PREFACE.
IN order to explain the particular object of this Treait will be necessary to give a brief account of the science on which it treats.
At the end of the seventeenth century, the theory of probabilities was contained in a few isolated problems, which had been solved by Pascal*, Huyghens, James Bernoulli, and others. They consisted of questions reto the chances of different kinds of play, beyond which it was then impossible to proceed : for the difof a question of chances depending almost enupon the number of combinations which may arise, the actual and exact calculation of a result be-comes exceedingly laborious when the possible cases are numerous. A handful of dice, or even a single pack of cards, may have its combinations exhausted by a mode-rate degree of industry : but when a question involves the chances of a thousand dice, or a thousand throws with one die, though its correct principle of solution would have been as clear to a mathematician of the sixcentury as if only half a dozen throws had been considered ; yet the largeness of the numbers, and the
Un probleme relatif aux jeux de hasard, propose a un austere jansepar un homme du monde, a ete l'origine du calcul des probabilites. Poisson.
A 4
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consequent length and tediousness of the necessary operations, would have formed as effectual a barrier to the attainment of a result, as difficulty of principle, or want of clear perception.
There was also another circumstance which stood in the way of the first investigators, namely, the not havconsidered, or, at least, not having discovered, the method of reasoning from the happening of an event to the probability of one or another cause. The questions treated in the third chapter of this work could not therefore be attempted by them. Given an hypothesis presenting the necessity of one or another out of a certain, and not very large, number of consequences, they could determine the chance that any given one or other of those consequences should arrive ; but given an event as having happened, and which might have been the consequence of either of several different causes, or explicable by either of several different hypotheses, they could not infer the probability with which the happening of the event should cause the different hypo-theses to be viewed. But, just as in natural philosophy the selection of an hypothesis by means of observed facts is always preliminary to any attempt at deductive discovery ; so in the application of the notion of probability to the actual affairs of life, the process of reasoning from observed events to their most probable antecedents must go before the direct use of any such antecedent, cause, hypothesis, or whatever it may he correctly termed. These two obstacles, therefore, the mathemadifficulty, and the want of an inverse method, pre-vented the science from extending its views beyond problems of that simple nature which games of chance present. In the mean time, it was judged by its fruits;
PREFACE vii
and that opinion of its character and tendency which is not yet quite exploded, was fixed in the general mind.
Montmort, James Bernoulli, and perhaps others, had made some slight attempts to overcome the mathemadifficulty ; but De Moivre, one of the most pro-found analysts of his day, was the first who made decided progress in the removal of the necessity for tedious operations. It was then very much the fashion, and particularly in England, to publish results and conmethods ; by which we are left without the know-ledge of the steps which led De Moivre to several of his most brilliant results. These however exist, and when we lock at the intricate analysis by which Laplace obthe same, we feel that we have lost some imlinks * in the chain of the history of discovery. De Moivre, nevertheless, did not discover the inverse method. This was first used by the Rev. T. Bayes, in Phil. Trans. liii. 370.; and the author, though now almost forgotten, deserves the most honourable rememfrom all who treat the history of this science.
Laplace, armed with the mathematical aid given by De Moivre, Stirling, Euler, and others, and being in possession of the inverse principle already mentioned, succeeded both in the application of this theory to more useful species of questions, and in so far reducing the difof calculation that very complicated problems may be put, as to method of solution, within the reach of an ordinary arithmetician. His contribution to the science was a general method (the analytical beauty and power of which would alone be sufficient to give him a high rank among mathematicians) for the solution of
The same may be said of several propositions given by Newton.
PREFACE.
all questions in the theory of chances which would otherwise require large numbers of operations. The instrument employed is a table (marked Table I. in the Appendix to this work), upon the construction of which the ultimate solution of every problem may be made to depend.
To understand the demonstration of the method of Laplace would require considerable mathematical know-ledge ; but the manner of using his results may be de-scribed to a person who possesses no more than a common acquaintance with decimal fractions. To reduce this method to rules, by which such an arithmetician may have the use of it, has been one of my primary objects in writing this treatise. I am not aware that such an attempt has yet been made : if, therefore, the fourth, and part of the fifth chapters of this work, should be found difficult, let it be remembered that the attainment of such results has hitherto been impossible, except to those who have spent a large proportion of their lives in mathestudies. I shall not, in this place, make any remark upon the utility of such knowledge. Those who already admit that the theory of probabilities is a desirstudy, must of course allow that persons who cannot pay much attention to mathematics, are benefited by the possession of rules which will enable them to obtain at least the results of complicated problems ; and which will, therefore, permit them to extend their inquiries further than a few simple cases connected with gambling. By those who do not make any such concession, it will readily be seen, that the point in dispute may be argued in a more appropriate place than with reference to the question whether others, who hold a different opinion,
PREFACE. 1X
should, or should not, be supplied with a certain arithmethod.
The first six chapters of this work (the fourth, and part of the fifth exclusive) may be considered as a treatise on the principles of the science, illustrated by questions which do not require much numerical comTo this must be added the first appendix, on the ultimate results of play. Omitting the first pages of the latter, the discussion on the noted game of rouge et noir will, with the problems in page 108. &c., serve to show the real tendency of such diversion. I am informed that this game is not played in England at any of the clubs which are supposed to allow of gambling : but it was permitted in the Parisian salons until the very recent suppression of those establishments; and the ac-count given of it will show what has taken place in our own day. The game of hazard is more used in this country; but I have been prevented from giving it the same consideration by the want of a clear account of the manner in which it is played. Nothing can be more unintelligible than the description given by the celeHoyle.
The fourth chapter has been already alluded to : it contains the method of using the tables at the end of the work in the solution of complicated problems. The seventh chapter, and the fourth appendix, contain the application of the preceding principles to instruments of observation in general.
The remainder of the work is devoted to the most common application of this theory, the consideration of life contingencies and pecuniary interests depending upon them, together with the main principles of the management of an insurance office. As this portion was
x PREFACE.
not written for the sake of the offices. but of those who deal with them, I have confined myself to such points as I considered most requisite to be generally known. Common as life insurance has now become, the present amount of capital so invested is trifling compared with what will be the case when its principles are better unprovided always that the offices continue to act with prudence until that time arrives. At present, while the public has little except results to judge by, the failure of an office would cause a panic, and perhaps refor half a century the growth of one of the most useful consequences of human association : but the time will come when knowledge of the subject will be so diffused, that even such an event as that supposed, if it could then happen, would not produce the same result.
There are, however, one or two things to which I should call the attention of those whose profession it is to calculate life contingencies : —
1. The notation for the expression of such contin(pp. 197—201.). This notation was suggested by that of Mr. Milne, from which it differs in what I believe to be a closer representation of the analogies which connect different species of contingencies. Thus, an annuity to last a number of years certain does not differ from a life annuity in any circumstance which requires a difference of notation ; nor an insurance from an ancertain of one year deferred till a life drops. Since writing the pages above referred to, I have learned that I was not the first who considered an insurance in that light. Some years ago the government granted t annuities for terms certain, to commence at the death of an individual ; but refused to insure lives: the consequence was, that, by a very obvious evasion, insur-
PREFACE. xi
ances were effected by buying annuities for one year certain, to commence at the death of a person named. This had the effect of putting an end to such annuities.
- The form of the rule for computing the value of fines, and its introduction into the method of calculating the present value of a perpetual advowson (pp. 231. 236. and Appendix the Second). It will be found that the rule of every writer on the subject is palpably wrong in principle, with the exception of that of Mr. Milne.
- The rule for the valuation of uniformly increasing or decreasing annuities, given in the fifth appendix. A simple application of the differential calculus is made a striking instance of the position, that the labour of a person of competent knowledge is seldom lost. The annuities given by Mr. Morgan and Mr. Milne, are for every rate of interest, from three to eight per cent.; and perhaps those gentlemen may have had some doubts as to the necessity of inserting the two last rates. It now appears, however, that, in consequence of the extent to which their tables are carried, the values of increasing or decreasing annuities, can be calculated with great accuracy for three and four per cent., and with sufficient nearness for five per cent. ; and with very little trouble, compared with that which it must have cost Mr. Morgan to calculate the table referred to in page xxviii. of the Appendix.
The rules, in page xxix. of the Appendix, contain a point which, as no demonstration is given, may cause some difficulty. In turning an annuity or insurance which cannot be extinguished during the life of the party into one which can, a direction to add is given which will at first sight, perhaps, be supposed to be a mistake, and that subtract should be written instead. But
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it must be remembered that an annuity of, say X3 a year, diminishing by X1 every year, is equivalent, by the first part of the rule, to an annuity of which the sucpayments are as follows :
£(-1), £(-2), £(-3), &c
That is, the first part of the rule, when the annuity is extinguished during the tabular life of the party, gives the value of his interest upon the supposition that he is to begin to pay as soon as he ceases to receive. If then, this is not to be the case, the value of his interest must be increased accordingly.
4. The method of the balance of annuities, or the determination of complicated annuities by the addition and substraction of simple ones. This has been done before ; but it has not, to my knowledge, been carried to the extent of making all the questions which commonly occur deducible from the fundamental tables, without the aid of any new series. It is desirable that the beginner should be accustomed to deduction by reasoning, without having recourse to the mechanism of algebra, which, as a quaint editor of Euclid observed, "is the paradise of the mind, where it may enjoy the fruits of all its former labours, without the fatigue of thinking." Of no part of algebra is this more true, than of the method by which complicated annuities are deduced from simple ones, by the resolution of the series which represent them into the simpler series of which they are composed. The education of an actuary does not necesimply the study of geometry ; and such processes, for instance, as those by which are found the values of a contingent insurance or a temporary insurance (pp. 222. 226.), will serve, as far as they go, to ac-
PREFACE. Xiii
custom him to make those efforts of mind, and to bear that tension of thought, the necessity for which is the distinction between a problem of geometry, and one of ordinary algebra.
The considerations contained in this volume have, in my opinion, a species of value which is not directly de-rived from the use which may be made of them as an aid to the solution of problems, whether pecuniary or not. Those who prize the higher occupations of intelsee with regret the tendency of our present social system, both in England and America, with regard to opinion upon the end and use of knowledge, and the purpose of education. Of the thousands who, in each year, take their station in the different parts of busy life, by far the greater number have never known real mental exertion; and, in spite of the variety of subjects which are crowding upon each other in the daily business of our elementary schools, a low standard of utility is gainground with the increase of the quantity of instrucwhich deteriorates its quality. All information be-gins to be tested by its professional value; and the know-. ledge which is to open the mind of fourteen years old is decided upon by its fitness to manure the money-tree.
Such being the case, it is well when any subject can be found which, while it bears at once upon questions of business, admits, at the same time, the application of strict reasoning; and by its close relation to knowledge of a more wide and liberal character, invites the student to pursue from curiosity a path not very remote from that which he entered from duty or necessity. Such a subject is the theory of life annuities, which, while it will attract many from its commercial utility, can hardly fail to be the gate through which some will find their
Xlv PREFACE
way to the general theory of probabilities, and, perhaps, from thence to the pursuit of other branches of science. There are strong instances in favour of such a suppoMany persons in this country have begun by the common studies of an accountant, have been led to an elementary knowledge of algebra and to the use of logarithms by seeing the value of such information in their particular pursuit, and have ended by becoming, in many cases well informed, and in some instances eminent, mathematicians.
Nothing is of more importance, as a help in holding out every bait by which students may be drawn to the exact sciences, than the co-operation of the universities ; which, though they do not possess much power of introsubjects into general study, yet have great influin the settlement of the manner in which those things shall be learned, the advantages of which have been, or may be, felt by the community at large. If ever it should happen that a particular branch of know-ledge becomes in request, it would be of much advanif those institutions would forthwith appropriate and liberalise it; to do which nothing more would be necesthan to promote the study of it among their aspir.. ants to distinction. The consequence would be, that it would find a place in the elementary works which so frequently appear; and not only a place, but its place; that is, in proper connection with other branches of learning, and treated by methods which would preserve that connection. Those who begin to study it in their younger days for professional purposes would be led to the method which bore the sanction of the universities, and not unfrequently to the pursuit of other subjects immediately connected with it.
PREFACE, xv
The theory of insurance, with its kindred science of annuities, deserves the attention of the academical bodies. Stripped of its technical terms and its comassociations, it may be presented in a point of view which will give it strong moral claims to notice. Though based upon self_interest, yet it is the most en-lightened and benevolent form which the projects of self_interest ever took. It is, in fact, in a limited sense, and a practicable method, the agreement of a community to consider the goods of its individual members as comIt is an agreement that those whose fortune it shall he to have more than average success, shall resign the overplus in favour of those who have less. And though, as et, it has only been applied to the reparaof the evils arising from storm, fire, premature death, disease, and old age ; yet there is no placing a limit to the extensions which its application might reif the public were fully aware of its principles, and of the safety with which they may be put in practice.
It is of great importance at the present moment that sound principles on the subject of insurance should be widely and rapidly disseminated. Within the last twenty years, many new institutions have been founded; and (luring the busy portion of the London year, seldom a month passes without the announcement of a novel plan. Of many of these projects I am unable to speak, from not having paid particular attention to them. But of one thing I am certain, that the magnificent style in which the prospectuses frequently indulge might often remind their readers of the unparalleled benefits which are promised by another description of traders, who vie with each other in describing the rare qualities of their several blackings. If there be in this country
a
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a person whose ambition it is to walk in the brightest boots to the cheapest insurance office, he has my pity: for, grant that he is ever able to settle where to send his servant, and it remains as difficult a question to what quarter he shall turn his own steps. The matter would be of no great consequence if persons desiring to insure could be told at once to throw aside every prospectus which contains a puff: unfortunately this cannot be done, as there are offices which may be in many circumstances the most eligible, and which adopt this method of adtheir claims. If these pompous announcebe intended to profess that every subscriber shall receive more than he pays, their falsehood is as obvious as their meaning ; if not, their meaning is' altogether concealed.
Public ignorance of the principles of insurance is the thing to which these advertisements appeal: when it shall come to be clearly understood that in every office some must pay more than they receive, in order that others may receive more than they pay, such attempts to persuade the public of a certainty of universal profit will entirely cease. To forward this result, I have enas much as possible, to free the chapters of this work which relate to insurance offices from mathedetails, and to make them accessible to all edupersons. Whether they act by producing convicor opposition, a step is equally gained : nothing but indifference can prevent the public from becoming well acquainted with all that is essential for it to know on a subject, of which, though some of the details may be complicated, the first principles are singularly plain.
August 3. 1838.