**first edition**

## 1756 · London:

by DE MOIVRE, Abraham (1667–1754).

London:: A. Millar, 1756., 1756. Quarto. 348 pp. Portrait medallion vignette on title. Original full calf; joints repaired. Fine. Third edition. THIS IS A KEY WORK BY THE FATHER OF PROBABILITY THEORY IN WHICH MAJOR STEPS IN THE MEASUREMENT OF UNCERTAINTY WERE ACHIEVED. DE MOIVRE IS "BEST KNOWN IN STATISTICAL CIRCLES FOR HIS FAMOUS LARGE-SAMPLE APPROXIMATION TO THE BINOMIAL DISTRIBUTION, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability theory." [Bellhouse-Genest, p.1]. It is the first systematic treatment of probability in English. Abraham De Moivre became, with Edmund Halley, a founder of English actuarial science. The author's dedicatory letter is address to Lord George Carpenter (1702-1749) (the first edition had been dedicated to Newton), where the author states emphatically "that this Doctrine is so far from encouraging Play, that it is rather a Guard against it..." [DNB, vol. 38, p.116].

"The first edition of this work contains 175 pages, the second edition 258 pages and the third 348 pages. The following list will indicate the parts which are new in the third edition: the Remark pages 30/33 and pages 48 & 49, the greater part of the second Corollary pages 64/66, the Examples page 88; the Scholium page 95, the Remark page 149 and pages 151/159, the fourth Corollary page 162, the second Corollary pages 176/179, the note at the foot of page 187, the Remark pages 251/254. The part on life annuities is very much changed. The Introduction is very much fuller than the corresponding part of the first edition. In his third edition De Moivre draws attention to the convenience of approximating to a fraction with a large numerator and denominator by continued fractions, which he calls "the Method proposed by Dr. Wallis, Huygens and others". He gives the rule for the formation of the successive convergents. This third edition contains 74 problems exclusive of those relating to life annuities (in the first edition there were 53 problems). The pages 220/229 contains one of De Moivre's most valuable contributions to mathematics, namely that of Recurring series. Pages 261/328 are devoted to Annuities on lives; an Appendix finishes the book, occupying pages 329/348: this also relates principally to annuities, but it contains a few notes on the subject of probability." – Todhunter. A very full account of the above third edition will be found in Todhunter's History of the theory of probability. "De Moivre's work on the theory of probability surpasses anything done by any other mathematician except Laplace. His principal contributions are his investigations respecting the duration of play, his theory of recurring series and his extension of the value of Bernouilli's theorem by the aid of Sterling's theorem". – Cajori.

Theodore Porter (UCLA) writes that De Moivre introduced the astronomer's law error to probability theory (p. 93). "Like most early probability mathematics, it first arose in the context of games of chance; it appeared as the limit of the binomial distribution. Because of its usefulness in combination and permutation problems, the binomial had become the heart of the doctrine of chances…. De Moivre then showed in a paper of 1733, reprinted in 1738 in the second edition of his Doctrine of Chances, that the exponential error function gave a very good approximation to the distribution of possible outcomes for problems like the result of 1,000 coin tosses Now, for the first time, it was practicable to apply probability theory to indefinitely large numbers of independent events."

REFERENCES: Babson 181 (1st ed.); Ball, A short account of the history of mathematics, pp. 383-4; BM Readex Vol. 17, p. 751; Cajori, History of Mathematics, pp. 229-30; DNB, vol. 38, p.116; Kress S.2793; Institute of Actuaries (1935) p. 39; Mansutti 504; Norman 1529 (1st ed.); Pearson, The History of Statistics in the 17th & 18th Centuries…, pp. 155-60, 165-66; Smith, Source book in mathematics, pp. 440-54; Stigler, The History of Statistics: The Measurement of Uncertainty before 1900 (1986), p. 70; Todhunter, History of the theory of probability; Walker pp. 12-13; Wellcome IV, p. 149; Westergaard pp. 104-5. Not in Goldsmiths or Hanson. See: Raymond Clare Archibald, "Abraham de Moivre"; David, F.N., Games, Gods and Gambling; The origins and history of probability and statistical ideas … (1962), pp. 161-178. [PLEASE CONTACT DIRECT FOR FURTHER INFORMATION]. (Inventory #: S13078)

"The first edition of this work contains 175 pages, the second edition 258 pages and the third 348 pages. The following list will indicate the parts which are new in the third edition: the Remark pages 30/33 and pages 48 & 49, the greater part of the second Corollary pages 64/66, the Examples page 88; the Scholium page 95, the Remark page 149 and pages 151/159, the fourth Corollary page 162, the second Corollary pages 176/179, the note at the foot of page 187, the Remark pages 251/254. The part on life annuities is very much changed. The Introduction is very much fuller than the corresponding part of the first edition. In his third edition De Moivre draws attention to the convenience of approximating to a fraction with a large numerator and denominator by continued fractions, which he calls "the Method proposed by Dr. Wallis, Huygens and others". He gives the rule for the formation of the successive convergents. This third edition contains 74 problems exclusive of those relating to life annuities (in the first edition there were 53 problems). The pages 220/229 contains one of De Moivre's most valuable contributions to mathematics, namely that of Recurring series. Pages 261/328 are devoted to Annuities on lives; an Appendix finishes the book, occupying pages 329/348: this also relates principally to annuities, but it contains a few notes on the subject of probability." – Todhunter. A very full account of the above third edition will be found in Todhunter's History of the theory of probability. "De Moivre's work on the theory of probability surpasses anything done by any other mathematician except Laplace. His principal contributions are his investigations respecting the duration of play, his theory of recurring series and his extension of the value of Bernouilli's theorem by the aid of Sterling's theorem". – Cajori.

Theodore Porter (UCLA) writes that De Moivre introduced the astronomer's law error to probability theory (p. 93). "Like most early probability mathematics, it first arose in the context of games of chance; it appeared as the limit of the binomial distribution. Because of its usefulness in combination and permutation problems, the binomial had become the heart of the doctrine of chances…. De Moivre then showed in a paper of 1733, reprinted in 1738 in the second edition of his Doctrine of Chances, that the exponential error function gave a very good approximation to the distribution of possible outcomes for problems like the result of 1,000 coin tosses Now, for the first time, it was practicable to apply probability theory to indefinitely large numbers of independent events."

REFERENCES: Babson 181 (1st ed.); Ball, A short account of the history of mathematics, pp. 383-4; BM Readex Vol. 17, p. 751; Cajori, History of Mathematics, pp. 229-30; DNB, vol. 38, p.116; Kress S.2793; Institute of Actuaries (1935) p. 39; Mansutti 504; Norman 1529 (1st ed.); Pearson, The History of Statistics in the 17th & 18th Centuries…, pp. 155-60, 165-66; Smith, Source book in mathematics, pp. 440-54; Stigler, The History of Statistics: The Measurement of Uncertainty before 1900 (1986), p. 70; Todhunter, History of the theory of probability; Walker pp. 12-13; Wellcome IV, p. 149; Westergaard pp. 104-5. Not in Goldsmiths or Hanson. See: Raymond Clare Archibald, "Abraham de Moivre"; David, F.N., Games, Gods and Gambling; The origins and history of probability and statistical ideas … (1962), pp. 161-178. [PLEASE CONTACT DIRECT FOR FURTHER INFORMATION]. (Inventory #: S13078)