplementary volume to his Histoire naturelle in which he discussed questions of probability and statistics Using an extensive collection of data on births marriages and deaths in Paris he developed mortality tables that make it possible to estimate the probability that a person aged n would live another x years It becomes clear that he had difficulty in making precise the notion of the probability of an outcome when we read that he gave 1 10000 as the probability that a 56 year old individual would die in the course of any one day and then asserted that in his opinion such a probability could not be distinguished from the probability zero He gave considerable attention to the St Petersburg paradox holding the problem for unrealistic because it could not be carried out in practice Moreover he repudiated games of chance fundamentally even

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Open archived version from archive - Mathematics In Europe - Georg Cantor (March 3, 1845–January 6, 1918); by Heinz Klaus Strick, Germany

case since the sequence does not contain the real numbers that differ in the first digit of their decimal expansion from x 1 and in their second digit from x 2 and in their third digit from x 3 and so on furthermore not all the digits can be equal to 9 CANTOR s second diagonalization procedure CANTOR realized that there must be at least two different types of infinity and he introduced the use of the Hebrew letter aleph to denote the cardinalities of the two sets aleph 0 N Q aleph 1 R Since the set of real numbers turned out not to uncountable today we call such a set uncountably infinite or simply uncountable whereas the subset of the real numbers comprising all algebraic numbers is countable it follows that the set of transcendental numbers is also uncountable or to put it somewhat sloppily that almost all real numbers are transcendental In 1874 CANTOR who at first considered it superfluous found time to use in his search a method by means of which to his great surprise one can establish a one to one correspondence between the points of the unit square and those of the unit interval 0 1 to a point x y with x 0 a 1a 2a 3 and y 0 b 1b 2b 3 is assigned the number z a 1b 1a 2b 2a 3b 3 of the interval 0 1 The inverse of this mapping is not quite one to one since for example one has 0 5 0 49999 however CANTOR found a way around this imperfection DEDEKIND who was the first to hear of these sensational results CANTOR Je le vois mais je ne le crois pas was surprised that a two dimensional surface should have the same cardinality as a one dimensional interval However he saw no flaw in CANTOR s reasoning The year 1874 also saw CANTOR s marriage to a friend of his sister their wedding journey took them to Interlaken where CANTOR met often with DEDEKIND When in 1877 CANTOR submitted a lengthy article on his researches to CRELLE s Journal including a proof that the unit interval has the same cardinality as the n dimensional unit cube opposition arose for the first time KRONECKER God created the integers all else is the work of man tried to prevent publication It was printed only after the intervention of DEDEKIND CANTOR submitted no further articles to CRELLE s Journal his next six articles appeared in the Mathematische Annalen supported by FELIX KLEIN In 1879 CANTOR finally was promoted to a full professorship in Halle following a long but vain hope of being called to the more prestigious University of Berlin When in 1881 HEINE died CANTOR hoped to bring DEDEKIND to Halle to occupy the now open chair as HEINE s successor But when DEDEKIND declined the offer the mutually fruitful correspondence between CANTOR and DEDEKIND came to an end CANTOR s publications in the Mathematische

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Open archived version from archive