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40-528: In 1691, John Dunton founded the Athenian Society (not to be confused with several other Athenian societies) in order to publish a journal. This group was originally composed of a small number of friends: John Dunton and mathematics teacher Richard Sault , then philosopher Dr. John Norris (though he declined to be part of the Society in writing and associated to profits), quickly joined by Dunton's brother-in-law

80-427: A translation into English from the third Latin edition of Breviarium Chronologicum , by Gyles Strauchius ( Aegidius Strauch II ), professor in the university of Wittenberg (the title page of the second (1704) edition has the by-then-dead Sault mistakenly as FRS). Abraham De Moivre Abraham de Moivre FRS ( French pronunciation: [abʁaam də mwavʁ] ; 26 May 1667 – 27 November 1754)

120-468: A whole volume for 2.5 shilling (about one month after the last issue collected was released), a more permanent form with indexes preferred by learned customers and distinguished women; this is why the journal is often referenced to by its original Athenian Gazette name rather than the Athenian Mercury issues. In 1695, a glut of new titles led to the journal temporarily pausing in early 1696; in 1697,

160-492: Is a common claim that De Moivre noted he was sleeping an extra 15 minutes each night and correctly calculated the date of his death as the day when the sleep time reached 24 hours, 27 November 1754. On that day he did in fact die, in London and his body was buried at St Martin-in-the-Fields , although his body was later moved. The claim of him predicting his own death, however, has been disputed as not having been documented anywhere at

200-426: Is directly related to its distance from the centre of the forces and reciprocally related to the product of the diameter of the evolute and the cube of the perpendicular on the tangent." In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P

240-588: Is no explicit contribution from Raphson) . It is unclear what relationship there might have been between Sault and Raphson, but the issue of Memoirs for the Ingenious for July, 1693 contains an exchange of letters on geometrically-inspired speculation of the sort Raphson treated in De Spatio Reali (1697) and Demonstratio de Deo (1710), followed by a letter dedicated to the Honoured Joseph Raphson, FRS ;

280-401: Is proportional to FM/(R*(FP) ) where R is the radius of the curvature at M. The mathematician Johann Bernoulli proved this formula in 1710. Despite these successes, de Moivre was unable to obtain an appointment to a chair of mathematics at any university, which would have released him from his dependence on time-consuming tutoring that burdened him more than it did most other mathematicians of

320-531: The Marquess of Normanby ; the second volume includes Sault's translation of a recent biography of Malebranche. In the Philosophical Transactions for 1698 is a short two-page note by Sault on Curvæ Celerrimi Descensus investigatio analytica, which shows that Sault was acquainted with Isaac Newton 's geometrical theory of vanishing quantities, and with the notation of fluxions . In 1699, Sault published

360-603: The Royal Exchange , London. John Dunton the publisher, learning of him and his skill in mathematics, supplied him with literary work. When the notion of establishing The Athenian Mercury occurred to Dunton, he sought Sault's aid as joint editor and contributor. The first number came out on 17 March 1691, and the second on 24 March. Before the third number Dunton and Sault had joined to them Dunton's brother-in-law, Samuel Wesley . There are Articles of agreement between Sam. Wesley, clerk, Richard Sault, gent., and John Dunton, for

400-622: The central limit theorem , a cornerstone of probability theory. Abraham de Moivre was born in Vitry-le-François in Champagne on 26 May 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at

440-450: The 4th in 1710. All four volumes were reprinted in 1728. Richard Sault Richard Sault (born around 1630s; died 1702) was an English mathematician , editor and translator , one of The Athenian Society . On the strength of his Second Spira he is also now credited as a Christian Cartesian philosopher. He kept in 1694 a mathematical school in Adam's Court, Broad Street, near

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480-566: The Die ), was written by Girolamo Cardano in the 1560s, but it was not published until 1663.) This book came out in four editions, 1711 in Latin, and in English in 1718, 1738, and 1756. In the later editions of his book, de Moivre included his unpublished result of 1733, which is the first statement of an approximation to the binomial distribution in terms of what we now call the normal or Gaussian function . This

520-574: The Royal Society, p. 219. As a result, some authors have argued that the Poisson distribution should bear the name of de Moivre. In 1707, de Moivre derived an equation from which one can deduce: which he was able to prove for all positive integers   n . In 1722, he presented equations from which one can deduce the better known form of de Moivre's Formula : In 1749 Euler proved this formula for any real n using Euler's formula , which makes

560-426: The book, he realised that it was far deeper than the books that he had studied previously, and he became determined to read and understand it. However, as he was required to take extended walks around London to travel between his students, de Moivre had little time for study, so he tore pages from the book and carried them around in his pocket to read between lessons. According to a possibly apocryphal story, Newton, in

600-674: The case of worms on the tongue mentioned in this latter letter was then taken up in correspondence in Philosophical Transactions in 1694 (where, however, Raphson is given as Ralphson , as also in Edmund Halley 's paper in the same volume). Sault, like Raphson, also worked on translations from the French. His translation of Nicolas Malebranche 's Concerning the Search after Truth (London, 1694, 1695) appeared in two volumes, both dedicated to

640-892: The coefficients of the terms in a binomial expansion. Specifically, given a positive integer n , where n is even and large, then the coefficient of the middle term of (1 + 1) is approximated by the equation: On June 19, 1729, James Stirling sent to de Moivre a letter, which illustrated how he calculated the coefficient of the middle term of a binomial expansion ( a + b ) for large values of n . In 1730, Stirling published his book Methodus Differentialis [The Differential Method], in which he included his series for log( n !): so that for large n {\displaystyle n} , n ! ≈ 2 π ( n e ) n {\displaystyle n!\approx {\sqrt {2\pi }}\left({\frac {n}{e}}\right)^{n}} . On November 12, 1733, de Moivre privately published and distributed

680-602: The controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House , St. Martin's Lane at Cranbourn Street, where he earned a little money from playing chess. De Moivre continued studying the fields of probability and mathematics until his death in 1754 and several additional papers were published after his death. As he grew older, he became increasingly lethargic and needed longer sleeping hours. It

720-415: The days of calculators calculating n ! for a large n was time-consuming. In 1733 de Moivre proposed the formula for estimating a factorial as n ! =  cn e . He obtained an approximate expression for the constant c but it was James Stirling who found that c was √ 2 π . De Moivre also published an article called "Annuities upon Lives" in which he revealed the normal distribution of

760-682: The death of Dunton's wife and the departure of Wesley after he received a promotion, led to a brief and aborted revival of the journal. It had run for 580 issues across nineteen volumes and a third: from 17 March 1691 to 8 February 1696 (19 full volumes of thirty issues, with a temporary closure between July and September 1692), then from May to 14 June 1697 (ten issues). In 1703, Dunton sold the Athenian Mercury to publisher Andrew Bell, who collected selected and abridged parts in larger volumes called The Athenian Oracle , 3 volumes in 1703–04, with multiple reprints. Dunton would go on to project compiling three more volumes (without serialization), releasing only

800-450: The first time had formal mathematics training with private lessons from Jacques Ozanam . Religious persecution in France became severe when King Louis XIV issued the Edict of Fontainebleau in 1685, which revoked the Edict of Nantes , that had given substantial rights to French Protestants. It forbade Protestant worship and required that all children be baptised by Catholic priests. De Moivre

840-548: The later years of his life, used to refer people posing mathematical questions to him to de Moivre, saying, "He knows all these things better than I do." By 1692, de Moivre became friends with Edmond Halley and soon after with Isaac Newton himself. In 1695, Halley communicated de Moivre's first mathematics paper, which arose from his study of fluxions in the Principia Mathematica , to the Royal Society . This paper

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880-666: The mortality rate over a person's age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person's age. This is similar to the types of formulas used by insurance companies today. Some results on the Poisson distribution were first introduced by de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus in Philosophical Transactions of

920-537: The poet Rev. Samuel Wesley (according to Dunton, it would eventually grow to 12 members; there is no evidence of such additional members, though). Its name, and all its subsequent related "Athenian" names, derived from a biblical reference to St. Paul in Athens: "For all the Athenians and strangers which were there spent their time in nothing else, but either to tell, or to hear some new thing." ( Acts 17:21 KJV ) The society

960-641: The proof quite straightforward. This formula is important because it relates complex numbers and trigonometry . Additionally, this formula allows the derivation of useful expressions for cos( nx ) and sin( nx ) in terms of cos( x ) and sin( x ). De Moivre had been studying probability, and his investigations required him to calculate binomial coefficients, which in turn required him to calculate factorials. In 1730 de Moivre published his book Miscellanea Analytica de Seriebus et Quadraturis [Analytic Miscellany of Series and Integrals], which included tables of log ( n !). For large values of n , de Moivre approximated

1000-466: The questions) were written anonymously by "a Member of the Athenian Society" (one of the four friends). The new journal received a tremendous response and generated several imitations. On 14 February 1692 a young Jonathan Swift sent them a letter of appreciation along with an "Ode to the Athenian Society", his first published work. Concurrently to the periodical, issues of the journal were bound in calf leather and sold as The Athenian Gazette , collecting

1040-466: The time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts. To make a living, de Moivre became a private tutor of mathematics , visiting his pupils or teaching in the coffee houses of London. De Moivre continued his studies of mathematics after visiting the Earl of Devonshire and seeing Newton's recent book, Principia Mathematica . Looking through

1080-518: The time of its occurrence. De Moivre pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. He also produced the second textbook on probability theory, The Doctrine of Chances: a method of calculating the probabilities of events in play . (The first book about games of chance, Liber de ludo aleae ( On Casting

1120-403: The time. At least a part of the reason was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of

1160-504: The time. When he was eleven, his parents sent him to the Protestant Academy at Sedan , where he spent four years studying Greek under Jacques du Rondel. The Protestant Academy of Sedan had been founded in 1579 at the initiative of Françoise de Bourbon, the widow of Henri-Robert de la Marck. In 1682 the Protestant Academy at Sedan was suppressed, and de Moivre enrolled to study logic at Saumur for two years. Although mathematics

1200-763: The writing the Athenian Gazette, or Mercury, dated April 10, 1691. Originally executed by the three persons. Sault was reputed to be a gentleman of courage and passion, and on one occasion about to draw his sword on Tom Brown, one of the editors of a rival publication, the Lacedemonian Mercury . In February 1695 the programme of a projected scheme of a new royal academy stated that the mathematics would be taught in Latin, French, or English by Sault and Abraham De Moivre . About 1700 Sault moved to Cambridge, where he died in May 1702 in poverty, supported by charitable scholars. He

1240-526: Was a French mathematician known for de Moivre's formula , a formula that links complex numbers and trigonometry , and for his work on the normal distribution and probability theory . He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau . He was a friend of Isaac Newton , Edmond Halley , and James Stirling . Among his fellow Huguenot exiles in England, he

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1280-399: Was a colleague of the editor and translator Pierre des Maizeaux . De Moivre wrote a book on probability theory , The Doctrine of Chances , said to have been prized by gamblers. De Moivre first discovered Binet's formula , the closed-form expression for Fibonacci numbers linking the n th power of the golden ratio φ to the n th Fibonacci number. He also was the first to postulate

1320-509: Was buried in the church of St Andrew the Great on 17 May 1702. Dunton published in 1693 The Second Spira, being a fearful example of an Atheist who had apostatized from the Christian religion, and died in despair at Westminster, Dec. 8, 1692. By J. S. Dunton obtained the manuscript from Sault, who professed to know the author. The original Spira was Francesco Spiera . The preface to Dunton's volume

1360-458: Was established in order to write and publish the Athenian Gazette , become The Athenian Mercury with its second issue due to a legal threat, a journal sold one penny twice weekly, then four times a week. It professed to answer in print all questions received from anonymous readers on "divinity, history, philosophy, mathematics, love, poetry", and things in general; the answers (and sometimes

1400-567: Was not part of his course work, de Moivre read several works on mathematics on his own, including Éléments des mathématiques by the French Oratorian priest and mathematician Jean Prestet and a short treatise on games of chance, De Ratiociniis in Ludo Aleae , by Christiaan Huygens the Dutch physicist, mathematician, astronomer and inventor. In 1684, de Moivre moved to Paris to study physics, and for

1440-599: Was only depicting his own mental and moral experiences. He printed in his memoirs a letter from Sault's wife, in which she accused her husband of loose living, as some proof of Sault's extramarital sex life, arguing this as a cause of his mental troubles. William Leybourne 's Pleasure with Profit (London, 1694) contains, as an appendix, Sault's A Treatise of Algebra (52pp), in which Sault's pays special attention to Joseph Raphson 's recent (1690) treatment of Converging Series for all manner of adfected equations , but prefaced by Sault's own notion of punctation of series (there

1480-496: Was published in the Philosophical Transactions that same year. Shortly after publishing this paper, de Moivre also generalised Newton's noteworthy binomial theorem into the multinomial theorem . The Royal Society became apprised of this method in 1697, and it elected de Moivre a Fellow on 30 November 1697. After de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. In 1705, de Moivre discovered, intuitively, that "the centripetal force of any planet

1520-548: Was sent to Prieuré Saint-Martin-des-Champs, a school that the authorities sent Protestant children to for indoctrination into Catholicism. It is unclear when de Moivre left the Prieure de Saint-Martin and moved to England, since the records of the Prieure de Saint-Martin indicate that he left the school in 1688, but de Moivre and his brother presented themselves as Huguenots admitted to the Savoy Church in London on 28 August 1687. By

1560-532: Was signed by Sault's initials, and the genuineness of the information supplied was attested by many witnesses. With it is bound up A Conference betwixt a modern Atheist and his friend. By the methodizer of the Second Spira, London, John Dunton, 1693. Thirty thousand copies of the Second Spira sold in six weeks. It is one of the seven books which Dunton repented printing, because he came to the conclusion that Sault

1600-399: Was the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the calculation of probable error . In addition, he applied these theories to gambling problems and actuarial tables . An expression commonly found in probability is n ! but before

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