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 , that no one had had the courage to undertake them. And, unfortunately, the mortality, according to the Loudon table, was so much above the common average, that the values of annuities in Mr Simpson’s tables were much too small for general use.

in the year 1746, M. Deparcieux published his Essai sur les probabilités de la durée de la vie humaine, in which he gave several valuable tables of mortality deduced from the mortuary registers of different religious houses, and from the lists of the Nominees in the French Tontines; also a table of the values of annuities on single lives, at three rates of interest, calculated from his table of mortality for the tontine annuitants. These tables were a great acquisition to the science, as, before their publication, there were only two extant that gave tolerably exact representations of the true law of mortality;—Dr Halley’s for Breslaw, and one constructed but a short time before by M. Kerseboom, principally from registers of Dutch annuitants. Those of M. Deparcieux for the Monks and Nuns, were the first ever constructed for the two sexes separately; and by them, the greater longevity of females was made evident.

The work commences with an algebraical theory of Annuities-certain; but the principal essay, On the Probabilities of the Duration of Human Life, is perfectly intelligible to those who have not studied Mathematics; it is written with great judgment and perspicuity, but contains very little more than the explanation of the construction of his tables, some of which relate to Tontines; and he did not avail himself to the extent he might have done, of the excellent tract of Thomas Simpson.

This work, however, appears to have been more read upon the Continent, and to have contributed more to the diffusion of this kind of information there, than all the other writings on the subject. The article Rentes viageres in the Encyclopédie, is acknowledged to have been taken entirely from it, as was also the article, durée de la; and these are proofs, among many others that might be produced, how little M. D’Alembert and the principal Mathematicians, his contemporaries, attended to the subject.

In the year 1752, Mr Simpson published, in his Select Exercises, a supplement to his Doctrine of Annuities; wherein he gave new tables of the values of annuities on two joint lives, and on the survivor of two lives, much mere copious than those he had inserted in the principal work; but these also were calculated from his London table of mortality.

The celebrated Euler, in a paper inserted in the Memoirs of the Royal Academy of Sciences at Berlin for the year 1760, gave a formula by which the value of an annuity on a single life of any age, may be derived from that of an annuity on a life one year older; which formula was included in that given by Mr Simpson eighteen years before, for effecting the same purpose in the case of any number of joint lives; and by this compendious method, M. Euler calculated a table of the values of single lives from M. Kerseboom’s table of mortality.

The first edition of Dr Price’s Observations on Reversionary Payments was published in 1769; and its chief object was, to give information to persons desirous of forming themselves into societies for the purpose of making provision for themselves in old age, or for their widows. When tables of the values of single lives, and of two joint lives are given, the methods of determining the terms on which such provisions can be made with safety to all the parties concerned, are very simple, and were, at that time, well understood in theory, by the Mathematicians who had studied the subject; but, for want of the requisite tables, the algebraical formulæ had, till then, been of little practical utility.

In the prosecution of this laudable design, Dr Price was obliged to have recourse to approximations. He informs us, that by following M. De Moivre too implicitly in his rules for determining the value of two joint lives, he was led into difficulties which convinced him that they were not only useless, but dangerous; he therefore calculated a table of these values upon M. De Moivre’s hypothesis of the decrements of life being equal, and its utmost limit 86 years, from a correct formula given by Mr Simpson in his Doctrine of Annuities (Cor. 5. Prob. I); by this, and a table of the values of single lives, calculated by Mr Dodson on M. De Moivre’s hypothesis, he was enabled to give answers tolerably near the truth, to some of the most interesting questions of this kind, and to show that the plans of several of the societies then recently established, were quite inadequate; and instead of the benefits they promised, could only, in the end, produce disappointment and distress, unless they either dissolved or reformed themselves.

The work also contained instructive dissertations on the probabilities and expectations of life, and on the mean duration of marriage and of widowhood; besides accounts of some of the principal societies which had then been formed for the benefit of old age, and of widows; with observations on the method of forming tables of mortality for towns, and two new tables of that kind, constructed from registers kept at Norwich and Northampton. Mr Morgan’s Doctrine of Annuities and Assurances was published in 1779, containing tables of the values of single lives, of two equal joint lives, and of two lives differing in age by 60 years, calculated from the Northampton table of mortality. And in the same year, M. De Saint-Cyran published his Calcul des Rentes viageres sur une et sur plusieurs têtes, wherein the valuation of annuities on lives is treated algebraically, but in a manner much inferior in all respects to that of Mr Simpson; and six tables are given of the values of annuities,—on single lives, on the survivor of two lives, and on the last survivor of three, calculated from M. Kerseboom’s table of mortality. Although the values in the cases of two, and of three lives, were only determined by approximation, these tables were, just then, a valuable acquisition to the science; but their use was entirely superseded only four years after, by the publication of others much more valuable.

The fourth edition of Dr Price’s Observations on Reversionary Payments appeared in 1783. One of the best effects of the preceding editions on the pro- Rh