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- Arbogast
- De Moivre
- Euler
- Kramp
- Stirling
- Taylor
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  Louis François Antoine Arbogast is a french mathematician, born in Mutzig (Alsace) on October 4th 1759, died in Strasbourg (France) on April 18th 1803.

  He was professor of mathematics at the Collège de Colmar and entered a mathematical competition which was run by the St Petersburg Academy. His entry was to bring him fame and an important place in the history of the development of the calculus. Arbogast submitted an essay to the St Petersburg Academy in which he came down firmly on the side of Euler. In fact he went much further than Euler in the type of arbitrary functions introduced by integrating, claiming that the functions could be discontinuous not only in the limited sense claimed by Euler, but discontinuous in a more general sense that he defined that allowed the function to consist of portions of different curves. Arbogast won the prize with his essay and his notion of discontinuous function became important in Cauchy's more rigorous approach to analysis.

  In 1789 he submitted in Strasbourg a major report on the differential and integral calculus to the Académie des Sciences in Paris which was never published. In the Preface of a later work he described the ideas that prompted him to write the major report of 1789. Essentially he realised that there was no rigorous methods to deal with the convergence of series, and Arbogast's career reached new heights. In addition to his mathematics post, he was appointed as professor of physics at the Collège Royal in Strasbourg and from April 1791 he served as its rector until October 1791 when he was appointed rector of the University of Strasbourg; in 1794 he was appointed Professor of Calculus at the Ecole Centrale (soon to become the Ecole Polytechnique) but he taught at the Ecole Préparatoire.

  His contributions to mathematics show him as a philosophical thinker that has to face his era. As well as introducing discontinuous functions, as we discussed above, he conceived the calculus as operational symbols. The formal algebric manipulation of series investigated by Lagrange and Laplace in the 1770s has been put in the form of operator equalities by Arbogast in 1800. We owe him the general concept of factorial as a product of a finished number of terms in arithmetic progression.

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De Moivre

De Moivre

  Abraham de Moivre is a french mathematician born in Vitry (France) on May 26th 1667 and died in London (England) on November 27th 1754.

  A French Protestant, de Moivre emigrated to England in 1685 following the revocation of the Edict of Nantes and the expulsion of the Huguenots; and In 1697, he was elected a fellow member of the Royal Society.

  De Moivre pioneered the development of analytic geometry and the theory of probability. He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities. In Miscellanea Analytica (1730) appears Stirling's formula (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre is also remembered for his formula for (cos x + isin x)^n which took trigonometry into analysis.

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  Leonhard Euler is a swiss mathematician, physicist, engineer and philosopher, born in Basel (Switzerland) on April 15th 1707 and died in St Petersburg (Russia) on September 18th 1783.

  Euler went there to the university and took lessons by Jean Bernoulli and passed his master's degree at sixteen. In 1727, invited by Catherine 1st, Empress of Russia, he becomes a member of the faculty of the Science Academy in Saint-Petersburg. He's appointed professor of physics in 1730 and professor of mathematics in 1733. In 1741, at the request of king of Prussia Frederic le Grand, he becomes professor of mathematics to the Science Academy of Berlin. He comes back to St Petersburg in 1766 and remains there until his death. Although handicapped before the thirty years age by a partial sight's loss and later by a quasi total blindness, Euler completed many significant mathematical works and hundreds of mathematical and scientific books.

  In his Introduction to the analysis of infinitely small (1748), Euler is the first to enter upon algebra, theory of equations, trigonometry and the analytical geometry, in an analytical and  global way. In this work, he deals with development in series of the functions and defines the rule in which only the convergent infinite series can be correctly evaluated. He discusses about 3D-surfaces and proves that the conic sections are represented by the general equation of second degree with two variables. Other work treats infinitessimal calculus, parts of which treat calculation of variations, numbers' theory, ireal numbers and specified and unspecified algebra. Euler gives also contributions in astronomy, mechanics, optics and acoustics. Engineer, he's the inventor of the first turbine. Among his works, we should quote Reflexions about space and time (1748), Treatise about differential calculus (1755), Etablishment of integral calculus (1768-1770), Introduction to nature's theory (1755-1759) and Introduction to algebra (1770).

  Euler had also a big interest in philosophy, fought againgst Wolff and Leinbniz's theories, was devoted to syllogistic of Aristote, and tried to formalize them with circles, precursors of Venn diagrams. In Chess, he studied the walk of the knight.

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  Christian Kramp is a french mathematician born on July 8th 1760 and died on May 13th 1826 in Strasbourg (France).

  Christian Kramp's father was has teacher at grammar school in Strasbourg. Kramp studied medicine and, after graduating, practised medicine in the area around where patients lived in a fairly wide area. However his interests certainly ranged outside medicine for, in addition to a number of medical publications, he published a work on crystallography in 1793. In 1795 France annexed the Rhineland area in which medical Kramp was carrying out his work and after this he became has teacher at Cologne (this city was a french one from 1794 to 1815), teaching mathematics, chemistry and physics.

  Kramp was appointed professor of mathematics at Strasbourg, the town of his birth, in 1809. He was elected to the geometry section of the Academy of Science in 1817. As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers. His work on factorials is independent of that of Stirling and Vandermonde. He was the first to uses the notation n!  (Elements d'arithmétique universelle, 1808). In fact the more general concept of factorial was found at the same time by Arbogast.

  Parts of Elements d'arithmétique universelle: http://members.aol.com/jeff570/stat.html

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  James Stirling is a scotish mathematician, born in May 1692 in Garden (near Stirling, Scotland), died on December 5th 1770 in Edinburg (Scotland).

  His talent was found by Newton. In 1717 Stirling taught in Venezia and published his first work in Roma Lineae Tertii Ordinis Neutonianae which extends Newton's theory of planes curves of third degree, adding a new standard of curves to the 72 given by Newton. The work was published in Oxford and Newton himself received a copy. Lineae Tertii Ordinis Neutonianae contains other results that Stirling had obtained. There are results on the curve of quickest descent, results on the catenation (in particular relating this problems to that of placing spheres in an arch), and results on orthogonal trajectories. The problem of orthogonal trajectories had been raised by Leibniz and many mathematicians worked on the problem in addition to Stirling, including Johann Bernoulli, Nicolaus(I) Bernoulli, Nicolaus(II) Bernoulli, and Leonard Euler. It is known that Stirling solved the problem early in 1716.

  In London in 1730, significant Stirling published his most significant work Methodus Differentialis . This book is his treatise on infinite series, summation, interpolation and squaring. At this time Stirling was in correspondance with de Moivre, Cramer and Euler. The asymptotic formula for n!, which made Stirling the most known, appears in Example 2 to Proposition 28 of the Methodus Differentialis. One of the principal goals of the book was to consider methods of speeding up the convergence of series. Stirling notes in the Preface that Newton had considered this problem. In Methodus Differentialis, many examples of his methods are given, including Leibniz's problem of pi/4=1-1/3+1/4-1/5+1/6-... and he gives also a theorem that  treats convergence of an infinite product. Included in this work on accelerating convergence is a discussion of de Moivre's methods. The book contains some other results of the Gamma function and the Hypergeometric function.

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  Brook Taylor is an eclectic english scientist born in Edmonton (England) on August 18th 1685, and died in London on December 29th 1731. He took an interest in music, paint and philosophy.   In 1712 Taylor was elected to the Royal Society of London. This was on 3rd of April and clearly it was an election based more on the expertise of Machin, Keill and others, rather than on the publication of his results. For example Taylor wrote to Machin in 1712 providing a solution to a problem concerning Kepler's second law of planetary motion. Also in 1712 Taylor was appointed to the committee set up to adjudicate on whether the claim of Newton or of Leibniz to cuts invented the calculus was correct.

  The year 1714 also marks the year Taylor was elected Secretary to the Royal Society. It was a position Taylor held from January 14th of that year until October 21st 1718, when he resigned, partly for health reasons, partly due to his lack of interest. The period during which Taylor was Secretary of the Royal Society marks his mathematically most productive period. Two books which appeared in 1715, Methodus incrementorum directa and reversed and Linear Perspective are extremely significant in the history of mathematics. Two later editions will appear in 1717 and 1719 respectively.

  Taylor made several visits in France. These were made partly for health let us reasons and partly to visit the friends. He meets Pierre Rémond de Montmort and corresponds with him over various mathematical topics after his return. In particular they discussed infinite series and probability. Taylor also corresponded with De Moivre over probability and there was a three-way discussion going one between these mathematicians.

  Taylor added to mathematics a new branch now called the calculus of finite differences, invented integration by shares, and discovered the famous series known as Taylor's expansion. These ideas appear in his book Methodus incrementorum directa and reversed of 1715 referred to above. In fact the first mention by Taylor of version of what is today called Taylor's theorem appears in a letter which he wrote to Machin on july 26, 1712. In this letter Taylor explains carefully where he got the idea from. It was, wrote Taylor, due to how Machin made in Child's Coffeehouse when he had commented in using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr. Halley's method of extracting roots" of polynomial equations. There are, in fact, two versions of Taylor's Theorem given in the 1715 paper. Taylor initially derived the version which occurs the Proposition 11, a generalization of Halley's method of approximating roots of the Kepler equation, goal soon discovered that it was a consequence of the Bernoulli series. This is the version which was inspired by the Coffeehouse conversation described above. The second version occurs the Corollary 2 to Proposition 7 and was thought of a method of expanding solutions of fluxional equations in infinite series. This result was one which Taylor was the first to discover!

  James Gregory, Newton, Leibniz, Johann Bernoulli and De Moivre had all discovered variable of Taylor's theorem. All these mathematicians made their discoveries independently and Taylor's work was also independent of the others' work. The importance of Taylor's theorem remained unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus! The term "Taylor's series" seems to cuts used for the first time by Lhuilier in 1786. Taylor also devised the basic principles of perspective in Linear Prospect (1715). The second edition has a different title, being called New principles of linear perspective.

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