## Annuities upon Lives: or, The Valuation of Annuities upon any Number of Lives; as also, of Reversions. To which is added, An Appendix concerning the Expectations of Life, and Probabilities of Survivorship.

London: Printed by W.P. and sold by F. Fayram; Benj. Motte; and W. Pearson, 1725.

A very nice copy of de Moivre’s influential study of annuities based upon the mortality statistics gathered by Edmund Halley in the 1690s. “De Moivre’s contribution to annuities lies not in his evaluation of the demographic facts then known but in his derivation of formulas for annuities based on a postulated law of mortality and constant rates of interest on money. Here one finds the treatment of joint annuities on several lives, the inheritance of annuities, problems about the fair division of the costs of a tontine, and other contracts in which both age and interest on capital are relevant. This mathematics became a standard part of all subsequent commercial applications in England” (DSB IX: 454).

❧Norman 1530; Garrison-Morton 1690.

“De Moivre’s preoccupation with matters concerning the conduct of a capitalist society, such as interest, loans, mortgages, pensions, reversions or annuities, dated back at least to the 1690s, from which time a short note survives in Berlin containing his answers to a client's questions. In 1693, using the lists of births and deaths in Breslau for each of the years 1687 to 1691, Edmond Halley had published in the *Philosophical Transactions* a life table together with applications to annuities on lives, but the amount of calculation involved in extending this to two or more lives turned out to be immense. De Moivre replaced Halley’s life table by a (piecewise) linear function, which allowed him to derive formulas for annuities of single lives and approximations for annuities of joint lives as a function of the corresponding annuities on single lives. He published these formulas, together with the solution of problems of reversionary annuities, annuities on successive lives, tontines, and other contracts that depend on interest and the ‘probability of the duration of life’, in his book *Annuities upon Lives*” (ODNB).

A French Huguenot, Abraham de Moivre (1667-1754) was jailed as a Protestant upon the revocation of the Edict of Nantes in 1685. When he was released shortly thereafter, he fled to London. In London he became a close friend of Sir Isaac Newton and the astronomer Edmond Halley. De Moivre was elected to the Royal Society of London in 1697 and later to the Berlin and Paris academies. Despite his distinction as a mathematician, he never succeeded in securing a permanent position but eked out a precarious living by working as a tutor and a consultant on gambling and insurance.

“In the *Philosophical Transactions* for 1711 de Moivre published a long article, ‘De mensura sortis’, which was followed by the *Doctrine of Chances*. The second, much extended, edition of the *Doctrine* (1738) contained his normal approximation to the binomial distribution that he had found in 1733. This special case of the central limit theorem he understood as a generalization and a sharpening of Bernoulli’s *Theorema aureum*, which was later named the law of large numbers by Poisson. De Moivre’s central limit theorem is considered as his greatest mathematical achievement and shows that he understood intuitively the importance of what was later called the standard deviation. Crucial to this theorem was a form of the so-called Stirling formula for *n!*, which de Moivre and Stirling had developed in competition which ended in 1730” (*ibid.*).

Kress 3595; Norman 1530; Garrison-Morton 1690. K. Pearson, History of Statistics in the 17th and 18th centuries, pp. 146-54; Stigler, The History of Statistics, pp. 70-85.

8vo (198 x 119 mm), pp. [ii], 4, viii, 108, [2], errata slip mounted on final blank leaf. Contemporary 'royal crest' stamped full calf (rubbed and spine joints tender but holding). Rare in such good condition.

Item #3291

**
Price:
$4,500.00
**