SKETCH OF DE MOIVRE.—HIS DOCTRINE OF CHANCES.—KERSSEBOOM.—DE PARCIEUX.—HODGSON.—DODSON.—FIRST FRAUD IN LIFE ASSURANCE—ITS ROMANTIC CHARACTER.—THOMAS SIMPSON.—CALCULATIONS OF DE BUFFON.

To the same year which witnessed the proposition for the new companies we are indebted for the work entitled the “Doctrine of Chances,” written by Abraham de Moivre, who, owing to the revocation of the Edict of Nantes, was compelled to seek shelter in England, where he perfected the studies he had commenced in his own country. In his boyhood he had neglected classics for mathematics, to the great surprise of his master, who often asked “what the little rogue meant to do with those ciphers.” In 1718, he published the first edition of the above book; and a few extracts from this, which led him afterwards to his hypothetical application of those chances to the survivorship of life, may not be unacceptable; as, though the author deemed it wise to apologise in his dedication for publishing a work which “many people in the world might think had a tendency to promote play,” yet his volume will prove the best apology. The book is very entertaining in its character, and is an evidence of an inquiring and mathematical mind employing itself upon trifling questions rather than remain idle. Thus, Case 1. is “To find the probability of throwing an ace in two throws of one die.” And this kind of problem he varied to almost every possible form. There is “the probability of throwing an ace in three throws,” of “throwing an ace in four throws,” of “two aces in two throws,” of “two aces in three throws,” worked out in a most exact and elaborate manner. From dice he proceeded to lotteries, and showed how many tickets ought to be taken to secure the probability of a prize. The volume, a considerable quarto, was nothing more than an amusing book on gambling and its various chances. But it produced a better effect. A few years later, he published something more worthy of him, in his “Doctrine of Chances, applied to the Valuation of Annuities on Lives,” in which he says, with some appearance of surprise, “Two or three years after the publication of the first edition of my ‘Doctrine of Chances,’ I took the subject into consideration; and consulting Dr. Halley’s tables of observation, I found that the decrements of life, for considerable intervals of time, were in arithmetical progression; for instance, out of 646 persons of 12 years of age, there remain 640 after 1 year; 634 after 2 years; 628, 622, 616, 610, 604, 598, 592, and 586, after 3, 4, 5, 6, 7, 8, 9, and 10 years respectively, the common difference of those numbers being 6. Examining afterwards other cases, I found that the decrements of life for several years were still in arithmetical progression, which may be observed from the age of 54 to the age of 71, where the difference for 17 years is constantly 10.”

The greatest difficulty which occurred to him was to invent practical rules that might readily be applied to the valuation of several lives, “which was, however, happily overcome, the rules being so easy that, by the help of them, more can be performed in a quarter of an hour, than by any method before extant in a quarter of a year.”

It was first published in 1725; and finding thus from Halley that, for several years together the decrement of life was uniform, it being only in youth and old age any considerable deviation took place, he founded an hypothesis that it was uniform from birth to extreme old age; in other words, that out of a given number of persons living at any age, “an equal number die every year until they are all extinct.” On this he gave a general theorem, by which the values of annuities on single lives might be easily determined. This was of great use at the time, no table of the real value of annuities having then been published, except a very contracted one founded on Halley’s paper; and if subsequent investigations proved that De Moivre was utterly wrong, his conclusions formed the basis of many a future calculation.

Although the ability of De Moivre was recognised by the Royal Society when it appointed him arbitrator in the contest betwixt Newton and Leibnitz, and although Newton, when applied to for an explanation of his own works, would often say “Go to De Moivre, he knows better than I do,” yet it is to be feared that golden opinions were won by him more freely than guineas.

It is sufficiently known that the coffee-houses of the eighteenth century were the resort of all who sought intelligence or loved the company of the wits and fine men about town. To one of these, in St. Martin’s Lane, De Moivre went, where it was customary to apply to him for the solution of many questions connected with annuities, and for answers to queries concerning games of hazard, which were propounded to him by those who hoped to turn the chance of loss into a certainty of gain. The payment of these questions was his chief mode of subsistence; and there is something unpleasant in the memory of this man, compelled, in his old age, to be at the bidding of gamesters, and to consort with men who lived on the town by their wits.

The opinion of posterity is divided upon his merits. “By the most simple and elegant formulÆ,” says Francis Baily, “he pointed out the method of solving all the most common questions relative to the value of annuities on single and joint lives, reversions, and survivorships.” The subsequent editions of his works prove that he was aware of his errors of detail, by correcting them. He enlarged the boundaries of the science which he loved, and encouraged others to follow in the same path. Although his hypothesis may not be applicable to all occasions and circumstances, and though later discoveries proved that it could not be always safely adopted, “nevertheless it is still of great use in the investigation of many cases connected with this subject, and will ever remain a proof of his superior genius and ability.” Such is the opinion of Baily on the merits of De Moivre; but it has been added by Morgan, that “on the whole the hypothesis of De Moivre has probably done more harm than good, by turning the attention of mathematicians from investigating the true laws of mortality.”

In 1737, an attempt was made to calculate the number of the people, which was estimated at 6,000,000, an amount probably not very far from the mark; as in 1688 the population was reckoned at a little over 5,000,000. Some important assistance was rendered in 1738, by the publication of Kersseboom’s tables, taken from the records of life annuities in Holland[10]; and as the ages of the annuitants had been there recorded for 125 years, they proved a considerable aid to those interested. So small was the progress made in England by 1746, that Dr. Halley’s Breslau Tables and those of M. Kersseboom were the only ones which gave anything like a representation of the true laws of mortality. In this year, however, the “Essai sur les ProbabilitÉs de la DurÉe de la Vie Humaine” of M. De Parcieux, with several valuable tables deduced from the mortuary registers of religious houses in France, and from the nominees in the French tontines, were an additional contribution to our information.

The first effort to show the value of annuities on lives from the London Bills of Mortality is attributable to James Hodgson. Nor was this endeavour uncalled for or unnecessary. Many assurance offices had arisen, undertaking to grant these annuities; and the tables principally in use were founded on the decrease of life at Breslau. But by the Breslau Tables, half the people lived till they were about 41 years of age, while in London half did not reach the age of 10. This was a vast difference in the estimate of mortality, and affected the price of annuities in a proportionate degree. But if the Breslau Tables calculated life at too high a rate, it was equally evident that the London Tables made them too low; it is obvious, therefore, that the value of a life annuity founded on any confined observations would be unsuitable to the general annuitant; and it is evident that a scale of prices should have been based on a more enlarged foundation.

The work of Mr. Hodgson deserves very great attention, and the notice of the reader is called to its investigation, as the conclusions were arrived at after great labour, and are a specimen of the time and trouble bestowed on the subject. “The easy way of raising money for public uses,” says Mr. Hodgson, “by granting annuities upon lives, has met with so great encouragement that there is no room to doubt it will be carried down to future times.” The following statements of this gentleman will be read with surprise by those who are acquainted with the chances of life as calculated at the present day. He estimated that “1000l. would purchase an annuity of 70l. per annum for a life of 29 years 10 months, when money is valued at 3 per cent. per annum; that the same sum will purchase the same annuity for a life of 23 years, when money is valued at 4 per cent. per annum; and that the same sum will purchase the same annuity for a life of 23 years, when money is valued at 5 per cent. per annum; and that it will purchase the same annuity for a life of 16 years 2 months, when money is valued at 6 per cent.

“It appears 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 7 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.”

Such was a portion of Mr. Hodgson’s contribution in 1747 to vital statistics. This work was followed in 1751 by the “Observations on the past Growth and present State of the City of London” of Corbyn Morris, containing tables of burials and christenings from 1601 to 1750. The tables were important in themselves, and the book is noticeable as containing a proposal to remodel the Bills of Mortality.

The topic was particularly interesting to mathematical men. In 1753, Mr. James Dodson pursued the subject, and solved in his “Repository” an immense variety of questions. Hitherto a table deduced by Simpson from the London Bills of Mortality, was the only one taken from real observation. But it need not be said that London was a very limited theatre on which to found the payment of premiums. The number of persons who died there in a given time, doubled that of other and more healthy cities. It was impossible to separate the casual visitors from the natives, in the record of deaths. It was equally difficult to divide those who had been born there, from those who were naturalised by virtue of a long and continued residence. The city, which has ever been the land of promise to the country, brought adventurers from the rural districts in a continued stream. The difficulties which prevented correct information from spreading may be judged by the statement that, from 1759 to 1768 a third more deaths than births were registered, the average annual burials being 22,956 to 15,910 of births. In the previous 10 years, the excess had been 10,500, or near half the burials. The baptismal registries were also very deficient in that large class denominated sectarians; Jews, Quakers, Roman Catholics, and all who refused to recognise the rites of the English Church being excluded. It required, therefore, care and calculation of no ordinary character to make any approximation to the truth; and Mr. Dodson believed he would be nearer it, by adopting the opinions of De Moivre as the ground work of his tables, rather than by entering on a sea of uncertain and hypothetical calculations.

In 1754, a further “valuation of annuities on lives,” deduced also from the London Bills of Mortality was published. By this it appeared that the work of Mr. Hodgson had not produced much effect in sending the Breslau Tables out of general use; for, says the author, “I think it very unreasonable that a poor citizen of London should be made to pay for an annuity according to the probability of the duration of life at Breslau, where, as appears from the bills of mortality, one-half of the people that are born live till they are about 41 years of age, whereas at London one-half die before they arrive at the age of 13.”

The first known fraud in assurance is one of the most singular in its annals. The reader must judge for himself of the circumstances attending it; but there is no doubt that others far more fearful in their results have since been practised.

About 1730, two persons resided in the then obscure suburbs of St. Giles’s, one of whom was a woman of about twenty, the other a man whose age would have allowed him to be the woman’s father, and who was generally understood to bear that relation. Their position hovered on the debatable ground between poverty and competence, or might even be characterised by the modern term of shabby genteel. They interfered with no one, and they encouraged no one to interfere with them. No specific personal description is recorded of them, beyond the fact that the man was tall and middle aged, bearing a semi-military aspect, and that the woman, though young and attractive in person, was apparently haughty and frigid in her manner. On a sudden, at night time, the latter was taken very ill. The man sought the wife of his nearest neighbour for assistance, informing her that his daughter had been seized with sudden and great pain at the heart. They returned together, and found her in the utmost apparent agony, shrinking from the approach of all, and dreading the slightest touch. The leech was sent for; but before he could arrive she seemed insensible, and he only entered the room in time to see her die. The father appeared in great distress, the doctor felt her pulse, placed his hand on her heart, shook his head as he intimated all was over, and went his way. The searchers came, for those birds of ill-omen were then the ordinary haunters of the death-bed, and the coffin with its contents was committed to the ground. Almost immediately after this the bereaved father claimed from the underwriters some money which was insured on his daughter’s life, left the locality, and the story was forgotten.

Not very long after, the neighbourhood of Queen Square, then a fashionable place, shook its head at the somewhat unequivocal connection which existed between one of the inmates of a house in that locality, and a lady who resided with him. The gentleman wore moustaches, and though not young, affected what was then known as the macaroni style. The lady accompanied him everywhere. The captain, for such was the almost indefinite title he assumed, was a visitor at Ranelagh, was an habituÉ of the Coffee-houses, and being an apparently wealthy person, riding good horses and keeping an attractive mistress, he attained a certain position among the mauvais sujets of the day. Like many others at that period, he was, or seemed to be, a dabbler in the funds, was frequently seen at Lloyd’s and in the Alley; lounged occasionally at Garraway’s; but appeared more particularly to affect the company of those who dealt in life assurances.

His house soon became a resort for the young and thoughtless, being one of those pleasant places where the past and the future were alike lost in the present; where cards were introduced with the wine, and where, if the young bloods of the day lost their money, they were repaid by a glance of more than ordinary warmth from the goddess of the place; and to which, if they won, they returned with renewed zest. One thing was noticed, they never won from the master of the house, and there is no doubt, a large portion of the current expenses was met by the money gambled away; but whether it were fairly or unfairly gained, is scarcely a doubtful question.

A stop was soon put to these amusements. The place was too remote from the former locality, the appearance of both characters was too much changed to be identified, or in these two might have been traced the strangers of that obscure suburb where as daughter, the woman was supposed to die, and as father, the man had wept and raved over her remains. And a similar scene was once more to be acted. The lady was taken as suddenly ill as before; the same spasms at the heart seemed to convulse her frame, and again the man hung over her in apparent agony. Physicians were sent for in haste; one only arrived in time to see her once more imitate the appearance of death, while the others, satisfied that life had fled, took their fees, “shook solemnly their powdered wigs,” and departed. This mystery, for it is evident there was some collusion or conspiracy, is partially solved when it is said, that many thousands were claimed and received by the gallant captain from various underwriters, merchants, and companies with whom he had assured the life of the lady.

But the hero of this tradition was a consummate actor; and though his career is unknown for a long period after this, yet it is highly probable that he carried out his nefarious projects in schemes which are difficult to trace. There is little doubt, however, that the soi-disant captain of Queen Square was one and the same person who, as a merchant, a few years later appeared daily on the commercial walks of Liverpool; where, deep in the mysteries of corn and cotton, a constant attender at church, a subscriber to local charities, and a giver of good dinners, he soon became much respected by those who dealt with him in business, or visited him in social life. The hospitalities of his house were gracefully dispensed by a lady who passed as his niece, and for a time nothing seemed to disturb the tenour of his way. At length it became whispered in the world of commerce, that his speculations were not so successful as usual; and a long series of misfortunes, as asserted by him, gave a sanction to the whisper. It soon became advisable for him to borrow money, and this he could only do on the security of property belonging to his niece. To do so it was necessary to insure their lives for about 2000l. This was easy enough, as Liverpool, no less than London, was ready to assure anything which promised profit, and as the affair was regular, no one hesitated. A certain amount of secresy was requisite for the sake of his credit; and availing himself of this, he assured on the life of the niece 2000l. with, at any rate, ten different merchants and underwriters in London and elsewhere. The game was once more in his own hands, and the same play was once more acted. The lady was taken ill, the doctor was called in and found her suffering from convulsions. He administered a specific and retired. In the night he was again hastily summoned, but arrived too late. The patient was declared to be beyond his skill; and the next morning it became known to all Liverpool that she had died suddenly. A decorous grief was evinced by the chief mourner. There was no haste made in forwarding the funeral; the lady lay almost in state, so numerous were the friends who called to see the last of her they had visited; the searchers did their hideous office gently, for they were, probably, largely bribed; the physician certified she had died of a complaint he could scarcely name, and the grave received the coffin. The merchant retained his position in Liverpool, and bore himself with a decent dignity; made no immediate application for the money, scarcely even alluding to the assurances which were due, and when they were named, exhibited an appearance of almost apathetic indifference. He had, however, selected his victims with skill. They were safe men, and from them he duly received the money which was assured on the life of the niece.

From this period he seemed to decline in health, expressed a loathing for the place where he had once been so happy; change of air was prescribed, and he left the men whom he had deceived, chuckling at the success of his infamous scheme.

It need not be repeated, that the poverty-stricken gentleman of the suburbs, the gambling captain of Queen Square, and the merchant of Liverpool, were identical. That so successful a series of frauds was practised appears wonderful at the present day; but that the woman either possessed that power of simulating death, of which we read occasional cases in the remarkable records of various times, or that the physicians were deceived or bribed, is certain. There is no other way of accounting for the success of a scheme which dipped so largely into the pockets of the underwriters.

The next movement in the scientific annals of life assurance was made by Thomas Simpson, a natural and self-taught mathematician, whose life prior to throwing himself on the world of London for support had been somewhat of a vagrant one. He had cast rustic nativities, told fortunes, advanced courtships, and occasionally varied his vagabondism by undertaking to raise the devil, an attempt in which he was so successful, that he sent his pupil mad, and was obliged himself to leave the village. In 1740, he produced a volume “On the Nature and the Laws of Chance;” in 1742, this was followed by his “Doctrine of Annuities and Reversions,” deduced from general and evident principles, with tables showing the value of joint and single lives. In 1752, he made an additional contribution to the statistics of annuities, as he published in his “Select Exercises” a supplement, wherein he gave new tables of the values of annuities on two joint lives, and on the survivor of two lives, more copious than hitherto. He first attempted to compute the value of joint lives; but as these were still taken from the London Bills of Mortality, they were by no means fit for general acceptance. He treated his subject, however, more broadly and clearly than it had been previously treated, giving some of the best tables of the values of life annuities, which were published for many years. Though the manner in which they might be computed had been shown by Dr. Halley, it is to the self-taught Simpson we are indebted for their practical application.

In 1760, M. Buffon published a further contribution to the statistics of assurance, in a table of the probabilities of life, estimated from the mortality bills of three parishes in Paris, and two country parishes in its neighbourhood.

The following are some of his calculations:—“By this table,” says the author, “we may bet 1 to 1 that a new-born infant will live 8 years; that a child of one year old will live 33 years more, that a child of full two years old will live 33 years and 5 months more, that a man of thirty will live 28 years more; that a man of forty will live 22 years longer, and so through the other ages.”

Buffon adds, “The age at which the longest life is to be expected is 7, because we may lay an equal wager, or 1 to 1, that a child of that age will live 42 years and 3 months longer. That at the age of twelve or thirteen, we have lived a fourth part of our life, because we cannot reasonably expect to live 38 or 39 years longer; that in like manner at the age of 28 or 29, we have lived one-half of our life, because we have but 28 years more to live; and lastly, that before fifty we have lived three-fourths of our life, because we can hope but for 16 or 17 years more.”

Some profound moral reflections followed these estimates; and as a critic of the day “thought all serious remarks out of place in an arithmetical calculation, and that M. Buffon had better reserve them for his book on beasts,” the reader will not be troubled with their repetition. He will not, however, be displeased to read the remarks on this table, by one of the annotators of the day.

“For insuring for 1 year the life of a child of three years old we ought to pay 10 per cent., for as it has by M. Buffon’s table an equal chance of living 40 years, it is 40 to 1 that it does not die in a year. In the same manner we ought to pay but 3 per cent. for insuring for 1 year the life of a lad of nineteen or twenty; but 4 per cent. for insuring for 1 year the life of a man of thirty-five; and 5 per cent. per annum for insuring for 1 year the life of a man of forty-three; after which the insurances ought to rise above 5 per cent. in proportion to the advance of a person’s age above forty-three. So that a man of seventy-seven ought to pay 25 per cent., and a man of eighty-five 33 1/2 per cent. for insuring his life for 1 year.”