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- Mathematics In Europe - Isaac Newton (January 4, 1643-March 31, 1727); by Heinz Klaus Strick, Germany

align 1 x 3 1 3x frac 3 cdot2 2 cdot x 2 frac 3 cdot2 cdot1 3 cdot2 cdot x 3 frac 3 cdot2 cdot1 cdot 0 4 cdot3 cdot2 cdot x 4 cdots 1 3x 3x 2 x 3 end align Such a series development is also possible for negative and fractional exponents begin align 1 x 3 1 3x frac 3 cdot 4 2 cdot x 2 frac 3 cdot 4 cdot 5 3 cdot2 cdot x 3 quad frac 3 cdot 4 cdot 5 cdot 6 4 cdot3 cdot2 cdot x 4 cdots 1 3x 6x 2 10x 3 15x 4 cdots end align Newton s proof of the correctness of this result follows by multiplying out In the product begin align 1 x 3 cdot left 1 3x 6x 2 10x 3 15x 4 cdots right end align all the terms with positive exponents add to zero and all that remains is the number 1 An analogous result is obtained with the square root function begin align 1 x 1 2 1 frac12 x frac frac12 cdot left frac12 right 2 x 2 frac frac12 cdot left frac12 right cdot left frac32 right 2 cdot3 x 3 cdots 1 frac12 x frac18 x 2 frac1 16 x 3 frac5 128 x 4 cdots end align In the case the proof of correctness of the series development is obtained by squaring the infinite sum Using this method the extraction of a root is considerably simplified For example an approximate value for sqrt 3 can be calculated in the following way begin align sqrt 3 sqrt 4 cdot frac34 2 cdot sqrt 1 frac14 approx 2 cdot left 1 frac12 cdot frac14 frac18 cdot frac1 16 right approx 1 734 end align Newton used this new technique of series development that he had devised to determine the areas under curves In his paper can be found a rule in which one can recognize a standard formula for the integral If ax frac m n y it shall be frac an m n cdot x frac m n n text Area Using this he also calculated among other things the area under the arc of a circle thereby determining the value of pi to fifteen decimal places North Korea 1993 The infinitesimal calculus developed by Newton method of fluxions and inverse method of fluxions is very strongly oriented toward physical ideas he conceptualizes a curve as a continuous motion of a point he calls the time dependent variables fluents flowing quantities and the rate of change of a quantity y is called its fluxion dot y For an infinitesimally small interval of time he employs the letter o while he notates the infinitesimally small increase in the fluent x in the infinitesimally small time interval o that is the instantaneous velocity in an infinitesimally small time period as dot x dot o To calculate the slope of a tangent to a curve one has only to compute the

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Open archived version from archive - Mathematics In Europe - Emmy Noether (March 23, 1882–April 14, 1935); by Heinz Klaus Strick, Germany

dissertation on the theory of invariants Later she called her thesis which had been awarded summa cum laude mere calculation and a tangle of formulas Indeed her dissertation reflected the style of her thesis advisor PAUL GORDAN who had made a few small advances in the theory of invariants in contrast to HILBERT who after a short acquaintance with that branch of mathematics had presented fundamentally new results about which GORDAN had only this to say That is not mathematics but theology She spent the following eight years teaching and doing research at the university but only unofficially even when she took over for her father who had taken ill Following sensational publications on various topics in algebra she was accepted into the Deutsche Mathematiker Vereinigung DMV the German Mathematical Society In 1909 she became the first woman to present a lecture at the DMV s annual conference In 1915 she was invited by KLEIN and HILBERT to continue her mathematical researches in Göttingen but the regulations governing habilitation at Prussian universities did not allow women as instructors and so her lectures could be given only with the following disclaimer Theory of Invariants Prof Hilbert with the Support of Fräulein Dr Noether In 1918 she published a theorem in which geometric properties of space were considered in connection with the conservation laws of physics This theorem now known in her honour as NOETHER s theorem proved in the years to come to be one of the most important foundations of theoretical physics Hilbert and Einstein Petitions by HILBERT and KLEIN to make an exception in the case of EMMY NOETHER and give her permission to teach were processed only grudgingly by the relevant ministry and were eventually turned down Even the intervention of ALBERT EINSTEIN who expressed his enthusiasm about NOETHER as a mathematician was unsuccessful The main argument against accepting these petitions was typical of the time no one had any experience to be able to tell whether over time a woman would be able to withstand the pressures of university teaching and for that reason it was impossible to give them permission to teach HILBERT is believed to have said in this regard The candidate s gender can surely be no argument against granting permission After all we are talking about a university not a bathhouse As part of the political reforms following the November Revolution of 1918 the laws governing the universities were changed in June 1919 EMMY NOETHER finally obtained the venia legendi the right to teach at a university Although she now was able to lecture under her own name and from 1922 on had the title of Außerordentlicher Professor associate professor she received no salary and depended on family for financial support It was not until 1923 that is when she was 41 years old that EMMY NOETHER began to be paid for giving lectures which is to say that she was paid only during the semester proper On account of her continued financial

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Open archived version from archive - Mathematics In Europe - Luca Pacioli (1445–1517); by Heinz Klaus Strick, Germany

pages The book appeared in the Italian language that is not in the usual Latin which led to its being widely disseminated And moreover as a printed book it could be reissued and indeed it was Later mathematicians frequently referred to the Summa and to that extent LUCA PACIOLI may be said to have influenced the further development of mathematics considerably One might even say that the publication of his book ushered in a new flowering of mathematics in Italy By the end of the fifteenth century the Arabic Indic numbers had largely replaced Roman numerals in Europe However sums and differences were not written as they are today PACIOLI proposed writing p as shorthand for plus italian più more and m for minus Italian meno Moreover he used the symbol R for the square root radix It was still a complicated matter to express relationships between quantities For example the equation x 4 x x 2 a would be expressed thus Censo de censo e cosa equale a censo e numero where cosa represented the unknown number censo the square of this number and censo de censo for the fourth power The Summa comprised eight chapters The first contains a summary of the books of EUCLID on fundamental geometric constructions calculations of areas and similarity theory The second is concerned with special lines in a triangle The third treats right triangles and the associated solution of quadratic equations theorem of PYTHAGORAS In contrast to the writings of AL KHWARIZMI 780 850 here the solutions of quadratic equations are found by calculations not by means of drawings In addition PACIOLI dealt with equations of the third and fourth degree which he held to be generally unsolvable impossibile an assertion that not long afterwards was refuted by SCIPIONE DEL FERRO 1515 but not published NICCOLÒ TARTAGLIA 1535 and GIROLAMO CARDANO 1545 The fourth chapter concerns the theory of the circle Tables of chords give information on the lengths of chords and their associated arcs For pi PACIOLI gives the approximation 3 33 229 In the fifth chapter the division of geometric figures is discussed theory of ratios The sixth chapter explains how to calculate the surface area and volume of geometric solids The seventh chapter introduces apparatus and methods of measurement The eighth chapter contains a number of applications of different types calculation of the volume of a barrel approximately described as two frusta of cones calculations on regular solids the inscribing of several equal circles of maximal size in a triangle and in a circle Finally PACIOLI provides an overview of the coinage and weights and measures of the various Italian city states Another postage stamp which also appeared in 1994 indicates an additional aspect of the content of chapter 8 PACIOLI gives an introduction to the so called Venetian method of bookkeeping which realizes the principle of the double entry accounting system Although PACIOLI assuredly did not invent the system he was the first to give a self contained

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Open archived version from archive - Mathematics In Europe - Blaise Pascal (June 19, 1623–August 19, 1662); by Heinz Klaus Strick, Germany

care of physicians whose success was more than medical they converted the entire family to the Catholic reform movement known as Jansenism From this time forward BLAISE PASCAL became deeply religious though at first he continued to work on mathematical problems One of his sisters decided to enter the Jansenist convent Port Royal in Paris In 1647 PASCAL turned once again to the subject of conic sections One of his works contains among other things the following theorem PASCAL s theorem If a hexagon has vertices A B C D E F that are points lying on a conic section here a circle then the intersection points of each pair of opposite sides that is AB and DE BC and EF CD and AF lie on a straight line PASCAL showed that this theorem remains valid even when the points are not taken in alphabetical order PASCAL s theorem The correspondence between BLAISE PASCAL and PIERRE DE FERMAT 1608 1665 from 1654 stimulated by ANTOINE GOMBAUD CHEVALIER DE MÉRÉ is considered the origin of probability theory PASCAL understood that one is more likely to roll at least one six on the throw of four dice 51 8 than at least one double six in twenty four throws of two dice 49 1 This had seemed to him illogical l arithmétique se démentoit since in each case one was dealing with the same numeric relationship 4 to 6 and 24 to 36 FERMAT and PASCAL discovered differing solutions to the following problem of LUCA PACIOLI How should the wagers of two players be divided if a game has to be terminated prematurely One of PASCAL s solutions involved a recursive calculation of the odds of winning Suppose that a game requiring three points to win is broken off with the score at 2 points to 1 Then the first player would win the entire wager were he to win the next round while if the second player won the next round then the odds for the two players would have become equal Therefore altogether the first player is three times more likely to have won than the second player If the game was broken off at a score of 2 to 0 then the first player would have won the entire wager if he had won the next round But if the second player had won the next round then the score would have stood at 2 to 1 the odds for which have already been determined above If the score was 1 to 0 when the game was terminated then if the first player had won the next round the score would have been 2 to 0 for which the odds have already been calculated while if the second player had won then their chances of winning would have been equal and so on Another of PASCAL s solutions relates to his intensive involvement with a special number triangle triangle arithmétique He showed that the correct distribution of the wager

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Open archived version from archive - Mathematics In Europe - Henri Poincaré (April 29, 1854–July 17, 1912); by Heinz Klaus Strick, Germany

effortlessly and allow nothing to divert his attention But if the beginning of an article would not fall properly into line he would stop work on it because he found the necessary intuition lacking He was convinced that his subconscious mind would continue working on the problem until the time came that it would reappear solved in his conscious mind Therefore he did not work on his own problems during the evening because he feared that his sleep might suffer if he did In 1906 he gave a lecture at a psychology convention in which he described how some of his inspired ideas had matured conscious thinking unconscious thought incubation inspiration and validation Poincaré was convinced that mathematical logic had no part in the development of ideas that it in fact limited the possibility of generating ideas He opposed the attempts of contemporary mathematicians such as Giuseppe Peano and David Hilbert to describe all of mathematics through a system of axioms suspecting the limitation of such an approach a limitation that was in fact revealed in 1931 by the Austrian logician Kurt Gödel Beginning with his doctoral dissertation Poincaré worked on problems in complex analysis investigating general properties of mappings as well as the structures that are preserved under certain mappings In this connection he introduced for a particular class of mappings in ignorance of prior work of Felix Klein at the time a professor of geometry at the University of Leipzig the name Fuchsian functions after the German mathematician Lazarus Fuchs even though the name Kleinian functions would have been more appropriate The exchange of letters on this topic between Poincaré and Klein that followed was not carried out in a cordial atmosphere When Poincaré eventually discovered connections between these functions and non Euclidian geometry which was in fact the special area of knowledge to which Felix Klein laid claim this led to the latter s suffering a collapse or perhaps the victory of a rival was just the trigger of a creative crisis for Felix Klein during a phase of extreme overwork Both Klein and Poincaré independently developed models of so called hyperbolic geometry in which in contrast to Euclidian geometry for a given line and point external to the line there are at least two lines that pass through the point and are parallel to the line In one of the models developed by Poincaré the lines are modelled by circular arcs and diameters that run perpendicular to the boundary a circle in the other it is semicircles and half lines that are perpendicular to the boundary line In 1885 King Oscar II of Sweden underwrote a prize competition The commission which comprised the Swedish mathematician Magnus Gustaf Gösta Mittag Leffler the German Karl Weierstrass and Charles Hermite quickly concluded that the 160 page contribution from Poincaré on the three body problem should be awarded the prize even though he had not completely solved the problem that had been posed In the three body problem one considers

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Open archived version from archive - Mathematics In Europe - Bernhard Riemann (September 17, 1826–July 20, 1866); by Heinz Klaus Strick, Germany

interval a b into subintervals and points t i in the subintervals x i 1 x i he considered sums of the form sum if t i x i x i 1 today called RIEMANN sums which give the area of the staircase figure associated with the decomposition and choice of intermediate points He declared a function f to be integrable on the interval a b if the sums for arbitrary choices of decomposition and intermediate points all converge to the same fixed number which is called the integral today RIEMANN integral of f over a b For his public habilitation lecture RIEMANN was required to offer the faculty three topics from which they could choose In parallel to his habilitation thesis he had worked on an article on the subject of electricity galvanism light and gravitation and he offered that as one of the topics The second topic was also related to questions in physics GAUSS however wanted to hear about the third topic Über die Hypothesen welche der Geometrie zu Grunde liegen on the hypotheses underlying geometry because he was curious to learn what sorts of insights into the subject so young a man as RIEMANN had come up with RIEMANN had the ambition of presenting a talk that would be accessible to a general audience even nonmathematicians However it is likely that GAUSS was the only one present who could truly follow the exposition He was deeply impressed and he praised something he did not often do the intellectual depth of RIEMANN s ideas It was only sixty years later that the importance of RIEMANN s approach was recognized when EINSTEIN relied on RIEMANN s work in the development of his general theory of relativity Einstein RIEMANN s lecture dealt with the mathematical possibility of the existence of non Euclidean geometries whether they actually exist is something for physicists to worry about RIEMANN defined space as an n dimensional manifold with a metric that allows the determination of the distance between points In this case the shortest points joining pairs of lines will not necessarily be lines but curves geodesics whose curvature can depend on the location of the points in space The angle sums of triangles can equal 180 but they can also be less than or greater than 180 After being granted the venia legendi or right to give lectures he was able to teach as a privatdozent beginning in the winter semester of 1854 1855 At first he had great difficulty in adjusting to the pace at which his students could master new material but gradually he succeeded in reading from their reactions whether an explanation needed to be repeated in different words and whether a proof had to be explained in greater detail Following the death of GAUSS DIRICHLET was called to Göttingen as his successor He tried in vain to have RIEMANN whom he valued highly to be named an associate professor Nevertheless DIRICHLET did manage to obtain for him an annual

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Open archived version from archive - Mathematics In Europe - Adam Ries (ca. 1492–March 30, 1559); by Heinz Klaus Strick, Germany

law This law presented a table that declared the weight that a loaf of bread costing a penny must have based on the current prices of grain and flour Three years later his book Ein Gerechnent Büch lein auff den Schöffel Eimer und Pfundgewicht a booklet for calculating by the scoop the bucket and the pound appeared in which it was explained how to convert between various weights and measures In 1539 he was named to the post of Kurfürstlich Sächsischen Hofarithmeticus an honorary appointment as ducal arithmetician as a reward for his services On his death in 1559 three of his sons continued his work as arithmeticians in Annaberg With his books written in German instead of the usual Latin and at a level understandable to the layman Adam Ries made a significant contribution to universal education in arithmetic in that now more people could learn to calculate than had previously been the case His books also accelerated the process of creating a unified German language The first book that Adam Ries wrote whose complete title is Rechenung auff der linihen gemacht durch Adam Riesen vonn Staffel steyn in massen man es pflegt tzu lern in allen rechenschulen gruntlich begriffen anno 1518 contains a large collection of exercises with solutions but not the reasoning oriented towards problems that arise in everyday life above all the calculation of prices according to the rule of three which involved conversions that were more complicated than what we have today 1 gulden 21 groschen 252 pfennigs To calculate on the lines one uses calculating pennies which are laid out on a cloth or board equipped with lines The lines represent from bottom to top ones tens hundreds and thousands corresponding to the Roman numerals I X C M A calculating penny placed between lines in the spacium corresponds respectively to 5 50 500 that is V L D In the figure the number 739 is represented For addition and multiplication one uses the technique of elevation when five coins appear on a line they are replaced with a single coin in the spacium above and when two coins are lying in the space they are replaced by a single coin on the line above For subtraction and addition one has to as necessary employ resolution In multiplying by single digit factors the number of coins on a line or in a space is first multiplied and then elevated The factor 10 causes the coins to jump to the next line or space as required Ries s second book whose complete title is Rechenung auff der linihen unnd federn in zal maß vnd gewicht auff allerley handierung gemacht vnd zusamen ge lesen durch Adam Riesen vö Staffelsteyn Rechenmeyster zu Erffurdt im 1522 Jar contains along with typical exercises from commercial life calculations in simple and compound interest problems with mixtures converting weights and measures exercises from recreational mathematics Moreover Ries uses the false position method or regula falsi Here is an example of the

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Open archived version from archive - Mathematics In Europe - Bertrand Russell (May 18, 1872–February 2, 1970); by Heinz Klaus Strick (Germany)

he discovered that the definition of a set introduced a few years earlier by GEORG CANTOR as a collection of objects of our intuition or thought grouped into a single object leads to an antinomy Greek incompatibility which has become known as RUSSELL s paradox a set that contains every set that does not contain itself as an element is a contradiction in terms Later RUSSELL offered the following popular formulation of the problem One can define a barber as someone who shaves all those who do not shave themselves The question then is does the barber shave himself The formation of a set must therefore be subject to certain restrictions To this end RUSSELL developed the theory of types in which there exist sets of various types This approach solves the problem but it did not take hold The German mathematician and logician GOTTLOB FREGE on whose work Die Grundlagen der Arithmetik the foundations of arithmetic RUSSELL had also initially relied fell into a deep depression on learning of RUSSELL s discovery In an epilogue to the second edition of his book FREGE acknowledged that through RUSSELL the foundations of his construction had been shattered From 1910 on RUSSELL worked as a lecturer in mathematics at Trinity College however the university fired him from that position when enraged at the general enthusiasm for war and the barbarity of the First World War he publicly declared himself a pacifist In 1918 he was sentenced to six months imprisonment for his antiwar activities While in prison he wrote a number of books including Roads to Freedom Socialism Anarchism and Syndicalism An initial sympathy for the socialist experiment ended after he visited the Soviet Union in 1920 and spoke with LENIN He called Russia an asylum for dangerous mental patients where the guards are the most dangerous of all The outbreak of the First World War meant for RUSSELL a departure of PYTHAGORAS even though he continued to write books and articles on philosophy such as the Introduction to Mathematical Philosophy 1919 written for a general audience and later the critical review A History of Western Philosophy 1946 In 1920 he accepted a guest professorship in Beijing and immersed himself in Chinese culture On his return he earned his living as the author of numerous books on a variety of subjects including popular scientific books on atomic physics and the theory of relativity but also on politics and education After a fruitless search for a suitable school for his two children he founded together with his second wife the experimental antiauthoritarian private Beacon Hill School about which he later admitted that his goals had not been realized He caused a stir with his books Why I Am Not a Christian 1927 in which the atheist RUSSELL dealt critically with religion Christianity in particular and his Marriage and Morals 1929 a plea for a freer sexual morality In 1936 he married for a third time and accepted a position at the University of

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Open archived version from archive