1994, ISBN: b4d2f28edf1b3869c1f5997555e1bca9

Edition reliée

Bologne: Stamperia di S. Tommaso d'Aquino, 1799. First edition. Soft cover. UNSOLVABILITY OF THE QUINTIC. First edition, very rare, of the first statement and proof that the general equa… Plus…

Bologne: Stamperia di S. Tommaso d'Aquino, 1799. First edition. Soft cover. UNSOLVABILITY OF THE QUINTIC. First edition, very rare, of the first statement and proof that the general equation of degree five or more cannot be solved algebraically. This is a remarkable author's presentation copy, uncut in the publisher's printed wrappers. "One of the most fascinating results in the realm of algebra - indeed in all of mathematics - is the theorem that the general polynomial of degree 5 is not solvable by radicals. Its discovery at the very end of the 18th century went counter to the belief and expectations of mathematical scholars; it came as a great surprise and was naturally met with scepticism ... this revolutionary idea was not accepted without a great deal of resistance" (Ayoub, p. 253). An exception was the great French mathematician Augustin-Louis Cauchy, who wrote to Ruffini in 1821: "Your memoir on the general resolution of equations is a work that has always seemed to me worthy of the attention of mathematicians and one that, in my opinion, demonstrates completely the impossibility of solving algebraically equations of higher than the fourth degree." In Ruffini's arguments one can now see the beginnings of modern group theory. "Ruffini's methods began with the relations that Lagrange had discovered between solutions of third- and fourth-degree equations and permutations of three and four elements, and Ruffini's development of this starting point contributed effectively to the transition from classical to abstract algebra and to the theory of permutation groups. This theory is distinguished from classical algebra by its greater generality: it operates not with numbers or figures, as in traditional mathematics, but with indefinite entities, on which logical operations are performed" (DSB). "Ruffini is the first to introduce the notion of the order of an element, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive. He proved some remarkable theorems [in group theory]" (MacTutor). Ruffini's proof did, in fact, have a gap which was filled in 1824 by Niels Henrik Abel (although Abel's proof also had a gap), and the insolvability of quintic equations is now known as the Ruffini-Abel theorem. This is a very rare book on the market in any form (ABPC/RBH list only a single copy) and we have never before seen nor heard of a copy in publisher's wrappers, or a presentation copy. Provenance: Author's presentation copy ("dono dell'autore" written on the front fly-leaf of both volumes). The method of solving quadratic equations was known to the Baghdad mathematician and astronomer Al-Khwarizmi (c. 780-850), and the formula involving square roots is now taught to every student in high school. Similar formulas for solving cubic and quartic equations were not found until the 16th century, by Scipione del Ferro (1465-1525), Lodovico Ferrari (1522-60), and Niccolo Tartaglia (1506-59), and were first published by Girolamo Cardano (1501-76) in his Ars magna (1545). These formulas expressed the solutions in terms of 'radicals,' i.e., expressions involving rational functions (ratios of polynomials) of the coefficients of the equation and their square-, cube-, and higher roots. The search for a similar formula for quintic equations proved fruitless. "For two centuries thereafter, the resolution of the enigma was regarded as one of the most important problems of algebra and occupied the attention of the leading mathematicians of this epoch" (Ayoub, p. 257). The most important work on the problem preceding Ruffini's was Lagrange's remarkable memoir 'Réflexions sur la résolution algébrique des equations, published in the Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin in 1770-71. Katz & Parshall (p. 298) remark that "his introduction of the notion of permutations proved crucial to the ultimate proof that there was no algebraic solution of a fifth-degree polynomial equation," but Lagrange himself still believed that a solution of the quintic would be found. "He concludes with this statement: 'There, if I am not mistaken, are the true principles of the resolution of equations, and the most appropriate analysis which leads to solutions; all reduces, as we see, to a type of calculus of combinations by which we find results which we might expect a priori. It would be pertinent to make application to equations of the fifth and higher degrees whose solution is, up to the present, unknown: but this application requires a large number of combinations whose success is, however, very doubtful. We hope to return to this question at another time and we are content here in having given the fundamentals of a theory which appears to us new and general.' So in spite of past failures in solving the quintic, Lagrange still harbors the hope that a careful analysis of his method will achieve the goal. "Did no one suspect that the solution of the quintic was impossible? Apparently not until 1799 when Ruffini published his book on the theory of equations: 'General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than 4 is impossible.' Parenthetically, we note that in the same year the young Carl Friedrich Gauss (1777-1855) wrote in his dissertation (in which he proved the fundamental theorem of algebra) as follows: 'After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that the resolution is impossible and contradictory ... Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place. Here it is enough to say that the general solution of equations understood in this sense [i.e., by radicals] is far from certain and this assumption [i.e., that any equation is solvable by radicals] has no validity at the present time.' Gauss published nothing more on the subject. "Ruffini begins the introduction of his book as follows: 'The algebraic solution of general equations of degree greater than 4 is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err); to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof'" (Ayoub, pp. 262-3). Ruffini's proof has been presented in modern dress by Ayoub and cannot be described in detail here. It is now viewed as being correct but with a gap: Ruffini assumed that the radicals that arise in the course of solving the equation are rational functions of the roots, and this assumption requires proof which Ruffini did not provide. This gap was filled 25 years later by Abel (1802-29). Abel's proof also had a gap, which was filled in 1849 by Leopold Königsberger. In the course of proving his remarkable theorem, Ruffini laid the foundations of group theory, which is now of central importance both in mathematics and in physics (symmetry groups). "Ruffini went beyond the mere recognition that there exists a connection between the solvability of algebraic equations and permutations. In his work the theory of permutations no longer plays the role of a mere computing device but is rather a structural component of solvability theory. "He begins with Lagrange's program of systematic investigation of permutations from the point of view of their effect on algebraic functions of n variables (a permutation can fix or change such a function) ... he prefaces his treatment of fifth- and sixth-degree equations in chapter 13 ('Riflessioni intorno all soluzione generale delle equazioni') with a classification of permutations. Apart from a different terminology, the modern permutation group concept appears in this chapter with full clarity. Not only is Ruffini - like Lagrange - concerned with permutations that leave a rational function of the roots invariant; he deals also with the totality of such permutations and their properties. He calls such a set of permutations 'permutazione.' Thus Ruffini's 'permutazione' coincides with what Cauchy later called a 'system of conjugate substitutions' and Galois called a (permutation) 'group.' "It is remarkable that Ruffini used consistently the fact that his 'permutazione' is closed, both in connection with the composition of the permutations reproducing the function, and even for the purpose of classifying groups in connection with the question of generators of the 'permutazione' ... "Ruffini calls the number p of permutations that leave invariant a given function of n roots of an equation the 'degree of equality' (grado di uguaglianza). Thus this concept coincides with that of the 'order' of a (permutation) group ... Ruffini goes on to determine the value of p for all groups that occur in connection with five quantities, the roots of a quintic ... In terms of content, this investigation comes down to an (almost) complete determination of all subgroups of the symmetric group S5. In this way Ruffini obtains the main result, formulated in article 275, to the effect that p can never be 15, 30, or 40, that is, that there are no (rational) functions of five quantities that take on 8, 4, or 3 different values when these quantities are permuted in all possible ways. On the basis of this correctly-proved group-theoretic result, Ruffini gives a proof - with some gaps - of the unsolvability of the general quintic in radicals" (Wussing, pp. 82-3). "What reception was accorded this remarkable discovery? In about 1801 Ruffini sent a copy of his 'Teoria' to Lagrange but received no response. Shortly thereafter, he wrote: 'Because of the uncertainty that you may have received my book, I send you another copy. If I have erred in any proof, of if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely.' Lagrange did not reply. "Again in 1802 he wrote to Lagrange: 'No one has more right ... to receive the book which I take the liberty of sending to you ... In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher then 4.' "Pietro Paoli, professor of analysis at Pisa, wrote in September 1799 with a certain chauvinism: 'I read with much pleasure your book ... and recommended greatly the most important theorem which excludes the possibility of solving equations of degree greater than 4. I rejoice exceedingly with you and with our Italy, which has seen a theory born and perfected and to which other nations have contributed little.' [It must be remembered that Lagrange was born in Turin and was considered Italian by Italians, though he had come of a French family.] "In 1803, Ruffini published a paper entitled 'On the solvability of equations of degree greater than 4.' This was written at the urging of his friend Pietro Abbati (1768-1842). Ruffini wrote: 'In the present memoir, I shall try to prove the same proposition [insolvability of the quintic] with, I hope, less abstruse reasoning and with complete rigor.' "To this proof Gian-Francesco Malfatti (1731-1807) raised certain objections which suggest that he did not understand the proof clearly ... In 1806 Ruffini published yet another proof with no visible reaction, and in 1813 he published a paper 'Reflections on the solution of general algebraic equations.' In the introduction, he expresses his disappointment, if not pique, at the reception accorded his work" (Ayoub, p. 269). "A further impulse to seek appraisal of his work came from a publication by Jean Baptiste Joseph Delambre (1749-1822). This was a report to 'His Majesty the Emperor and King' called 'Historical report on the progress of the mathematical sciences since 1789.' In it Delambre says 'Ruffini proposes to prove that it is impossible ...' Ruffini replied: 'I not only proposed to prove but in reality did prove ... and I had in mind presenting the proof to the institute to have it examined and to have the institute pronounce on its validity.' Ruffini was informed that Lagrange, Legendre and Lacroix had been appointed to a committee to examine his memoir. He was told, however, that 'if a thing is not of importance, no notice is taken of it and Lagrange himself 'with his coolness' found little in it worthy of attention.' "Ruffini wrote again to Delambre asking about the status of his paper, and noted that the Italian minister had spoken to Lagrange who told the minister that because of the character and manner of expression, he had understood nothing and no longer wished to undertake the reading of his memoir. Ruffini asked Delambre to speak to Lagrange and if the latter did not want to read it, Delambre was to appoint a new board of examiners. "As it turns out, Lagrange, who was old at the time, reported to Gaultier de Claubry that he had read Ruffini's memoir, had found it good but, since it treated of a difficult matter and since Ruffini had not given sufficient proof of certain things which he claimed, Lagrange did not want to create excitement among the mathematicians of the institute and, therefore, did not want to publish his approval. "Ruffini also sent his memoir to the Royal Society in London. The reply said that the Society itself does not give official approval of any work but reported that those who had read it were quite satisfied that he had proved what he claimed to prove. "His greatest advocate, however, was no less a person than A. L. Cauchy (1789-1857). Cauchy found in Ruffini's work a veritable gold mine. In the years 1813-1815, Cauchy wrote a lengthy paper on the theory of permutation groups generalizing some of Ruffini's results. This paper was assessed by a committee of the French Academy of Sciences and this committee mentions Ruffini by name. "Cauchy acknowledged his indebtedness to Ruffini in a letter dated 1821 about 6 months before Ruffini's death: '... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgment, proves completely the insolvability of the general equation of degree > 4. If I have not discussed it in my course, it is because this course was directed at the students of the École Royale Polytechnique and I could not deviate too much from the syllabus ... In another memoir which I read last year to the Academy of Sciences, I cited your work and reminded the audience that your proofs establish the impossibility of solving equations algebraically ... I add moreover, that your work on the insolvability is precisely the title of a lecture which I gave to several members of the academy ...' "In view of the endorsement of the Royal Society (admittedl, Stamperia di S. Tommaso d'Aquino, 1799, 0, London: W[illiam]. Pearson for the Author, 1718. First edition. Hardcover. THE FOUNDING WORK OF THE FIELD OF PROBABILTY AND STATISTICS. First edition of this classic on the theory of probability, the first original work on the subject in English. "De Moivre's book on chances is considered the foundation for the field of probability and statistics" (Tomash). "De Moivre's masterpiece is The Doctrine of Chances" (DSB). "His work on the theory of probability surpasses anything done by any other mathematician except P. S. Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli's theorem by the aid of Stirling's theorem" (Cajori, A History of Mathematics, p. 230). "He was among the intimate friends of Newton, to whom this book is dedicated. It is the second book devoted entirely to the theory of probability and a classic on the subject" (Babson 181). De Moivre's interest in probability was raised by Pierre-Rémond de Montmort's Essay d'analyse sur les jeux de hazard (1708), the first separately-published work on probability. "The [Doctrine] is in part the result of a competition between De Moivre on the one hand and Montmort together with Nikolaus Bernoulli on the other. De Moivre claimed that his representation of the solutions of the then current problems tended to be more general than those of Montmort, which Montmort resented very much. This situation led to some arguments between the two men, which finally were resolved by Montmort's premature death in 1719 ... De Moivre had developed algebraic and analytical tools for the theory of probability like a 'new algebra' for the solution of the problem of coincidences which somewhat foreshadowed Boolean algebra, and also the method of generating functions or the theory of recurrent series for the solution of difference equations. Differently from Montmort, De Moivre offered in [Doctrine] an introduction that contains the main concepts like probability, conditional probability, expectation, dependent and independent events, the multiplication rule, and the binomial distribution" (Schneider, p. 106). Provenance: Charles Meynell (early engraved bookplate). The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the 'Problem of points'. Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. "Huygens heard about Pascal's and Fermat's ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae ... essentially followed Pascal's method of expectation. ... At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are [Jakob] Bernoulli's Ars conjectandi (1713), Montmort's Essay d'analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivre's Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat's combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal's method of expectations." (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296). "De Moivre's earliest book on probability, the first edition of the Doctrine of Chances, was an expansion of a long (fifty-two pages) memoir he had published in Latin in the Philosophical Transactions of the Royal Society in 1711 under the title 'De mensura sortis' (literally, 'On the measurement of lots'). De Moivre tells us that in 1711 he had read only Huygens' 1657 tract De Ratiociniis in Ludo Aleae and an anonymous English 1692 tract based on Huygens' work (now known to have been written by John Arbuthnot). By 1718 he had encountered both Montmort's Essay d'analyse sur les jeux de hazard (2nd ed., 1713) and Bernoulli's Ars Conjectandi (1713), although the latter had no pronounced effect on De Moivre at that early date" (Stigler, p. 71). The Doctrine consists of an introduction with definitions and elementary theorems, followed by a series of numbered problems. "De Moivre begins with the classical measure of probability, 'a fraction whereof the numerator be the number of chances whereby an event may happen, and the denominator the number of all the chances whereby it may either happen or fail'. He gives the summation rule for probabilities of disjunct events explicitly only for the case of the happening and the not happening of an event. Expectation is still on the level of Huygens defined as the product of an expected sum of money and the probability of obtaining it, the expectation of several sums is determined by the sum of the expectations of the singular sums. He defines independent and dependent events and gives the multiplication rule for both. But whereas today the criterion for independence of two events is the validity of the multiplication rule in the [Doctrine], the multiplication rule follows from the independence of the events, which seems to be a self-evident concept for De Moivre ... "With these tools 'those who are acquainted with Arithmetical Operations' (as De Moivre remarked in the preface) could tackle many problems, in part already well known but which he gradually generalized. Because the majority of the solved problems depends on rules 'being entirely owing to Algebra' and to combinatorics, De Moivre tried to convince those readers who had not studied algebra yet to 'take the small Pains of being acquainted with the bare Notation of Algebra, which might be done in the hundredth part of the Time that is spent in learning to write Short-hand'. Remarks of this kind are typical of the private teacher of mathematics De Moivre, who was accustomed to ask his clients before he began with his instructions about their mathematical knowledge" (Schneider, pp. 107-9). Following the introduction are 53 numbered problems: I-XIV are various problems solvable with the rules contained in the introduction including problems dealing with the games of Bassette (XIII) and Pharaon (XIV); XV-XXXII are problems solvable by combinatorial methods, including some dealing with lotteries (XXI and XXII), and of Pharaon (XXIII); XXXIII-XLVI are concerned with the problem of the duration of play, or the ruin problem; and XLVII-LIII are further problems solvable by combinatorial methods, including Hazard (XLVII, LIII), Whisk (XLVIII), Raffling (XLIX) and Piquet (LI, LII). "Some problems, as already stated by Jakob Bernoulli (1654-1705) in his Ars conjectandi, can be solved more easily by the use of infinite series. As an illustration de Moivre offers the problem to determine the amounts each of two players A and B has to stake under the condition that the player who throws the first time an Ace with an ordinary die wins the stake and that A has the first throw. He considers it as reasonable that A should pay 1/6 of the total stake in order to have the first throw, B should pay 1/6 of the rest which is 1/6.5/6 for having the second throw, A should pay 1/6 of the remainder for having the third throw, etc. The part that A has to stake altogether is the sum of a geometrical series with 1/6 as the first term and the quotient 25/36, which is 6/11 of the total stake. Accordingly B's share is 5/11 of the total stake. De Moivre claims that in most cases where the solution affords the application of infinite series the series are geometrical [in which each term is a fixed multiple of the preceding term]. The other kind of infinite series which relate to the problem of the duration of play are recurrent series the terms of which can be connected with the terms of geometrical series. Other problems depend on the summation of the terms of arithmetical series of higher orders and a 'new sort of algebra'" (Schneider, pp. 109-110). Recurrent series - those in which each term of the series is related to a fixed number of preceding terms by a fixed (linear) relation - are needed in the solution of the problem of the duration of play. "It resulted from a generalization of the last problem that Huygens had posed to his readers at the end of his treatise De ratiociniis in ludo aleae (1656). The first to deal with the problem in the new form seems to be Montmort, and after him Nikolaus Bernoulli. De Moivre concerned himself with it at about the same time. His formulation of the problem in the [Doctrine] of 1718 is nearly the same as he used in the third edition (p. 191): 'Two gamesters A and B whose proportion of skill is as a to b, each having a certain number of pieces, play together on condition that as often as A wins a game, B shall give him one piece; and that as often as B wins a game, A shall give him one piece; and that they cease not to play till such time as either one or the other has got all the pieces of his adversary: now let us suppose two spectators R and S concerning themselves about the ending of the play, the first of them laying that the play will be ended in a certain number of games which he assigns, the other laying to the contrary. To find the probability that S has of winning his wager'" (Schneider, p. 112). De Moivre gave a complete solution of the problem of duration of play in Doctrine, but he did not indicate how he had obtained the results, and this became a challenge to the next generation of probabilists, notably Laplace (see Hald, pp. 361 et seq). One of the most important devices introduced by De Moivre is that of a 'generating function', later developed extensively by Laplace. De Moivre introduces generating functions in his solution of Problem III. "It asks after the number of chances to throw a given number p + 1 of points with n dice, each of them of the same number f of faces. Here the word 'dice' or 'die' is used in the more general sense of, for example, a roulette wheel with f sectors" (Schneider, p. 110). De Moivre introduced a series whose coefficients are the chances sought, and was able to determine the sum of the series, from which the chances were easily found. Indeed, he formed the series f(r) = 1 + r + r2 + ... + rf-1 = (1 - rf)/(1 - r) and noted that the number of chances required is equal to the coefficient of the term with exponent p + 1 - n in the expansion of f(r)n = (1 - rf)n.(1 - r)-n, an expansion which is easily obtained using the binomial theorem. "An early reaction to the book which surely accounts for the high estimation it was held at least in England is its exploitation by the Englishman Thomas Simpson, who in his Treatise on the nature and laws of chance (1740) just repeated the results achieved in the [Doctrine]. The fact that De Moivre had specialized in the theory of probability, for which he had prepared appropriate tools and to which he had contributed the solutions of the most interesting problems posed to him by his competitors and by his clients for some decades, made [Doctrine], especially the last edition of 1756, the most complete representation of the new field in the second half of the 18th century. "This was felt by the leading mathematicians of the next generation. In particular, J.L. Lagrange and Laplace had planned a French translation of the book which however was never realized. Their interest goes back to De Moivre's solution of the problem of the duration of play by means of what he called 'recurrent series' and what amounts to the solution of a homogeneous linear difference equation with constant coefficients. In fact, the most effective analytical tool developed by Laplace for the calculus of probabilities, the theory of generating functions, is a consequence of his concern with recurrent series. Indeed, the most important results of the book reappear in Laplace's probability theory in a new mathematical form and in a new philosophical context. This, more than anything else, confirms de Moivre's status as a pioneer in the field and as a predecessor of Laplace" (Schneider, p. 119). At the top of page 1 of the text is an engraving which De Moivre himself had designed. It shows Minerva, on the left of the picture, pointing to a piece of paper with a circle on it; this alluded to his solution to the problem of the duration of play, the details of which he had withheld in the book. The piece of paper is held by Fortuna, the goddess of fortune. She is identified by the wheel of fortune behind her and the cornucopia at her feet. With Minerva standing at a dominant position over Fortuna, the interpretation is that De Moivre's mathematical results dominate fickle fortune or fate. The paper under the cornucopia has some illegible writing on it. It may represent some previous work that has borne fruit, perhaps referring to Huygens' original results in De ratiociniis. On the right of the picture four men stand around a table with dice and a dice box on it. The clean-shaven man is De Moivre; he is instructing the other men on the theory of probability. A similar engraving is found at the beginning of Montmort's Essay, but there it is the God Mercury standing at the table watching a man and a woman play a game of dice. Thus De Moivre is taking a swipe at Montmort, expressing through the engraving that he does not have the effrontery to speak directly to the gods and instruct them. The middle part of the engraving has two additional swipes at Montmort. Two naked boys are sitting with a pair of dice at their feet. A short distance away are some discarded cards and further yet is a chessboard of size 4 x 6 squares rather than the standard 8 x 8 shown in Montmort's engraving. One of the boys is reading a book, perhaps Doctrine of Chances, to the other explaining De Moivre's newly discovered results in probability. The discarded chessboard, being incomplete, is an indication that the work in Montmort's Essay is also incomplete. Abraham Moivre stemmed from a Protestant family. His father was a surgeon from Vitry-le-François in the Champagne. From the age of five to eleven he was educated by the Catholic Péres de la doctrine Chrètienne. Then he moved to the Protestant Academy at Sedan were he mainly studied Greek. After the latter was forced to close in 1681 for its profession of, W[illiam]. Pearson for the Author, 1718, 0<

1994, ISBN: b4d2f28edf1b3869c1f5997555e1bca9

London: W[illiam]. Pearson for the Author, 1718. First edition. Hardcover. THE FOUNDING WORK OF THE FIELD OF PROBABILTY AND STATISTICS. First edition of this classic on the theory of pro… Plus…

London: W[illiam]. Pearson for the Author, 1718. First edition. Hardcover. THE FOUNDING WORK OF THE FIELD OF PROBABILTY AND STATISTICS. First edition of this classic on the theory of probability, the first original work on the subject in English. "De Moivre's book on chances is considered the foundation for the field of probability and statistics" (Tomash). "De Moivre's masterpiece is The Doctrine of Chances" (DSB). "His work on the theory of probability surpasses anything done by any other mathematician except P. S. Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli's theorem by the aid of Stirling's theorem" (Cajori, A History of Mathematics, p. 230). "He was among the intimate friends of Newton, to whom this book is dedicated. It is the second book devoted entirely to the theory of probability and a classic on the subject" (Babson 181). De Moivre's interest in probability was raised by Pierre-Rémond de Montmort's Essay d'analyse sur les jeux de hazard (1708), the first separately-published work on probability. "The [Doctrine] is in part the result of a competition between De Moivre on the one hand and Montmort together with Nikolaus Bernoulli on the other. De Moivre claimed that his representation of the solutions of the then current problems tended to be more general than those of Montmort, which Montmort resented very much. This situation led to some arguments between the two men, which finally were resolved by Montmort's premature death in 1719 ... De Moivre had developed algebraic and analytical tools for the theory of probability like a 'new algebra' for the solution of the problem of coincidences which somewhat foreshadowed Boolean algebra, and also the method of generating functions or the theory of recurrent series for the solution of difference equations. Differently from Montmort, De Moivre offered in [Doctrine] an introduction that contains the main concepts like probability, conditional probability, expectation, dependent and independent events, the multiplication rule, and the binomial distribution" (Schneider, p. 106). Provenance: Charles Meynell (early engraved bookplate). The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the 'Problem of points'. Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. "Huygens heard about Pascal's and Fermat's ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae ... essentially followed Pascal's method of expectation. ... At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are [Jakob] Bernoulli's Ars conjectandi (1713), Montmort's Essay d'analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivre's Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat's combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal's method of expectations." (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296). "De Moivre's earliest book on probability, the first edition of the Doctrine of Chances, was an expansion of a long (fifty-two pages) memoir he had published in Latin in the Philosophical Transactions of the Royal Society in 1711 under the title 'De mensura sortis' (literally, 'On the measurement of lots'). De Moivre tells us that in 1711 he had read only Huygens' 1657 tract De Ratiociniis in Ludo Aleae and an anonymous English 1692 tract based on Huygens' work (now known to have been written by John Arbuthnot). By 1718 he had encountered both Montmort's Essay d'analyse sur les jeux de hazard (2nd ed., 1713) and Bernoulli's Ars Conjectandi (1713), although the latter had no pronounced effect on De Moivre at that early date" (Stigler, p. 71). The Doctrine consists of an introduction with definitions and elementary theorems, followed by a series of numbered problems. "De Moivre begins with the classical measure of probability, 'a fraction whereof the numerator be the number of chances whereby an event may happen, and the denominator the number of all the chances whereby it may either happen or fail'. He gives the summation rule for probabilities of disjunct events explicitly only for the case of the happening and the not happening of an event. Expectation is still on the level of Huygens defined as the product of an expected sum of money and the probability of obtaining it, the expectation of several sums is determined by the sum of the expectations of the singular sums. He defines independent and dependent events and gives the multiplication rule for both. But whereas today the criterion for independence of two events is the validity of the multiplication rule in the [Doctrine], the multiplication rule follows from the independence of the events, which seems to be a self-evident concept for De Moivre ... "With these tools 'those who are acquainted with Arithmetical Operations' (as De Moivre remarked in the preface) could tackle many problems, in part already well known but which he gradually generalized. Because the majority of the solved problems depends on rules 'being entirely owing to Algebra' and to combinatorics, De Moivre tried to convince those readers who had not studied algebra yet to 'take the small Pains of being acquainted with the bare Notation of Algebra, which might be done in the hundredth part of the Time that is spent in learning to write Short-hand'. Remarks of this kind are typical of the private teacher of mathematics De Moivre, who was accustomed to ask his clients before he began with his instructions about their mathematical knowledge" (Schneider, pp. 107-9). Following the introduction are 53 numbered problems: I-XIV are various problems solvable with the rules contained in the introduction including problems dealing with the games of Bassette (XIII) and Pharaon (XIV); XV-XXXII are problems solvable by combinatorial methods, including some dealing with lotteries (XXI and XXII), and of Pharaon (XXIII); XXXIII-XLVI are concerned with the problem of the duration of play, or the ruin problem; and XLVII-LIII are further problems solvable by combinatorial methods, including Hazard (XLVII, LIII), Whisk (XLVIII), Raffling (XLIX) and Piquet (LI, LII). "Some problems, as already stated by Jakob Bernoulli (1654-1705) in his Ars conjectandi, can be solved more easily by the use of infinite series. As an illustration de Moivre offers the problem to determine the amounts each of two players A and B has to stake under the condition that the player who throws the first time an Ace with an ordinary die wins the stake and that A has the first throw. He considers it as reasonable that A should pay 1/6 of the total stake in order to have the first throw, B should pay 1/6 of the rest which is 1/6.5/6 for having the second throw, A should pay 1/6 of the remainder for having the third throw, etc. The part that A has to stake altogether is the sum of a geometrical series with 1/6 as the first term and the quotient 25/36, which is 6/11 of the total stake. Accordingly B's share is 5/11 of the total stake. De Moivre claims that in most cases where the solution affords the application of infinite series the series are geometrical [in which each term is a fixed multiple of the preceding term]. The other kind of infinite series which relate to the problem of the duration of play are recurrent series the terms of which can be connected with the terms of geometrical series. Other problems depend on the summation of the terms of arithmetical series of higher orders and a 'new sort of algebra'" (Schneider, pp. 109-110). Recurrent series - those in which each term of the series is related to a fixed number of preceding terms by a fixed (linear) relation - are needed in the solution of the problem of the duration of play. "It resulted from a generalization of the last problem that Huygens had posed to his readers at the end of his treatise De ratiociniis in ludo aleae (1656). The first to deal with the problem in the new form seems to be Montmort, and after him Nikolaus Bernoulli. De Moivre concerned himself with it at about the same time. His formulation of the problem in the [Doctrine] of 1718 is nearly the same as he used in the third edition (p. 191): 'Two gamesters A and B whose proportion of skill is as a to b, each having a certain number of pieces, play together on condition that as often as A wins a game, B shall give him one piece; and that as often as B wins a game, A shall give him one piece; and that they cease not to play till such time as either one or the other has got all the pieces of his adversary: now let us suppose two spectators R and S concerning themselves about the ending of the play, the first of them laying that the play will be ended in a certain number of games which he assigns, the other laying to the contrary. To find the probability that S has of winning his wager'" (Schneider, p. 112). De Moivre gave a complete solution of the problem of duration of play in Doctrine, but he did not indicate how he had obtained the results, and this became a challenge to the next generation of probabilists, notably Laplace (see Hald, pp. 361 et seq). One of the most important devices introduced by De Moivre is that of a 'generating function', later developed extensively by Laplace. De Moivre introduces generating functions in his solution of Problem III. "It asks after the number of chances to throw a given number p + 1 of points with n dice, each of them of the same number f of faces. Here the word 'dice' or 'die' is used in the more general sense of, for example, a roulette wheel with f sectors" (Schneider, p. 110). De Moivre introduced a series whose coefficients are the chances sought, and was able to determine the sum of the series, from which the chances were easily found. Indeed, he formed the series f(r) = 1 + r + r2 + ... + rf-1 = (1 - rf)/(1 - r) and noted that the number of chances required is equal to the coefficient of the term with exponent p + 1 - n in the expansion of f(r)n = (1 - rf)n.(1 - r)-n, an expansion which is easily obtained using the binomial theorem. "An early reaction to the book which surely accounts for the high estimation it was held at least in England is its exploitation by the Englishman Thomas Simpson, who in his Treatise on the nature and laws of chance (1740) just repeated the results achieved in the [Doctrine]. The fact that De Moivre had specialized in the theory of probability, for which he had prepared appropriate tools and to which he had contributed the solutions of the most interesting problems posed to him by his competitors and by his clients for some decades, made [Doctrine], especially the last edition of 1756, the most complete representation of the new field in the second half of the 18th century. "This was felt by the leading mathematicians of the next generation. In particular, J.L. Lagrange and Laplace had planned a French translation of the book which however was never realized. Their interest goes back to De Moivre's solution of the problem of the duration of play by means of what he called 'recurrent series' and what amounts to the solution of a homogeneous linear difference equation with constant coefficients. In fact, the most effective analytical tool developed by Laplace for the calculus of probabilities, the theory of generating functions, is a consequence of his concern with recurrent series. Indeed, the most important results of the book reappear in Laplace's probability theory in a new mathematical form and in a new philosophical context. This, more than anything else, confirms de Moivre's status as a pioneer in the field and as a predecessor of Laplace" (Schneider, p. 119). At the top of page 1 of the text is an engraving which De Moivre himself had designed. It shows Minerva, on the left of the picture, pointing to a piece of paper with a circle on it; this alluded to his solution to the problem of the duration of play, the details of which he had withheld in the book. The piece of paper is held by Fortuna, the goddess of fortune. She is identified by the wheel of fortune behind her and the cornucopia at her feet. With Minerva standing at a dominant position over Fortuna, the interpretation is that De Moivre's mathematical results dominate fickle fortune or fate. The paper under the cornucopia has some illegible writing on it. It may represent some previous work that has borne fruit, perhaps referring to Huygens' original results in De ratiociniis. On the right of the picture four men stand around a table with dice and a dice box on it. The clean-shaven man is De Moivre; he is instructing the other men on the theory of probability. A similar engraving is found at the beginning of Montmort's Essay, but there it is the God Mercury standing at the table watching a man and a woman play a game of dice. Thus De Moivre is taking a swipe at Montmort, expressing through the engraving that he does not have the effrontery to speak directly to the gods and instruct them. The middle part of the engraving has two additional swipes at Montmort. Two naked boys are sitting with a pair of dice at their feet. A short distance away are some discarded cards and further yet is a chessboard of size 4 x 6 squares rather than the standard 8 x 8 shown in Montmort's engraving. One of the boys is reading a book, perhaps Doctrine of Chances, to the other explaining De Moivre's newly discovered results in probability. The discarded chessboard, being incomplete, is an indication that the work in Montmort's Essay is also incomplete. Abraham Moivre stemmed from a Protestant family. His father was a surgeon from Vitry-le-François in the Champagne. From the age of five to eleven he was educated by the Catholic Péres de la doctrine Chrètienne. Then he moved to the Protestant Academy at Sedan were he mainly studied Greek. After the latter was forced to close in 1681 for its profession of, W[illiam]. Pearson for the Author, 1718, 0<

1718, ISBN: b4d2f28edf1b3869c1f5997555e1bca9

Edition reliée

[PU: W[illiam]. Pearson for the Author, London], AUG2022, THE FOUNDING WORK OF THE FIELD OF PROBABILTY AND STATISTICS. First edition of this classic on the theory of probability, the firs… Plus…

[PU: W[illiam]. Pearson for the Author, London], AUG2022, THE FOUNDING WORK OF THE FIELD OF PROBABILTY AND STATISTICS. First edition of this classic on the theory of probability, the first original work on the subject in English. "De Moivre's book on chances is considered the foundation for the field of probability and statistics" (Tomash). "De Moivre's masterpiece is The Doctrine of Chances" (DSB). "His work on the theory of probability surpasses anything done by any other mathematician except P. S. Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli's theorem by the aid of Stirling's theorem" (Cajori, A History of Mathematics, p. 230). "He was among the intimate friends of Newton, to whom this book is dedicated. It is the second book devoted entirely to the theory of probability and a classic on the subject" (Babson 181). De Moivre's interest in probability was raised by Pierre-Rémond de Montmort's Essay d'analyse sur les jeux de hazard (1708), the first separately-published work on probability. "The [Doctrine] is in part the result of a competition between De Moivre on the one hand and Montmort together with Nikolaus Bernoulli on the other. De Moivre claimed that his representation of the solutions of the then current problems tended to be more general than those of Montmort, which Montmort resented very much. This situation led to some arguments between the two men, which finally were resolved by Montmort's premature death in 1719 . De Moivre had developed algebraic and analytical tools for the theory of probability like a 'new algebra' for the solution of the problem of coincidences which somewhat foreshadowed Boolean algebra, and also the method of generating functions or the theory of recurrent series for the solution of difference equations. Differently from Montmort, De Moivre offered in [Doctrine] an introduction that contains the main concepts like probability, conditional probability, expectation, dependent and independent events, the multiplication rule, and the binomial distribution" (Schneider, p. 106). Provenance: Charles Meynell (early engraved bookplate). The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the 'Problem of points'. Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. "Huygens heard about Pascal's and Fermat's ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae . essentially followed Pascal's method of expectation. . At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are [Jakob] Bernoulli's Ars conjectandi (1713), Montmort's Essay d'analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivre's Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat's combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal's method of expectations." (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296). "De Moivre's earliest book on probability, the first edition of the Doctrine of Chances, was an expansion of a long (fifty-two pages) memoir he had published in Latin in the Philosophical Transactions of the Royal Society in 1711 under the title 'De mensura sortis' (literally, 'On the measurement of lots'). De Moivre tells us that in 1711 he had read only Huygens' 1657 tract De Ratiociniis in Ludo Aleae and an anonymous English 1692 tract based on Huygens' work (now known to have been written by John Arbuthnot). By 1718 he had encountered both Montmort's Essay d'analyse sur les jeux de hazard (2nd ed., 1713) and Bernoulli's Ars Conjectandi (1713), although the latter had no pronounced effect on De Moivre at that early date" (Stigler, p. 71). The Doctrine consists of an introduction with definitions and elementary theorems, followed by a series of numbered problems. "De Moivre begins with the classical measure of probability, 'a fraction whereof the numerator be the number of chances whereby an event may happen, and the denominator the number of all the chances whereby it may either happen or fail'. He gives the summation rule for probabilities of disjunct events explicitly only for the case of the happening and the not happening of an event. Expectation is still on the level of Huygens defined as the product of an expected sum of money and the probability of obtaining it, the expectation of several sums is determined by the sum of the expectations of the singular sums. He defines independent and dependent events and gives the multiplication rule for both. But whereas today the criterion for independence of two events is the validity of the multiplication rule in the [Doctrine], the multiplication rule follows from the independence of the events, which seems to be a self-evident concept for De Moivre . "With these tools 'those who are acquainted with Arithmetical Operations' (as De Moivre remarked in the preface) could tackle many problems, in part already well known but which he gradually generalized. Because the majority of<

1718, ISBN: b4d2f28edf1b3869c1f5997555e1bca9

Edition reliée

Engraved vignette on title and engraved head- & tailpieces. 2 p.l., xiv, 175 pp. Large 4to, cont. mottled calf (expertly rebacked & recornered by Aquarius), spine richly gilt, red morocco… Plus…

Engraved vignette on title and engraved head- & tailpieces. 2 p.l., xiv, 175 pp. Large 4to, cont. mottled calf (expertly rebacked & recornered by Aquarius), spine richly gilt, red morocco lettering piece on spine. London: W. Pearson for the Author, 1718. First edition and a fine copy of this classic on the theory of probability; it is dedicated to Isaac Newton who was a personal friend of the author. "His work on the theory of probability surpasses anything done by any other mathematician except P.S. Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli's theorem by the aid of Stirling's theorem."-Cajori, A History of Mathematics, p. 230. Nice copy. ? Babson 181-"He was among the intimate friends of Newton, to whom this book is dedicated. It is the second book devoted entirely to the theory of probability and a classic on the subject." Stigler, The History of Statistics, pp. 70-85. Tomash M 114.<