examples and instructions for valuing of single lives; two or more lives; lives taken in with other lives; annu. in expectation; an instance in the Mercers Co.; estates for any certain term of years, as freeholds, leaseholds, and reversions, without any decimals, etc. The whole being made easy to a common capacity.

There is nothing in this ed. calling for special remark.

In the same year M. De Parcieux pub. his famous essay on the probable duration of human life, in which he gave several valuable tables of mort. deduced from the Mortuary Registers of different religious houses in France, and from the lists of the nominees in the French Tontines; also a table of the Values of annu. on single lives, at three rates of int., calculated from his table of mort. for the tontine annuitants. The work commences with an algebraical theory of annuities certain; and became very popular.

We give here the annuity values deduced from his table, for quinquennial periods: [DE PARCIEUX's Table of Mort.]

In 1747 Mr. James Hodgson, F.R.S., pub.: The Valuation of Annuities upon Lives deduced from the London Bills of Mort. The work is chiefly of a demonstrative character in opposition to merely imaginary methods; and the annuity values are extended to four places in decimals. He says by way of preface:

It is now some years since I undertook the trouble of computing the following Tables, and the principal motive that induced me to do it was, that at that time the Tables most in use were founded upon the Bills of Breslaw.

TABLE-4 p.c.

Age.

Years' Purchase.

19'008

15

18.502

20

17.938

25

17'420

30

16.810

35

16.084

40

15'133

45

13'904

At Breslaw, one half of the people that are born live till they are about 41 years of age; at Lond. one-half die before they arrive at the age of ten years, which must necessarily make a considerable alteration in the chances of life, as well as in the value of the annu. The method of putting down the ages of the several classes of people that die within the bills of mort. (for which we are indebted to Mr. Valens Comyn), has furnished us with ample matter to found the computations upon.

The easy way of raising money for publick uses by granting annu. upon lives, has met with so great encouragement, that there is no room to doubt that it will be carried down to future times.

The frequent entails upon estates by wills; the granting of leases upon the lives of persons of different ages and differently interwoven, have rendered a true estimate of the values of lives, according to the present circumstances of the times, of very great consequence; and to this end the utmost care and pains have been taken to render the tables true and exact; and if any mistake may have happened (which is not impossible in a work that requires such a vast number of calculations), it is to be hoped the candid reader will pass a favourable censure upon it.

After having explained in detail the methods by which his values were deduced, he says:

It is remarkable that the highest value of a life is when the person is about 6 years of age, and that from the birth to that time the value of lives decrease, as they do from that time to the utmost extremity of old age; that a life of 1 year old is nearly equal in value to a life of 25 years old; that a life of 2 years old is nearly equal in value to a life of 77 [years] old; that a life of 3 years old is nearly equal in value to a life of 12 years old; that a life of 4 years old is nearly equal in value to a life between 9 and 10 years; and that a life of 5 years is nearly equal in value to a life of 7 years of age; and hence arose the custom of putting the value of the lives of minors upon the same value with those of a middling age, which at the best is but a bold guess, and made use of for no better reason, than that they knew of no better way to find the true value.

The above is an abstract of his Tables of life annu. values, 4 p.c. We shall have occasion to notice this work again under B. OF MORT., and LOND. T. OF MORT.

In 1748 Mr. James Dodson, F.R.S., pub. the 1st vol. of the Mathematical Repository, in which was contained various problems and solutions bearing upon annu. He strongly recommended the hypothesis of De Moivre as the most eligible means for determining the value of annu. until the results derivable from the Lond. bills should be more worthy of confidence (see again 1750 and 1755).

In the great usury case of Chesterfield v. Janssen, heard before Lord Hardwicke in 1750, it was said by counsel: "This extravagance [of borrowing] has estab. a trade of annu. and post-obits, universally exclaimed against. The ruin of a man who falls into this method is declared not to be far off: he ruins his estate without spending half." We shall have occasion to notice this case further in other parts of this work.

A 3rd ed. of De Moivre's work on annu. was printed in 1750. There were no material alterations from the 2nd ed.

In 1751 Mr. Weyman Lee pub. a valuation of annu. and leases certain for a single life. In this work he attempted to refute the various charges of error, etc., which his former

publication (1737) had elicited. It seems but fair to hear what he has to say after the benefit of the controversy which he had raised :

Some time ago, I pub. an essay on the method of ascertaining the value of annu., and of leases reduced to annu. certain for one or more lives; and therein I laid down this position as the groundwork, viz., that the even chance of the duration of a life, computed once only for the whole life, was the sole true measure of the value of an annu. depending on that life. But since some persons have made obs. on this my method, and others have form'd valuations of these estates, and have defended and still adhere to the method prescribed by Dr. Halley, I think it will not be amiss to make some remarks upon them, partly to answer and remove the objections which have been offer'd, but chiefly, because it gives me an opportunity to maintain, and further confirm the fundamental position. Amongst my adversaries, Mr. H. B. appears to be most potent, and indeed the only one who deserves any serious consideration; and the question between him and me is, whether Dr. Halley's rule or mine be the righter for forming an estimate of the value of annu. on a single given life. The Dr.'s rule is to find the value of the annu. for each year of the given life to an 100: which is done and can be no otherwise done but by computing the chances of mort. arising in each year, and deducting the value thereof out of the value of an annu. absolute for that year; and then to put together all the values so collected, and the sum total is the value of the annu, for the given life. Mine is, to find to what term of years any life is equal, or an even chance may prob, continue; and the value of an annu, for the given life will be the same as the value of an annu. for that term of years.

It may be said of this writer; that he was a gentleman who refused to be convinced. [MORT. OBS.]

In 1752 Mr. Simpson pub. his Select Exercises, being a supplement to his doctrine of annu.; wherein he gave new tables of the values of annu. 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 were also calculated from his Lond. table of mort.

In the same year the 4th ed. of De Moivre's Treatise on Annu. was pub. It contained two extra tables for single life annu. at the rate of 3 and 3 p.c. int. ; and some directions as to the employment of int. tables in general.

In 1753 a 2nd vol. of the Mathematical Repository was pub. ; and it contained a numerous collection of annu. questions resolved in accordance with De Moivre's hypothesis. Mr. Dodson, its editor, had calculated to 3 places of decimals a complete set of annu. tables for single lives, at the several rates of 3, 3, 4, 4, 5, and 6 p.c. int., thus incidentally offering the fullest embodiment of De Moivre's hypothesis extant.

In 1753 Mr. James Hardy pub. A Complete System of Int. and Annu., Founded upon New, Easy, and Rational Principles; containing the whole bus. of int. and annu. for a fixed time, in perpetuity, or depending on single lives or any number of joint or successive lives either in possession or reversion, with a great variety of questions, etc., "useful for landed and monied men, lawyers, stewards, and all such as have any concern with annu." In the questions so proposed to be discussed is mentioned "Insurances on Lives”—almost the first time any work had made distinct mention of the bus. ; but there was nothing in the work itself to justify the expectation of the title-page on that subject. Mr. Hardy may be properly called a disciple of Weyman Lee's.

In 1754 Mr. Stonehouse pub. The Valuation of Annuities on Lives deduced from the Lond. Bills of Mort. He says:

My principal motives to undertake the calculating a fresh set of tables from the Lond. Bill of Mort. (which most certainly are the properest foundation for the city of Lond. as the Bills of any other place would also be for that place) were, that the state of the bills of 20 years hereunto annexed, affords conclusions somewhat different from those formerly obtained from the same bills of 7, 8, or 10 years; that all the tables founded upon the Lond. Bills (except Mr. Simpson's, in which there is an alteration made from the bills by increasing the number of the living in all the ages under 25) are pub. without practical rules, or with such as are false, uncertain, and inconclusive; and I think it very unreasonable that a poor citizen of London should be made to pay for an annu. according to the prob. of the duration of life at Breslaw, where, as appears from the bills, onehalf of the people that are born live till they are about 43 years of age, whereas at Lond. one-half die before they arrive at the age of 13.

Age.

Annu. for each £100.

Age.

Annu. for each £100.

I

8.1566 50

10.8576

5

6:5703

55

11.6414

6.6445

60

12.8205

15

7'0423

14'5138

20

7.6278

16.6113

25

8.1037

75

19.8413

30

8.5689

24'9376

35

9'1075 85

35.8423

40

9.6628

72.4638

45

10.1823

The annexed is an abstract of his table of annu. values, 5 p.c. deduced as stated.

In the same year Mr. Dodson addressed a letter to the Royal Society, Concerning the Value of an Annu. for Life, and the Prob. of a Survivorship; in which he refers to De Moivre's calculation of the proportionate ann. payment in relation to the exact date of death, and offers a simple method for determining the value. This was pub. in Phil. Trans. In 1754 also Mr. Edmond Hoyle pub. An Essay towards making the Doctrine of Chances easy to those who understood vulgar mathematics only. To which was added some useful tables on annu. for lives, viz.:

1. The value of an annu. of £1 a year upon a single life, int. of money at 3 p.c. calcu

lated according to the Lond. Bills of Mort. 2. Ditto at 4 p.c. 3. Ditto at 3 p.c. ac

cording to the Breslau Table. 4. Ditto, 4 p. c. Then follow some interest tables.

In 1755 the 3rd vol. of Dodson's Mathematical Repository was pub., containing further problems relating to annu. on lives, many prob. relating to leases for lives, and

finally some questions relating to Life Ins.-the first which had distinctly treated of the subject; see LIFE INS., HIST. of.

The 3rd ed. of De Moivre's Doctrine of Chances, pub. in 1756, after the author's death, again contained a species of appendix, relating to life annu.

In 1756 also Mr. Dodson contributed to the Phil. Trans. a Table of Annu. Values based upon a Compound Table of Mort. constructed by Dr. Brackenridge from the Breslau and Lond. Bills of Mort.

In 1757 all subs. of £100 to the Loan on 3 p.c. annu. then raised, were allowed by way of bonus a life annu. of £1 2s. 6d. ; and a like annu. in respect of each add. £100 subs., irrespective of age. These lives were included in Mr. Finlaison's Obs., 1829.

In the same year Richard Gaddesbury pub. a Treatise on Decimal Arithmetic, in which was contained some obs. on annu. for years certain, and also for lives.

In 1760 the learned Euler gave a formula by which the value of an annu. on a single life of any age may be derived from that of an annu. 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 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 mort.Milne. He (Euler) pointed out the great necessity that asso. granting annu. should be founded on sound principles, and managed with prudence.

In this same year a writer in the Gentleman's Mag., with Buffon's table before him, said:

Then with regard to the purchase or sale of annu. for life, we may from this table, and the tables of compound int., easily see what a person of any age ought to pay for an annu. for life; because in this table we may see what number of years a person of any age has an equal chance to live, and in the tables of compound interest we may see what is the present value of any annu, for that number of years at the common rate of int. Thus, a person of 30 has by this table an equal chance to live 28 years, and by the table of compound int. we may see that the present value of £1 p.a. for 28 years, reckoning int. at 3 p.c., is a little above £18 15s. Therefore a person of that age ought to pay, at the present low rate of int., nearly 19 years' purchase for an annu. for life: whereas if the common rate of int. were still at 5 p.c. he ought to pay full 15 years' purchase; and as there are always more sellers than buyers, the common price was generally under this rate.

The fallacies which lurk in the above were fully pointed out by a Mr. W. Chapple, in the following number of the Mag.

In 1762 the Equitable So. was founded, and appears upon the authority of Dr. Price to have entered upon annu. bus. He says, "It assu. any sums or rev. annu. on any lives, for any number of years, as well as for the whole continuance of the lives, at rates settled by particular calculation." We believe that in later years the So. has not transacted any annu. bus.

66

In 1762 Mr. Benjamin Webb pub. The Complete Annuitant, consisting of Tables of Int. Simple and Compound; which were declared by their author to be the most complete, extensive, and useful set hitherto pub.," and to be constructed on a true and familiar plan. [MORT. TABLES.] Martin pub., in his work on Decimal Arithmetic, Tables of Among the latter a reprint of Halley's table of values of annu.

In 1763 Mr. Benjamin compound int. and annu. on lives.

About 1763 an Annu. So. was founded in the City of Salisbury, for the purpose of raising a fund sufficient to allow the widow of every person who had been a member three years an annuity of £30 during life.

In 1765 Herr Lambert pub. in Germany the first part of a treatise on annu. The third part was not pub. until 1772. [MORT. TABLES.]

On the 1st January, 1765, the Lond. Annu. So. was founded, with a view to make ann. provision for widows. The contribution was to be £5 5s. p.a., payable half-yearly; and for this an annu. of £20 was to be given to every widow during widowhood, provided her husband lived one year after admission; of 30 if the husband live 7 years; or £40 if he lived 15 years. There was a £5 5s. admission fee for every member not exceeding 45 years of age; with £5 5s. extraordinary for every year above 45.

In 1766 the Laudable So. of Annuitants was founded. The subs. was the same as in the London. If the husband lived but one year, the widow was entitled to an annu. of £10; if 2 years, £15; if 3 years, £20; 4 years, £25; 7 years, £30; 10 years, £35; 13 years, £40. No member was admitted beyond 45 years of age. The entrance fee was £55.; there were no other payments.

Dr. Price in the first instance (1771), and Mr. Wm. Dale in 1772, drew very marked attention to the promised advantages held forth by each of these sos. They proved to demonstration that the annu. proposed to be granted were much larger than the subs. justified; and by means of the controversy which ensued the science of life contingencies became in some measure popularized. This subject is fully discussed later in this art. In this year the Gov. raised a further sum by the sale of life annu. and a lottery. There were no special features. Some of the lives of this year were included in Mr. Finlaison's Obs. No. 2 (1829).

A parliamentary return pub. in 1767 showed that the nine-shillings life annu. given by way of bonus on lottery tickets in 1745 had orig. involved an ann. charge on the Treasury

of £22,500, reducible as the annuitants died off. The charge was then reduced to 17,314. The eighteen-shillings annu. which constituted the lottery bonuses in 1746 involved an orig. charge of £45,000 p.a. That had become reduced to £34,465. The £1 2s. 6d. bonus on the 3 p.c. annu. of 1757 had involved an ann. charge of £33,750; but it had become reduced by lives falling in to £31,558.

In 1768 Mr. J. Rowe, of Exeter, pub. Letters relative to Societies of Annuitants, etc., by which the members were informed of the insufficiency of their payments in relation to the advantages promised by the founders and expected by the subs.

In 1769 was founded the So. for Annu. Encreasing to the Survivors: in other words, a scheme of Tontine annu. The So. had some special features, but a complete outline of it will be given under its alphabetical title.

About this period there were estab. in Lond., in add. to the sos. we have particularly mentioned, the following: The Amicable So. of Annuitants; The Provident So.; The So. of Lond. Annuitants; The Equitable So. of Annuitants; The Westminster Union So.; The London Union So.; The Consolidated So.; The Public Annu. So.; The Rational So.; The Friendly So. of Annuitants; and prob. various others whose names are now lost or forgotten. Similar sos. were also founded in the Provinces. We have the articles of one estab. at Romsey in Hants, in 1770. The So. was founded at the Dolphin Inn, and its principal art. were printed for circulation. Members were to be above 21 and under 40 years of age. There was to be an entrance fee, commencing at £2 25. for the early members, and increasing to £5 5s. or more when a 100 members had been obtained. The ann. subs. was to be £3, payable quarterly, with small extra payments to Sec. and Treasurer, and 2s. 6d. towards the "annual feast.' The advantages promised in return were the following:

VII.-That the widow of every person who shall have been a member of the So. for the space of three years, and shall have regularly paid his quarterly and ann. payments during that time, shall be entitled to receive from the stock of the said So. an annu. or clear yearly sum of £30 during her widowhood, provided she shall have been lawfully married six calendar months before his death; which annu. shall be paid by equal sums payable quarterly. But if any such widow shall be married again to any person of good character (not being a soldier or seaman), and he shall be willing to become a member of the So. and shall notify the same to the Treasurers at the next monthly meeting after such marriage, he shall be admitted a member upon paying the then settled entrance fee; and whenever he shall happen to die, such widow shall be restored to her annuity of £30; and in case he shall happen to survive her, he shall enjoy the same advantages as any other member, but in no other degree. VIII. That every member when he shall have been admitted full 15 years, and shall have in every respect complied with the estab. rules of the So., shall be entitled to receive from the said fund an annu. or yearly payment of £15 in case such member be then unmarried, and at least 45 years of age; or in case he shall afterwards become a widower, which annu. shall commence, etc., and be paid quarterly. [The member might relinquish his personal annu., and any future wife, over 40, would be entitled on her widowhood to receive an annu. of £30. The member to continue his subscriptions.]

By art. No. ix., if the current income of the So. was not sufficient to provide the annu., the members to pay such add. quarterly or ann. sums as would make them good; and by art. xiv., members might pay double subs., etc., and secure double advantages. Mr. Baily, writing in 1813, and clearly referring to this period, says:

Nothing perhaps tended so much to destroy the numerous Bubble Societies which spang up about forty years ago: nothing, probably, opened so effectually the eyes of the public with respect to their delusive schemes, as the pub. of more correct and comprehensive tables of the values of life annuities, whereby the true value, which ought to be given in such cases was more accurately determined. And as one improvement in science generally leads on to another, this naturally opened the way to a more complete and comprehensive investigation of the subject; so that at the present day a new, a distinct, and an interesting branch of analysis has arisen, which was unknown to mathematicians of a former period.

About this period also the two following adv. appeared very frequently in the daily

newspapers:

Eleven p.c. for life given to all ages.-Any persons desirous of purchasing annu. for their own or any other life or lives, may have the same secured on Freehold Estate, and regularly paid at a Banker's in London. For farther particulars apply to W.R., at Tom's Coffee-house, Cornhill.

Eleven p.c. for life given to all ages.-Annuity office, numb. & Pope's Head Alley, Cornhill,Persons inclined to purchase annu. for their own lives, or the lives of others, of any age, may have the same well secured, and regularly paid half yearly, at a Banker's in Lond. Further particulars may be had by applying personally or by letter to this office.

We have now to mark another, and an important epoch in the hist. of life annu. This dates from 1771, the year in which Dr. Price-the compiler of the NORTHAMPTON TABLE of Mort.-pub. the first ed. of his famous work: Obs. on Reversionary Payments; on Schemes for Providing Annu. for Widows, and for Persons in Old Age; on the Method of Calculating the Values of Assu. on Lives, etc.; to which was added, "Four Essays on different subjects in the doctrine of life annu. and political arithmetic. The date of this work is erroneously given by some writers as 1762; by several others as 1769. The first ed. now lies before us dated 1771.

Dr. Price, in the preface to the work just named, tells us how his attention became drawn to the subject:

A few years ago, many gentlemen of the first eminence in the law formed themselves into a society for providing annu. for the widows of all such persons in judicial offices, barristers, civilians, and solicitors, as should choose to become members. A plan was agreed upon and printed; but some doubts happening to arise with respect to it, the directors resolved to ask the opinion and advice of

three gentlemen, well known for their skill in calculation. This occasioned a further reference to me; and the issue was, that the plan being found to be insufficient, the whole design was laid aside. About the same time, several other societies were formed with the same views; but all on plans alike improper and insufficient. Finding, therefore, that the public wanted information on this subject, I was led to undertake this work, imagining that it might be soon finished, and that all I could say might be brought into a very narrow compass. But in this I have been much mistaken. A design which I at first thought would give little trouble, has carried me far into a very wide field of inquiry; and engaged me in many calculations that have taken up much time and labour. I shall, however, be sufficiently rewarded for my labour, should it prove the means of preventing any part of that distress, which is likely to be hereafter produced by the sos. now subsisting for the benefit of widows. I have proved the inadequateness of their plans, by undeniable facts and mathematical demonstration. I have further given an account of some of the best plans, which are consistent with sufficient prob. of permanency and success. Should, therefore, any of these sos. determine to reform themselves, or should any institutions of the same kind be hereafter established, they will here find direction and assistance.

He adds:

The general disposition, which has lately shown itself to encourage these sos. is a matter of the most serious concern, and ought I think to be taken under the notice of the Legislature. The leading persons amongst the present members will be the first annuitants; and they are sure of being gainers: and the more insufficient the scheme is on which the so. is formed, the greater will be the gains of the first annuitants. The same principle, therefore, that has produced and kept up other bubbles has a tendency to preserve and promote these; and for this reason, it is to be feared, that in the present case no arguments will be attended with any effect.

He adds the following foot-note by way of confirmation to the preceding :

This apprehension has been verified by fact. At the beginning of last winter a letter was pub. to the Provident So. containing a clear proof of the insufficiency of the plans of all these sos. It was at least to be expected that such a pub. would prevent the rise of new sos., formed on more inadequate plans. But this was so far from being the effect, that soon afterwards a so. sprung up which calls itself the Rational Annu. So.; and which, though it does not take half the values of the annuities it promises, has had the shamelessness to assure the public that it is formed on a plan incontestably durable. The Consolidated, the Public Annuitant, and the Westminster Union Sos. are yet worse inst. which have been since formed; and there may, for ought I know, be many more; for indeed all Lond. seems now to be entering into asso. of this kind.

He returns to his text, and continues:

The consideration that "the gain made by some of these sos. will be so much plunder taken from others," ought immediately to engage all to withdraw from them who have any regard to justice and humanity; but experience proves, that this argument, when opposed to private interest, is apt to be too feeble in its influence.

It cannot be said with precision how long these sos. may continue their payments to annuitants after beginning them. A continued increase, and a great proportion of young members, may support them for a longer time than I can foresee. But the longer they are supported by such means, the more mischief they must occasion. So a tradesman, who sells cheaper than he buys, may be kept up for many years by increasing bus. and credit; but he will be all the while accumulating distress; and the longer he goes on, the more extensive ruin he will produce at last.

It is impossible that any one could have spoken more earnestly, and more to the point than this. He proceeds to give the reader an outline of the contents of the work:

In the first essay I have made many obs. on the expectations of lives, the pernicious influence of great towns on health and manners and pop.; the increase of mankind; and other subjects in the doctrine of annu. and political arithmetick. In the last essay I have stated carefully the proper method of forming tables of the prob. of human life, from given obs., and in the appendix, besides several new tables, I have thought it necessary to give Mr. Simpson's Tables of the Values and expectations of Lond. lives; and all the other tables which can be wanted in the perusal of this work. The plan of the work, and its method, may be gathered by those who are not already familiar with these, by the following:

Question 1. A set of married men enter into a so. for securing annu. to their widows. What sum of money, in a single present payment, ought every member to contribute, in order to entitle his widow to an annu. of £30 p.a. for her life, estimating int. at 4 p.c.?

Answer. It is evident that the value of such an expectation is different, according to the different ages of the purchasers, and the proportion of the age of the wife to that of the husband. Let us then suppose that every person in such a so. is of the same age with his wife, and that, one with another, all the members when they enter may be reckoned 40 years of age, as many entering above this age as below it.

He then reasons out the value as demonstrated by the hypothesis of De Moivre, and the table of Simpson; and he offers the following obs. regarding the labours of these gentlemen :

of life.

Mr. De Moivre has calculated the values of single lives on the supposition of an equal decrement of life thro' all its stages till the age of 86, which he has considered as the utmost possible extent This hypothesis eases very much the labour of calculating the values of lives; and it is so conformable to Dr. Halley's Table of Obs. that there is little or no reason for distinguishing between the values of lives as deduced from this table, and the same values deduced from the hypothesis.

In order to avoid putting the reader to trouble, I have given this table at the end of this work, and I have also given two other tables, which I have formed from the B. of Mort. at Northampton and Norwich. These last tables answer more nearly to Mr. De Moivre's hypothesis than even Dr. Halley's table; and the difference between the values of single and joint lives by the hypothesis, and the same values computed strictly from the tables, is generally less in these tables than in Dr. Halley's, as will be shown in the last essay. When, therefore, in the course of this work the values of single and joint lives are mentioned, as given agreeably to Dr. Halley's, it must be understood that they are taken from Tables VI. and VII. [De Moivre's single and joint lives] in the appendix, and given in strict agreement only to the hypothesis; that for this reason they are still more conformable to the Northampton and Norwich Tables.

It was in this quiet manner that the NORTHAMPTON TABLE, which has since played