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.¹
The doctrine of Compound Interest and Annuities—certain is too simple ever to have occupied much of the attention of mathematicians: inquiries into the values of interests dependent upon the continuance or the failure of human life, being more interesting and difficult, have occupied them more, but yet not so much as their importance would seem to demand; the discoveries both in Pure Mathematics and Physics, especially those of Newton, which distinguished the close of the seventeenth century, having provided them with ample employment of a more interesting kind, ever since the subjects of this article were submitted to calculation.
Fermat, Pascal, and Huygens, by laying the foundation of the doctrine of probabilities, about the middle of that century, first opened the way to the solution of problems of this kind. The earliest mathematical publication on probabilities, the little tract of Huygens, De Ratiociniis in Ludo Aleæ, appeared in 1658; and in 1671 his celebrated countryman John de Witt published a treatise on Life-Annuities in Dutch. (Montucla, Hist. des Math. tome iii. p. 407.) This, however, appears to have been very little known or read, and to have had no sensible influence on the subsequent progress of the science, the origin of which may be properly dated from the publication of Dr Halley's paper on the subject, in the Philosophical Transactions for the year 1693 (No. 196). That celebrated mathematician there first gave a table of mortality, which he had constructed from observations made at Breslaw, and showed how the probabilities of life and death, and the values of annuities and assurances on lives, might be determined by such tables; which, he informs us, had till then been only done by an imaginary valuation. Besides his algebraical reasonings, he illustrated the subject by the properties of parallelograms and parallelopipeds: there are, perhaps, no other mathematical inquiries, in the prosecution of which algebra is entitled to so decided a preference to the elementary geometry as in these, and this example of the application of geometry has not been followed by any of the succeeding writers.
In the year 1724 M. de Moivre published the first edition of his tract entitled Annuities on Lives. In order to shorten the calculation of the values of such annuities, he assumed the annual decrements of life to be equal; that is, that out of a given number of persons living at any age, an equal number die every year until they are all extinct; and upon that hypothesis he gave a general theorem, by which the values of annuities on single lives might be easily determined. This approximation, when the utmost limit of life was supposed to be 86 years, agreed very well with the true values between 30 and 70 years of age, as deduced from Dr Halley's table; and the method was of great use at the time, as no tables of the true values of
¹ Hardly any terms are made use of in this article which may properly be considered technical. But since it is desirable that the reader should have perfectly clear and well-defined ideas of the terms that are employed, in the demonstrative part, which follows the historical, a few have been defined in the paragraphs where they are first introduced; and we here give those terms in alphabetical order, with the numbers of the paragraphs in which their definitions are given:
| Term | Paragraph | |-----------------------|-----------| | Annuity | | | Annuity, Certain | 3 | | Annuity, Deferred | 4 | | Annuity, Life | 20 | | Annuity, Temporary Life| 31 | | Annuity on any Life or Lives | 68 | | Assurance on any Life or Lives | 62 | | Mortality, Table of...| 77 | | Years' Purchase, No. of; that an Annuity is worth | 32 |
Terms annuities had then been calculated, except a very contracted one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre has 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 annuities.
The same distinguished analyst also endeavoured to approximate 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 Doctrine of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more general 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 deduced from the London observations, and four others calculated from it, of the values of annuities on lives, each at three rates of interest: the first for single lives, the three others for two and three equal joint lives, and for the longest 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 facilitate 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 according to the London 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 1736 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. 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 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 Viagères in the French Encyclopédie is acknowledged to have been taken entirely from it, as was also the article Vie, 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 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 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 18 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. Kersseboom's table of mortality.
The first edition of Dr Price's Observations on Revolutionary Payments was published in 1770, 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 formulae 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. 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 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 Viagères 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 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 Atkin 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 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 Masereon 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. Depareieux constructed from the lists of the nominees in the French tontines. 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 precedency, 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 it 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 Milne 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. This 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 History, which Dr Price derived from the Northampton table. It is the author's opinion that the values of interests dependent upon the continuance or the failure of life may be derived from them more correctly than from any others then extant, and he has taken considerable pains to assist his readers in judging of this for themselves.
Besides the tables, the principal contents of the second volume are explanations of their construction and uses. Many of them relate to the progress of population,—the comparative mortality of different diseases, of different seasons,—and of the two sexes at every age, the proportion of the sexes at birth, and that of the born alive to the still-born of each sex.
It will be found that the author has collected records of facts and observations of great value, and that he has endeavoured to present the information they afford in the forms best calculated for the further prosecution of these inquiries.
Mr Gompertz's Sketch of an Analysis and Notation applicable to the estimation of the value of Life Contingencies was read at a meeting of the Royal Society on the 29th of June 1820, and printed in the Society's Transactions for that year.
The second edition of Mr Morgan's Principles and Doctrine of Assurances and Annuities on Lives was published in 1821. The formulae for determining the values of contingent reversions, first given by the author in the Philosophical Transactions, as above stated, were in this second edition substituted for the approximations given in the first; and it contains tables of the values of annuities on single and joint lives, calculated from the Northampton and Sweden tables of mortality, and taken from Dr Price's Observations on Reversionary Payments.
This edition contains nothing new, except a table on the last two pages, showing the number of persons whose lives were insured in the Equitable Society, who died of each disease in each decade of age from 10 years to 80, and above 80 years of age, during a term of 20 years commencing with 1801. Also (in the last line of the table) "the number assured during the same term" in each of these intervals of age. The obvious meaning of this expression would seem to be, the number of persons on whose lives assurances were effected by the Society during these 20 years, in each interval of age. This is the sense in which Mr Babbage understood it, and therefore drew wrong inferences from it (in his Comparative View of Life Insurance Institutions, p. 63 and table 13). The late Dr Young was also misled by it, and appears to have taken those for the mean numbers of the lives on which the society had policies in force in the several intervals of age during these 20 years; and thence concluded that during these 20 years, according to that table, the mean number of lives of all ages on which the society had policies in force was more than 150,000, and that only one out of 1500 of them died annually. (Philosophical Transactions, 1826, p. 287.)
Indeed Mr Morgan himself, on the page preceding his table, stated that it contained an account of all the deaths which had happened in the society during 20 years, "among a population exceeding 150,000 persons."
If the numbers only in that last line of the table had been given, without any explanation of them, it would, upon due consideration, have been conjectured that they could be no others than the sums of the numbers of lives insured by the society which were found to be in those intervals of age at one and the same period in each of those 20 years; that is, 20 times the mean numbers of them in the several intervals during the same term. But History, neither of Mr Morgan's explanations of them would admit of that construction.
The present article was first published in the Supplement to this Encyclopaedia in 1816.
In the article on the Law of Mortality in that Supplement, which appeared in 1822, it was shown that, according to Mr Morgan's statement, made at a general court of the Equitable Society in the year 1800, of the mortality which had taken place among the lives insured by that society as compared with the Northampton table, the mortality among those lives in each decade of age from 10 years to 50 was very nearly the same as in the Carlisle table of mortality; also that, above 50 years of age, the difference, upon the average, was not great. And early in 1826 were published Tables of Life Contingencies, by Mr Davies, and A Comparative View of the various Institutions for the Assurance of Lives, by Mr Babbage; in each of which works was given a table of the mortality which had prevailed among the lives insured in the Equitable Society at all ages above 10 years, constructed from that statement of Mr Morgan.
Mr Babbage gave also a table of the values of annuities on single lives of all ages above ten years, derived from his table of mortality above mentioned; and the indefatigable Mr Davies gave tables of the values of annuities on single and joint lives, calculated both from his table of mortality above mentioned and from the Northampton table, rather fuller and more complete than any that had previously been published, except that those derived from the law of mortality in the Equitable Society necessarily included no ages under 10 years. The values according to the Northampton table were given only at the rates of 3 and 4 per cent. interest; but Mr Davies, not content to take them on Dr Price's authority, has, like Mr Barrett, calculated them anew, and, as well as the other values of annuities, has carried them to four places of decimals. This author's comprehensive little book also contains many other tables of the values of annuities and assurances on lives and survivorships, with one of the values of policies of assurance, calculated from the Northampton table of mortality at 3 per cent. interest, which, as well as many another single table contained in it, must have cost him much time and labour; and those derived from the Northampton table of mortality must be very valuable to such of the assurance companies as take that table for their guide in transacting business.
Mr Babbage and Mr Davies also gave formulae and tables similar to, or not materially different from, those of Mr Barrett above mentioned, for determining the values of temporary and deferred, as well as increasing or decreasing annuities and assurances on single lives; the tables of Mr Babbage being derived from his table of mortality in the Equitable Society, and from the Carlisle table; that of Mr Davies from the Northampton table of mortality alone.
Mr Morgan, in the statement on which those tables of Mr Babbage and Mr Davies were founded, contented himself with the use of two simple digits only, to express the proportion of the mortality in the Equitable Society to that in the Northampton table in each decade of age; and although that gentleman for twenty-five years after 1800, when that statement of the thirty years' previous observations was made, continued, in Notes on Dr Price's Observations on Reversionary Payments, and in his addresses to the general courts of the Equitable Society, to state that the proportions still remained the same, the
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1 See the note at the end of this Historical Introduction. History, observations extending over a period of fifty-five years, he never gave any statement more full or distinct.
These were but scanty materials certainly to construct a table of mortality from; and yet the agreement between the tables formed from them, and the best of other existing tables of mortality, is very remarkable.
A committee of the House of Commons on friendly societies having been appointed in 1827, chiefly for the purpose of inquiring into the law of mortality, and the values of life-annuities and assurances in this country, the report of that committee, by bringing the subject prominently before the public, and exciting attention to well-established but much neglected results of inquiries into it, had the effect of correcting to a considerable extent opinions upon it, taken upon trust without due examination, and generally diffused. The establishment of many new assurance companies, and the increasing prevalence of life-assurance for 15 or 20 years before, by exciting discussion and examination of their rates, had also contributed to produce that effect. At length the members of the Equitable Assurance Society, as the period of the decennial investigation of their affairs in 1829 approached, expressed a desire to avail themselves of the information respecting the law of mortality in the society which the office books might afford, for estimating and dividing their profits; and in 1828 Mr Morgan published a pamphlet entitled *A View of the Rise and Progress of the Equitable Society*, in which (p. 42) he gave the following "table of the decrements of life in the society during the preceding 12 years."
### TABLE (a).
| Age | Number | Died | Should have died | |-----|--------|------|------------------| | 20 to 30 | 4,720 | 29 | 68 | | 30 to 40 | 15,951 | 106 | 243 | | 40 to 50 | 27,072 | 201 | 506 | | 50 to 60 | 23,307 | 339 | 545 | | 60 to 70 | 14,705 | 426 | 502 | | 70 to 80 | 5,056 | 289 | 290 | | 80 to 95 | 701 | 99 | 94 |
In the same place, the author informs his readers that "in his former statement (without mentioning which of them) he was not aware of the great number of instances in which there were several policies on one and the same life." Also that "the present is, in fact, the only correct table of the decrements of life in the society."
In the same pamphlet the author has given a table to show the law of mortality among the lives insured in the Equitable Society, founded upon his last statement; also a table of the expectations of life, and one of the values of annuities on single lives at three per cent. per annum interest, both derived from that table of mortality, none of them including any age under 20 years.
The author of the present article, when preparing this historical sketch, not clearly understanding the import of the numbers in the second column of this table (a), nor the last line in the table of deaths by the different diseases, given at the end of Mr Morgan's work on annuities (2d edit. 1821)—having also some doubts about the statement made in 1800—wrote to Mr Morgan, and requested the desired information respecting them, when that gentleman forwarded him the following:
1. That the second column should have been headed, "Number of persons living at the beginning of each year during the last 12 years;" and the fourth, "the number which should have died, according to the Northampton table." 2. That as to the table at the end of his treatise on annuities, he knew the numbers of deaths by the different diseases to be correct; but "the numbers in the last line of that table were those of the policies, which he had since found greatly to exceed the numbers of the members." 3. Upon his statement of 1800 Mr Morgan made no observation.
From the first part of the information thus obtained, it appears that the number in the second column on any line in table (a), is twelve times the mean number of lives in the interval of age set against it, on which the Equitable Society had policies in force during these 12 years; so that, in a society similar in all other respects, in which the law of mortality was constantly the same, and the number of lives in each interval of age was always just 12 times as great as in the Equitable during these 12 years, the population and mortality would be as in columns 2 and 3 of the following table (b).
### TABLE (b).
| Age | Number | Died | Should have died | |-----|--------|------|------------------| | 20 & 30 | 4,720 | 29 | 68 | | 30 & 40 | 15,951 | 106 | 243 | | 40 & 50 | 27,072 | 201 | 506 | | 50 & 60 | 23,307 | 339 | 545 | | 60 & 70 | 14,705 | 426 | 502 | | 70 & 80 | 5,056 | 289 | 290 | | 80 & 95 | 701 | 99 | 94 |
If our inferences in the 4th and 5th columns of this table from Mr Morgan's data in the 2d and 3d be correct, neither Mr Morgan's in the 6th column, nor his table of mortality, nor consequently those of the expectations and values of lives above mentioned, can have been accurately derived from the same data.
But even with these data we have reason to believe that Mr Morgan is not well satisfied, and that that gentleman has since been able to form a table from much more unexceptionable documents, which shows the probabilities
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1 It may be proper to show here, how our numbers in columns 4 and 5 have been determined. For this purpose we give as an example the manner in which the 2d in col. 5 has been calculated.
\[ \text{Expectation of life at } \begin{cases} 30 = 28.27 \times 4385 \\ 40 = 23.08 \times 3635 \end{cases} \]
number annually attaining that age \(= \begin{cases} 123,964 \\ 83,296 \end{cases}\)
number living above \(= \begin{cases} 30 \\ 49 \end{cases}\)
The differences are the numbers between 30 and 40; 750 dying annually, and 40,068 constantly living.
And 40,068 : 750 : 15,951 : 299. See article on Law of Mortality. It is satisfactory to find Mr Morgan at last coming into an opinion, of the truth of which, evidence had previously been adduced sufficient to convince almost all who attended to it; and, trusting that he will publish the results of his latest inquiries on the subject, we shall take no further notice here of the table of the values of annuities in his pamphlet above mentioned, which, otherwise, we could not with propriety have avoided.
Mr Morgan (Pamph. p. 42 and 43) asserts that a comparison of columns 3 and 6 of this table (b) affords a striking proof of the accuracy of the Northampton table; but, to judge of that, we consider it is not column 6, but 5, which should be compared with column 3.
In the Transactions of the Cambridge Philosophical Society, vol. iii. part 1, are two papers by Mr Lobbuck: the first On the Calculation of Annuities, and on some Questions in the Theory of Chances (read May 26, 1828); the other On the Comparison of various Tables of Annuities (read March 30, 1829).
In the year 1808 government commenced granting life-annuities at prices calculated from the Northampton table of mortality, and continued so to grant them for 20 years, at a great loss to the nation, especially when the lives were young, as was well known at the time to those who understood the subject; and was mentioned more than once by the author of the present article, to a very able and well-informed member of the government, not long after they commenced granting them. But none were then granted on lives under 35 years of age; and the gentleman alluded to only observed, that the applicants for annuities were principally aged persons, and it was desirable that a safe and advantageous mode of employing their savings should be afforded them. After the year 1816 those annuities were granted to persons of all ages above 21 years.
Although the data necessary for determining the law of mortality among the people, and the values of pecuniary interests dependent upon the continuance or the failure of human life, cannot be obtained without the active concurrence of many persons of influence and authority; yet, for all the tables containing information of that kind relative to this country, and published before the year 1829, the public were indebted to the zeal and industry, and the separate efforts, of a few individuals.
But in March 1819 Mr Finlaison was appointed by government, with all the aids they could afford him, including proper assistants, and access to the registers of the nominees in tontines, and others on whose lives annuities had been granted by government for more than a hundred years before; in which registers the exact ages at which the annuitants were nominated, and those at which they died, were stated.
Thus the data not otherwise accessible being provided, and the labour lessened by the number of calculators employed—the expense also being defrayed by the public—at the end of ten years, viz. in March 1829, Mr Finlaison made a report to the Lords of the Treasury, which was printed by order of the House of Commons, and, in tables filling 50 folio pages, shows the rates of mortality, and the values of annuities on single lives at all ages, among many different classes of annuitants, both separate and combined; the sexes being generally distinguished both in exhibiting the law of mortality and the values of annuities.
These, from the number and accuracy of the data, are more valuable than any thing of the same kind that had previously been published; but it is the values of annuities only which we have to notice here.
The lives on which annuities depend will generally be somewhat better (by which we here mean, will attain to greater longevity) than the general average of the population, though probably not nearly so much better as many believe them to be. The prevailing error in the popular estimate on this subject appears to have arisen in great measure from comparing the mortality among annuitants and assured lives, with that represented to take place by tables of mortality erroneously considered to correspond with the general average of the people; while, from being constructed on erroneous principles, and from insufficient data, or else being derived from observations made where the mortality was and is much greater than in Britain, the mortality according to these tables was considerably greater than that which actually prevails among the bulk of the people here. Proofs of this will be found under the article Law of Mortality.
That the lives on which annuities and assurances depend cannot be so very select or so much better than the common average as has generally been supposed, might reasonably be expected on these grounds:
1. As to annuitants. The lives are not all chosen on account of their presumed goodness; for many persons who have no occasion to provide for others who may survive them, purchase annuities on their own lives, only that they may themselves enjoy the whole benefit of the purchase-money, both principal and interest, during their lives.
And the greatest recommendation of these lives seems to be, that they are generally prudent persons, of temperate and regular habits.
Many other persons, especially females, spendthrifts, and faithful servants, enjoy annuities bequeathed to them by their deceased relatives, masters, or mistresses, as the most eligible provision for their future comfort and security from want; and there seems little ground to suppose them to be better lives than the common average of the same age and sex.
2. In such cases as tontines, where most of the lives are selected for their presumed goodness, the best criterion probably is, hereditary longevity in the family of the nominee; but partiality for their own friends or kindred often has considerable influence in biasing the judgment of those who select them.
That they will generally be persons of good constitutions and regular habits when selected, is all that is likely to be obtained under these circumstances; and that is also the case with the average of the population in comfortable circumstances.
Whatever the constitutions and habits of annuitants may be, the annuities held by them during their own lives, by protecting them from many of the wants, cares, and anxieties which the bulk of the people are exposed to, no doubt contribute to their longevity. But where powerful motives to raise money by the sale of an annuity on a person's own life exist, it is extremely difficult to prevent him from parting with it, whatever precautions may have been taken with that view; and with it, he also loses that help to longevity.
3. Insured lives are also generally supposed to be much better than the average of the population, as it is incumbent upon the insurance offices to be cautious in selecting them.
But bad lives, by the failure of which persons interested in them would sustain loss, are most likely to be offered, and are continually offered, for insurance; and there is reason to believe that all the caution in selection which the offices in general can exercise, is necessary to keep the lives insured up to the average goodness of the bulk of the population;—supposing always that people in gene- The two following Tables, A and B, show the number of Years' Purchase Annuities on Lives of different ages are worth in present money, according to More Correct Tables of Mortality.
### TABLE A.
Without distinction of Sex.
| Column | a | b | c | d | e | f | g | h | i | |--------|---|---|---|---|---|---|---|---|---| | Description of Lives. | Nominees in the French Tontines. | Population of Carlisle and environs. | Equitable Assurance Society. | Selected by Contributors | Chosen by Lot. | The two last together. | Various combined. | Population of Sweden and Finland. | First English Tontine. | | Observations began... | 1690 | 1779 | 1770 | 1789 | 1789 | 1789 | 1773 | 1755 | 1693 | | Ended... | 1742 | 1787 | 1800 | 1826 | 1826 | 1826 | 1826 | 1776 | 1783 | | Years' duration... | 50 | 9 | 30 | 37 | 37 | 37 | 53 | 21 | 90 | | Greatest number of lives... | 9260+ | 10,517+ | 6344+ | 3518 | 4831 | 8349 | 18,798 | 4,051,116 | 1002 | | Mean number... | 5993± | 8177 | 2529± | 2860± | 3920± | 6780± | 15,459± | 2,310,160 | 501± | | Number of deaths... | 7933± | 1840 | 1220± | 1315 | 1823 | 3138 | 6,679 | 1,401,989 | 1002 | | Table of Mortality published in... | 1746. | 1815. | 1827. | 1839. | 1788. | 1829. | | Constructed by... | Deparcieux. | Milne. | Davies. | Finlaison. | Price. | Finlaison. |
Age | 10 | 19-008 | 19-585 | 19-647 | 19-167 | 19-068 | 19-118 | 19-242 | 18-891 | 17-128 | |----|----|-------|-------|-------|-------|-------|-------|-------|-------|-------| | | 15 | 18-502 | 18-956 | 18-944 | 18-475 | 18-422 | 18-448 | 18-532 | 18-336 | 16-207 | | | 20 | 17-938 | 18-363 | 18-242 | 18-011 | 17-946 | 17-979 | 17-954 | 17-603 | 15-349 | | | 25 | 17-420 | 17-645 | 17-494 | 17-526 | 17-530 | 17-528 | 17-534 | 16-839 | 14-976 | | | 30 | 16-810 | 16-852 | 16-701 | 16-889 | 16-925 | 16-907 | 16-995 | 16-006 | 14-624 | | | 35 | 16-084 | 16-041 | 15-867 | 16-098 | 16-099 | 16-099 | 16-314 | 15-138 | 14-023 | | | 40 | 15-133 | 15-074 | 14-939 | 15-195 | 15-124 | 15-160 | 15-316 | 14-034 | 13-193 | | | 45 | 13-904 | 14-104 | 13-845 | 14-061 | 13-985 | 14-023 | 14-533 | 12-959 | 12-199 | | | 50 | 12-526 | 12-869 | 12-599 | 12-671 | 12-528 | 12-599 | 13-295 | 11-658 | 11-183 | | | 55 | 11-173 | 11-300 | 11-349 | 12-163 | 11-915 | 10-320 | 10-141 | 9-772 | 8-789 | | | 60 | 9-713 | 9-663 | 10-052 | 9-772 | 10-491 | 8-789 | 8-836 | 8-308 | 7-328 | | | 65 | 8-039 | 8-307 | 8-635 | 8-308 | 8-896 | 7-328 | 7-342 | 6-729 | 5-783 | | | 70 | 6-394 | 6-709 | 7-167 | 6-729 | 7-316 | 5-783 | 5-823 | 5-122 | 4-534 | | | 75 | 4-945 | 5-239 | 5-670 | 5-122 | 5-837 | 4-534 | 4-456 | 5-122 | 4-534 |
The numbers for the two classes separately were deemed insufficient at these ages. ### TABLE B.
The Sexes distinguished.
#### RATE OF INTEREST FOUR PER CENT.
| Age | Male | Female | Excess in Value of a Female above a Male Life. | Male | Female | Age | |-----|------|--------|-----------------------------------------------|------|--------|-----| | | | | | | | | | 10 | 18-674 | 19-109 | 0-435 | 18-782 | 19-701 | 10 | | 15 | 18-105 | 18-568 | 0-463 | 18-004 | 19-059 | 15 | | 20 | 17-335 | 17-872 | 0-537 | 17-295 | 18-613 | 20 | | 25 | 16-592 | 17-087 | 0-495 | 16-940 | 18-127 | 25 | | 30 | 15-751 | 16-261 | 0-510 | 16-444 | 17-546 | 30 | | 35 | 14-812 | 15-465 | 0-653 | 15-749 | 16-880 | 35 | | 40 | 13-668 | 14-401 | 0-733 | 14-875 | 16-156 | 40 | | 45 | 12-535 | 13-383 | 0-848 | 13-798 | 15-269 | 45 | | 50 | 11-267 | 12-049 | 0-782 | 12-430 | 14-161 | 50 | | 55 | 9-998 | 10-642 | 0-644 | 11-039 | 12-790 | 55 | | 60 | 8-540 | 9-039 | 0-499 | 9-721 | 11-261 | 60 | | 65 | 7-090 | 7-566 | 0-476 | 8-216 | 9-577 | 65 | | 70 | 5-670 | 5-897 | 0-227 | 6-775 | 7-858 | 70 | | 75 | 4-487 | 4-582 | 0-095 | 5-410 | 6-264 | 75 |
The three following Tables, M, N, and O, show the Values of Annuities on Lives, according to Less Correct Tables of Mortality.—All without Distinction of Sex.
#### M.
#### N.
#### O. The sign (+) after the number of lives and of deaths in column a signifies that the real number was greater than is there stated. Those stated were the numbers of the nominees in the two tontines which commenced in 1689 and 1696, the particulars of which have been given by M. Deparcieux; but he also made all the use he could of the tontine which commenced in 1734 (less than eight years before his observations terminated), without stating any of the numbers in his essay. And whatever may have been the whole number of nominees, or of their deaths, which he availed himself of in this tontine, M. Deparcieux's observations were actually made on so many more than 9260 nominees, and 7933 deaths, among them.
The number of persons living in the two Carlisle parishes at the end of the observations was 8677; but besides them, the observations were made upon the 1840 persons who died in the place in the term of nine years during which they were continued; and these numbers together amount to 10,517, the greatest number stated in column b. But the real number the observations were made upon was greater still, by the number who left the place and did not return during the observations, which is the reason of the mark (+) being put after 10,517 in column b.
For information respecting the number of lives insured, and of deaths, in the Equitable Assurance Society, given at the head of column c, the reader is referred to the note at the end of this historical introduction.
The better to enable the reader to judge of the comparative extent of the observations made upon the nominees in tontines, and other annuitants, by M. Deparcieux and Mr Finlaison, and of those made upon the population of the two Carlisle parishes, the lives insured in the Equitable Society, and the population of Sweden and Finland, the mean number of living annuitants has been assumed to have been an arithmetical mean proportional between the numbers of them at the commencement and at the end of the term, which can only be precisely true if they died off by equal numbers in equal times; and that is the reason why the double sign (=) has been placed after the mean number of the nominees or other annuitants in each column. Thus 2860= in column d shows that the mean number of living nominees of that description was 2860 more or less. The deviation from precision in this case is of no importance.
The values in column a have been taken from the work on Annuities of Mr Baron Maseres; those in columns d and e from Mr Finlaison's report (obs. 4 and 5); that in column f at each age to 50 inclusive is a mean between those in columns d and e;—after 50 they are taken from Mr Finlaison's 5th observation. The value in column g at each age is a mean between the two against the same age in columns e and f of table B; the values in column i are from Mr Finlaison's first observation.
Of Table B.
The values in columns e and f have been taken from the 20th and 13th observations respectively in Mr Finlaison's report, and were calculated from the rates of mortality for the two sexes, which have been adopted for use by government.
They were deduced from observations on the mortality among the nominees in the three Irish tontines which commenced in 1773, 1775, and 1778 respectively, on the tontine of 1789, and those of the sinking fund from 1808 to 1822.
It will be observed that the excess of the value of an annuity on a female life above that of a male is, according to the table for Sweden, in many cases not half, and in some less than one third as much as according to Mr Finlaison's, derived from the government annuitants. The cause of this cannot but be an object of interest, and deserves further investigation. It may arise in great measure from the ages of many females being stated below the truth in the Swedish returns, while they were accurately ascertained among the government annuitants.
Of Tables M, N, and O.
The values according to Demoivre's hypothesis were taken from Dodson's Mathematical Repository (vol. ii. p. 169); those in column d of table M, founded upon Duvillard's table of mortality for France before the Revolution, published in his work on the Mortality from Small-Pox (4to, Paris, 1806), were taken from The Doctrine of Compound Interest by M. Corbaux (8vo, London, 1825); the values according to Dupré and St Cyran from the Calcul des Rentes Viagères of the latter.
The authorities for the rest appear sufficiently from the preceding historical sketch.
The values of annuities according to M. Kersseboom's table of mortality are not given here, that table being of doubtful character, as he neither published the whole of the data from which he formed it, nor explained the manner of its construction.
It would have been desirable to include the values according to the tables of Dr Halley for Breslaw, and Dupré de St Maur for the Parisian and French country parishes in table M; but as the values of annuities have not been calculated from these tables at four per cent., we have added tables N and O, and have given in each of them the values from the Northampton table, with the view of facilitating the comparison of the values in N and O with those in M and A.
Observations on the above Tables.
All the tables of mortality from which the values of annuities in tables M, N, and O, have been deduced, were calculated from bills of mortality alone of places where the population was variable, and the numbers of the people at the different ages were not ascertained. And therefore, notwithstanding the attempts to supply their defects, which were made by the eminent mathematicians who constructed them, none of them represented truly the laws of mortality in the places where the respective observations were made; as will be evident to those who understand the article on the Law of Mortality in this work, and pay the necessary attention to the materials and manner of construction of those tables. Consequently the values of annuities derived from them cannot be correct, but will in general be considerably less than the truth, even for the general average of the whole population of the places in which the observations were made.
But those values of annuities are also objectionable on this ground—that the places they were intended for, and understood to be adapted to, were generally populous towns, containing a large proportion of poor persons dependent upon their daily labour for their supply of food from day to day, often with little forethought, and many of them engaged in unwholesome employments, amongst whom great distress is often endured by the comparatively high prices of bread and potatoes, or the low rate of wages, when the unwholesome and scanty food they are reduced to produces typhus fever, and sometimes the dysentery among them, which carry them off in great numbers. And these visitations were much more common at the times when the observations were made from which most of those tables were constructed, than they have been of late years. None of those causes of mortality operate sensibly upon the general average of those persons upon whose lives leases or annuities, and reversions or assurances, depend, they being generally in the higher and middle classes. Neither do they produce much effect among the more deserving persons in the lower class, such as the members of friendly societies, and others who are both industrious and frugal enough to live within their incomes; nor indeed upon any who are in comfortable circumstances.
Hence it follows that the values of life-annuities, and consequently, those of any pecuniary interests dependent upon the continuance or the failure of human life, cannot be correctly determined from observations made on a whole population similar to those of the places these tables were constructed from.
But this was not distinctly seen till of late years, and appears to be very imperfectly understood at present (in the year 1830), even by some who might be expected to possess correct information on the subject.
The tables constructed by Dr Price, both from the Swedish observations, and those made by Dr Haygarth at Chester, threw valuable light on this subject. But deficient crops in Sweden operate powerfully in raising the mortality there, in comparison with the more fruitful parts of Europe; therefore, the values of annuities copied into table B, and col. h of table A, from Dr Price’s work, must only be understood as sufficiently correct for the period and place in which the observations were made. And the Chester table is in some degree liable to the same objections as the others above mentioned.
On Table A.
For nearly 70 years after its publication, M. Deparcieux’s table, from which the values given in col. a of table A were derived, was the only one from which the values of life-interests and of reversions depending upon lives could be determined with considerable accuracy. But the comparatively high values of annuities according to that table were always supposed to arise from the careful selection of the lives; notwithstanding that they were almost all inhabitants of Paris and its environs. At that time (1689-1696) the Parisians were much worse lives than during the last 50 years, and a judicious selection was much less likely to be made then than now.
It is to be regretted that the Carlisle observations were only continued nine years, commencing with 1779; but the less so since Mr Milne has shown in his work on annuities (p. 429), that during the term of 22 years commencing with 1779, the proportion of the annual average number of deaths to the mean number of the people was the same as in these first nine years, viz, that of 1 to 40.
In comparing the values in column e of table A with the rest, it should be borne in mind that a great majority of insured lives are males, on which account the values are somewhat lower, especially from 15 to 55 years of age, than they would have been had there been nearly equal numbers of both sexes.
Columns d and e are very instructive. In the session of 1789, an act (29 Geo. III. cap. 41) was passed for raising the sum of L1,002,500 by the sale of shares in a tontine; but the scheme did not succeed, the persons who in the first instance had taken the whole of the shares with the expectation of selling them at a profit not having been able to dispose of half of them; and to afford those persons relief, an act (30 Geo. III. cap. 45) was passed in the next session, allowing them, before the 20th September of that year (1790), to exchange each of the tontine shares they had not been able to dispose of, for an annuity-certain payable during 69½ years. And in order that those who had taken shares, and fixed upon their nominees in the tontine, might be placed in the same situation with regard to the benefit of survivorship as if the scheme had completely succeeded, this act empowered the commissioners of the treasury to select tontine nominees for the exchanged shares, from the Peers of Great Britain and Ireland, their children or grandchildren; baronets, lords of manors, justices of the peace in England and Wales, or their children; the dignitaries of the church or beneficed clergy, fellows of colleges, the governors of the Charter House, the Foundling Hospital, or Christ’s Hospital, and those who were registered in the books of the Amicable Life Assurance Society.
The commissioners of the treasury were to distribute their nominees into six classes according to their ages, in proportion to the nominees of the contributors in the same classes. Tickets with the names of the nominees were then to be put into six boxes, set apart for the respective classes, and drawn out till a sufficient number to complete the tontine was obtained for each class. All of which was performed accordingly.
The values of annuities on the lives chosen by the contributors are given in col. d, on those thus drawn by lot in col. e, and the two combined in col. f.
Thus it appears that, of the lives in col. e, there was no selection except their being taken from the descriptions of persons above mentioned, who were all in the upper or middle classes of society. But, as might have been anticipated, a considerable majority of female lives were chosen by the contributors, and a considerable majority of males were drawn by the commissioners of the treasury, the descriptions of persons they were restricted to, consisting principally of males.
It will be observed that the values in columns d and e agree very nearly—they would probably have agreed better still had the proportion of the two sexes been the same in both.
And this shows how little advantage the contributors derived from choosing their nominees, beyond what was secured to them by the classes of society they were selected from.
The values of annuities in column i are much less than in any other of table A; they are even less than those derived from the Northampton table of mortality.
But it would be equally precipitate and unphilosophical to conclude from thence, without further investigation, that the bulk of the people of England 100 years ago were so much shorter lived than they are now.
That the prolongation of life among the bulk of the population, from and after every age, has been very considerable during the last century, no unprejudiced person who has paid sufficient attention to the subject to qualify himself for judging of it can entertain a doubt; or that it has also been somewhat lengthened among the upper and middle classes of society; but not nearly to the extent which a comparison of columns g and i of table A would seem to imply.
There is certainly no rational ground for supposing that the physical constitution of man has altered: any change that has taken place can only have been produced by changes in the habits of the people, and the circumstances in which they have been placed.
All this might have been reasonably concluded in the absence of further information; but an examination of the circumstances under which the tontine of 1693 was
---
1 As M. Deparcieux states in his Essai, p. 62. In that year, the same in which Dr Halley's paper on the Breslaw Bills was published in the Philosophical Transactions, this branch of knowledge was in its infancy. By the 22d section of the act (4 William and Mary, cap. 3) for raising a million of money by the sale of shares in this tontine, it was enacted that, if the sale of the tontine shares did not produce the whole sum wanted by the first of May 1693, then between that day and the 29th September following, for any sum contributed towards the completion of the million, the contributor should receive 14 per cent. per annum on such sum during the life of any person he might choose to nominate; the common interest of money at that time being 6 per cent. per annum.
And by an act passed in the next session (5 William and Mary, cap. 5), the term for granting annuities on these terms, and for the same purpose, was extended to the 1st of May 1694. This was selling annuities at half their true value.
The age of a nominee is never mentioned in either of these acts, and those in the tontine were not distinguished into classes. These things were, at the same time, managed better in France.
Even Dr Halley was not aware of the greater mortality of males, and the consequent greater proportion of females in the population, as appears by his paper above mentioned; for after calculating from his table of mortality the number of inhabitants in Breslaw between 18 and 56 years of age to be 18,053, he says, "at least one half of these are males."
The contributors to the tontine could not be expected to be better informed on these subjects than the parliament and Dr Halley; and, with respect to ages and sexes, the following appears by Mr Finlaison's statement to have been their selection of nominees.
| Aged | Number of | |------|-----------| | | Males | Females | Both | | Under 6 | 178 | 113 | 291 | | Between 6 and 11 | 178 | 118 | 296 | | Under 11 | 356 | 231 | 587 | | Between 11 and 16 | 119 | 96 | 215 | | Under 16 | 475 | 327 | 802 | | Between 16 and 21 | 49 | 39 | 88 | | Under 21 | 524 | 366 | 890 | | Above that age | 70 | 42 | 112 | | Total | 594 | 408 | 1002 |
In the absence of better information, it would seem reasonable to conclude that the younger a life was, the longer it would be likely to last; and accordingly we find that more than half the lives were under 11 years of age at the time of their nomination. And as males are more robust than females, it was also natural to conclude that a less rate of mortality would prevail among them; and accordingly three fifths of the nominees were males.
But subsequent observations have shown that, in both instances, the contributors made a bad choice.
Besides, of all the nominees, only one ninth had completed their 21st year at the time of their nomination. Not only the constitutions of these young nominees were not then fully formed or developed, but the mortality among them would depend greatly upon their destiny in after-life, or the circumstances of their respective situations, and upon their moral conduct: all of which must History, be very uncertain, and difficult to judge of at such early ages.
It is not improbable, too, that, on account of their beauty and healthy appearance, many children of scrofulous constitutions were selected; and they, on an average, would be short-lived.
It is probable that a great majority of the contributors, and therefore of their nominees, resided in London or other large and crowded towns, which have always been peculiarly unfavourable to the health of children, but were much more so then than they are now.
All circumstances considered, there appears sufficient evidence to show that the mortality among these nominees must have been much greater than among the general average of selected lives, or the general average of the people in comfortable circumstances at that time; and that, if the nominees in the English tontine of 1693 had been distributed into classes according to their ages, and a larger annuity for the same purchase-money had been allowed to the older classes, always with benefit of survivorship, the lives would have been more judiciously chosen, and would not probably have differed materially from the nominees in the French tontines of 1689 and 1696, which M. Deparcieux's observations were made upon,—they, it has been shown, were but little inferior in goodness to our present annuitants and insured lives.
The values of annuities on single lives given in Mr Finlaison's report are only specimens at one rate of interest of the results of calculations made at several rates. And extensive calculations have also been made at the Government Life-Annuity Office, of the values of annuities on two and on three joint lives, with distinction of the sexes. But none of these have been published, nor do they now (in July 1830) at that office grant any annuity depending upon more than one life, nor expect to do so for two or three years to come.
In this country, cases are continually occurring, in which an equitable adjustment of the rights and interests of different parties in property depending upon the continuance or the failure of human life, is of great importance; and in many cases it cannot be made with sufficient accuracy without tables of the values of all possible combinations of two and even of three joint lives of different ages and of both sexes. But not unfrequently, especially in cases of contingent reversions, we have been hitherto, and are still, obliged to use approximations to the values of all the possible combinations of the lives involved, not only with each other, but also of them with others one year younger than each of them respectively. In these cases it sometimes happens that the whole value sought, being but a small part of a year's purchase, is less than the probable error of several of these approximations considered separately; and then it is very difficult to give with confidence even a near approximation to the value sought.
But in a great many other cases of frequent occurrence and less difficulty, the want of a complete set of good tables of this kind is much felt. It is therefore highly desirable that, numerous calculations for them having been made at the expense of the public, they should be completed at the public expense, and rendered accessible to persons having occasion to use them; by printing, if the expense would not be too great; otherwise, by having several manuscript copies accurately made, and deposited in convenient places for inspection, upon payment of a small fee.
A few writers on these subjects, of late years, have employed the differential and integral calculus in their in-