in *Scottish Law*, denotes the yearly interest or profit due by a debtor in a sum of money to a creditor for the use of it.—A Right of Annualrent was the original method in Scotland of burdening lands with a yearly payment for the loan of money, before the taking of interest was allowed. annuities had then been calculated, except a very con- tected one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre probably contributed to retard the progress of the science, by turning the attention of mathematicians from the investigation of the true law of mortality, and the best methods of constructing tables of the real values of an- nuities.

The same distinguished analyst also endeavoured to ap- proximate the values of joint lives; but it has since been found that the formulae he gave for that purpose are too incorrect for use. Mr Thomas Simpson published his Lettres of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more gen- eral and perspicuous than it had been previously. His formulae are adapted to any table of mortality; and, in the seventh corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe all the best tables of the values of life-annuities that have since been published.

In the same work he also gave a table of mortality de- duced from the London observations, and four others cal- culated from it, of the values of annuities on lives, each at the rates of interest: the first for single lives, the three others for two and three equal joint lives, and for the largest of two or of three lives.

These were the first tables of the values of joint lives that had been calculated; for although Dr Halley had shown, half a century before, how such tables might be computed, and had taken considerable pains to facili- tate the work, the necessary calculations by the methods known previous to the publication of Mr Simpson's treatise were so very laborious that no one had had the courage to undertake them. And unfortunately the mortality ac- cording to the London table was so much above the com- mon 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 la Probabilité de la Durée de la Vie Humaine, which included several valuable tables of mortality deduced from all mortality registers of different religious houses, and from the lists of the nominees in the French tontines; and a table of the values of annuities on single lives, at the rates of interest, calculated from his table of mortal- ity for the tontine annuitants. These tables were a great addition to the science, as, before their publication, there were only two extant that gave tolerably exact re- presentations of the true law of mortality—Dr Halley's of Breslaw, and one constructed but a short time before by M. Kersseboom, 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 an- nuities-certain; but the principal essay, On the Probabili- ty 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 will himself to the extent he might have done, of the ex- cellent tract of Thomas Simpson.

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

In the year 1732 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 more 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 Me- moirs 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 18 years be- fore 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. Kersseboom's table of mortality.

The first edition of Dr Price's Observations on Rever- sionary Payments was published in 1770, and its chief object was, to give information to persons desirous of form- ing 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 formulae had till then been of little prac- tical utility.

In the prosecution of this laudable design, Dr Price was obliged to have recourse to approximations. He in- forms 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 calcu- lated a table of these values upon M. de Moivre's hypo- thesis of the decrements of life being equal, and its utmost limit 86 years, from a correct formula given by Mr Simp- son in his doctrine of Annuities (Cor. 5, Prob. 1). 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 ob- servations 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 mor- tality. And in the same year M. de Saint-Cyran pub- lished his Calcul des Rentes Viagères sur Une et sur Plu- sieurs 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 History from M. Kersseboom'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 progress of the science had been, to direct the public attention to these inquiries, by showing their important uses in the affairs of life; and to procure the requisite data for forming tables of mortality, that should illustrate the laws according to which human life wastes under different circumstances, by exciting the curiosity of intelligent men who had the necessary leisure and means of information. The ingenious author had accordingly been furnished with the necessary abstracts of mortuary registers which had been kept with these views, by Dr Haygarth at Chester, Dr Aikin at Warrington, and the Rev. Mr Gorsuch at Holy-Cross, near Shrewsbury, since the publication of the first edition; also by Mr Wargentin, with the mean numbers both of the living and the annual deaths in all Sweden and Finland for 21 successive years; in all of which the sexes were distinguished; and from these data he constructed tables of mortality that threw great light on the subject. He also inserted in this edition an improved table of mortality for Northampton; and, what had been so long wanted, a complete set of tables of the values of annuities on single lives at six rates of interest, and on two joint lives at four, all calculated from the new Northampton table. The combinations of joint lives were sufficiently numerous to admit of all the values not included being easily interpolated. Besides these, he also gave tables of the values of annuities on single lives from the Swedish observations, both with and without distinction of the sexes, and on two joint lives without that distinction.

The values given in these tables are too low for the general average of lives at all ages under 60; but in the treatise of Mr Baron Maseres on the Principles of the Doctrine of Life Annuities, which was published in the same year (1783), others were given, calculated from the table of mortality which M. Deparcieux constructed from the lists of the nominees in the French tentines. The tables for single lives are calculated at twelve different rates of interest from 2 to 10 per cent., but those for joint lives only at 3½ and 4½ per cent.; and the combinations they include are only those of ages that are equal, or that differ by 5 or 10 years, and the multiples of 10.

There is reason to believe that the values in these tables, at all ages under 75 or 80 years, are nearer the truth, for the average of this country, than any others then extant; but certainly for the average of lives on which annuities and reversions depend. After that period of life, however, they are too small; and, in most cases, it is difficult to derive the values of joint lives from them with sufficient accuracy, on account of the contracted scale they have been calculated upon.

It was not Dr Price's object to deliver the elements of the science systematically; but he treated most parts of it with great judgment, enriched it with a vast collection of valuable facts and observations, and corrected several errors into which some of the most eminent writers upon it had fallen. The mathematical demonstrations (which are given in the notes) are much inferior to the rest of the work.

The values of reversionary sums and annuities, which depend upon some of the lives involved failing according to assigned orders of precedence, had been approximated by Mr Simpson in his Select Exercises, and by Mr Morgan in his Doctrine of Annuities; but the latter gentleman first gave accurate solutions of problems of this kind, in the Philosophical Transactions for the years 1788, 1789, 1791, 1794, and 1799.

Mr Baily's Doctrine of Life Annuities and Assurances was published in 1810. In it the whole subject is treated except the construction of tables of mortality, on which the practical application of all the rest depends. In consequence of the author having adopted Mr Simpson's notation, this work presented a more perspicuous exposition of the whole theory, especially of the improvements made in between the time when Mr Simpson wrote and the date of its publication, than had previously appeared. And in an appendix to it, published in 1813, principally for the purpose of explaining the construction and uses of tables for determining the values of life-annuities, calculated at a vast sacrifice of time and labour by Mr George Barrett, since deceased, formulae were given for calculating from tables of that kind the values of temporary and deferred life-annuities and assurances, and also for determining the values of annuities and assurances when the annuity or the sum assured, instead of remaining always the same, increases or decreases from year to year by equal differences, with considerably greater facility and expedition than the same things could have been effected with by the tables and methods of calculation in previous use.

Except by these improvements, and the solution of the problems above stated to have been first given by Mr Morgan, which were severely criticised and given anew, with some amendments besides the important one of the notation in Mr Baily's work, the science had not been materially advanced, during a period of more than 30 years, which had elapsed since the appearance of the fourth edition of Dr Price's observations, when Mr Mine published his Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships, in the year 1815.

The work consists of two volumes; the first is mathematical, the second entirely popular, except the notes and a few of the tables. The algebraical part of this article is merely a short abstract of the first volume, and may serve as a specimen of the manner in which the subject has been treated there; but the construction of tables of mortality, which forms the subject of the third chapter, has not been noticed here; neither is the valuation of reversionary sums or annuities depending upon assigned orders of survivorship treated in the present article; and these are parts of the work which will not be found the least interesting to mathematicians.

The second volume contains upwards of 50 new tables, with a few others that had been published before, but have been reprinted either on account of their value or scarcity, or both. Four of the new ones are tables of mortality constructed by the author, from registers kept at Carlisle and Montpellier, and in all Sweden and Finland, since the period of the observations Dr Price made use of: the sexes are distinguished in the tables for Sweden and Montpellier, but not in that for Carlisle. The last is the only table, besides those for Sweden and Finland, applicable to the mass of the people, that has been formed from the necessary data,—enumerations of the living, as well as registers of the deaths, in every interval of age.

Twenty-one of these tables, being the seventeenth to the thirty-seventh inclusive, in the collection at the end of the work, render it easy to apply the algebraical formulae to practical purposes, and numerous examples of such applications are given. They have all been calculated from the Carlisle table of mortality; those of the values of life-annuities on the same extensive scale with those