In treating of the above subject, it cannot be expected that any new principles will be put forward, or that any great amount of novelty will be found in the method of handling the question. All that I have aimed at is, to apply acknowledged principles to the discussion of a question of considerable practical interest at all times, and more peculiarly so at the present time.

In bringing before the members of the Institute of Actuaries a question so important to the prosperity of Assurance Companies as the investment of the funds, they will be prepared to admit that it is one which could not be fully treated within the limits usually allowed for a paper for discussion at the sessional meetings. All that I could hope for is, to throw out a few hints such as experience or reflection may suggest to me, and trust to elicit, from the practical knowledge and abilities of those present, the materials for a fuller debate of the subject.

The late Mr. William Morgan, in the second edition of his Principles and Doctrine of Assurances and Annuities on Lives, published in the year 1821, has, in the 7th Problem of that work, investigated “the method of determining the amount of an annuity forborn and improved at compound interest during the continuance of a given life.” This problem he has subsequently discussed in an algebraical form, in Note V., page 154; and he has moreover given a table (Table VII.) showing the sum to which an annuity for-born and improved at 4 per cent, compound interest will amount, on the extinction of a given life, computed according to the foregoing method, from the Northampton Table of Observations.

No law has hitherto been discovered, by which a correct table of mortality could be constructed, but attempts have been made by De Moivre and others to obtain an approximation.

If, in conformity with the opinion now generally entertained, and in which I concur, we adopt the Carlisle Table of Mortality, as formed by Milne from the observations made by Dr. Heysham, as a nearer approximation to the actual duration of human life than any other table of mortality hitherto in use, it is remarkable that the following simple hypothesis should enable us to obtain the expectation of life almost identical with Milne's table for a long series of years up to the age of 60.

The subject of the division of surplus in a Life Assurance Company is one that cannot fail to be interesting to all who are engaged in the business of life assurance, as well as to the public at large; and while there is so great a diversity as at present exists in the methods of division in use, it must be useful occasionally to draw attention to it, as a means of inviting discussion and the interchange of opinions. This must be my apology for bringing before this Society this evening a subject which has more than once been discussed here; for I can scarcely hope to lay before you in my remarks much that has the merit of novelty, on a matter that has occupied the attention of the most eminent men in our profession.

When an observation has been made for the purpose of determining the mortality which has prevailed during a specified time in the community observed upon, it is usually found that the series of probabilities of living a year, at the various ages thence deduced, is very far from conforming to our ideas of what an exponent of the law of mortality ought to be. Instead of a series exhibiting throughout its extent a gradual and progressive change of value in passing from each term to the next, we have one showing but a faint approach to the desired regularity. This is just what, from the circumstances in which such observations are usually made, we ought to expect. It is not often that the community subjected to observation is sufficiently large, or that the period during which it is so subjected is sufficiently extended, to furnish us with such a series of average results as can alone fitly figure forth the law of mortality.

In a paper which was read before the Institute of Actuaries on the 8th January, 1855, the idea was first suggested by Mr. Farren of introducing into the theory of life contingencies the hypothesis of a constantly fluctuating rate of interest, in lieu of the supposition hitherto adopted of a uniform rate.

Abundant reasons are assigned, in the paper referred to, for making the proposed change; and no one, at all acquainted with the practice of life assurance, can fail to see that the effect of Mr. Farren’s suggestion, if carried out, will be to base all computations upon an assumption which is in the closest possible accordance with actual facts.

It appears to be an admitted principle in the commercial world, that great evils have a tendency to work their own cure. Over-trading will bring about a state of things by which eventually it must be checked, and excessive speculation has its limit by the same invariable law.

More than two years ago, it was quite evident that the rapid manner in which new Companies were being formed for the purpose of assuring lives would ere long produce sad and humiliating results.

The subject which I am about to bring under the notice of the members of the Institute is one as to which a good deal of misapprehension exists. An impression, I believe, very commonly prevails, that combinations of the nature in question are little better than rude and arbitrary arrangements, in which the rights of the several parties concerned are not much regarded, and which are usually carried out by sacrificing more or less the interests of one class to those of another. I propose now to show that such is by no means the case—that these arrangements are susceptible of the nicest and most accurate adjustment—that the advantages derived from them are appreciable with great exactness—and that the rights of all concerned may be most scrupulously maintained. It is true that, the details being somewhat complicated, persons unfamiliar with such subjects are not very well able to satisfy themselves of this; but the fact is so, nevertheless, as we will now endeavour to demonstrate.

On a former occasion I had the honour to lay before the Institute a historical sketch of this subject, from the earliest period of authentic records down to that of the legal establishment of the practice of taking interest, in the reign of Elizabeth.

The facts then adduced showed that very high nominal rates of interest were obtained in the ancient world, and during the middle ages; but such rates are, by no means, to be taken as measures of the profits made upon loans, which must always have been seriously diminished, and often entirely swallowed up, by the exactions and spoliations to which the lender was exposed, and the difficulties he experienced in enforcing his claims, from defective laws, and the prejudices entertained against him by the community.

Another mode of easily appreciating the regularity and the analogies of different tables is, to construct a diagram, in the form of a curve, of which the absciss represents the age, and the ordinates the corresponding decrements of life (see opposite). The inspection of such a diagram is sufficient to convince us of the great irregularity of the Carlisle Tables of Mortality, which must obviously have been formed, as they confessedly were, from observations on a very limited number of individuals, so that they exhibit a succession of different climacterics, after which the mortality is diminished; while about the age of 74 the curve that represents them towers to an incredible height, affording an expectation of longevity which some of the strongest advocates of those tables have abandoned in their practical applications, since they take their estimate of life, in advanced age, even lower than it is represented in the Northampton Tables.

It will no doubt be in the recollection of many of our readers, that some few years ago attempts were made by the Council of the Institute to collect, from the Offices willing to contribute the results of their experience, such information connected with the extra risks undertaken by them as might afford the means of determining, if not exactly, at least with some approach to accuracy, the proper rates of premium to meet them. The Council made application to all the existing Companies; but did not succeed, unfortunately, in obtaining returns from more than fifty-two of them. These fifty-two Companies, however, were at the pains to supply as many as 6,154 cases—each of them of a special nature, as regarded unusual hazard to life arising from residence in foreign climates, or from particular pursuits; but so infinitely varied in their conditions and circumstances were they, that it became immediately obvious that considerable difficulty would arise as to grouping them into classes which should be sufficiently well defined to yield results of a useful and practical character. If, on the one hand, the conditions of the risk were kept within limits sufficiently restricted, the numbers in each class were all but insignificant; and if, on the other, attempts were made to remedy this last inconvenience, the groups comprised risks which were altogether heterogeneous.

John Rowe was the author of an Introduction to the Doctrine of Fluxions, which reached its third edition in 1776. The work was then advertised with an extract of a letter to the author “from the learned and well known Reverend J. Stubbs, M.A., Fellow of Queen's College, Oxford,” setting forth that he had received Rowe's valuable present, and was much surprised to find it so prodigiously improved. Indeed, it so much resembled a new work, when compared with the first edition, that he wished no mention had been made of its being the third, but that the two former editions should be left to be forgotten.

The following observations and tables are extracted from a pamphlet recently published by Mr. John James Downes, the actuary of the above-mentioned Society. The length of the period embraced, the number of lives concerned, and the well known ability of the compiler, render the statement of considerable value and importance, and we have inserted it not doubting but that it will be looked upon by our readers in that light.

It was intended that the short essay descriptive of the existing system of settlements of claims under average fire policies, which appeared in this Journal as far back as No. X. (January, 1853), should have been promptly followed by some remarks on the most obvious defects of the system, with a few practical suggestions for their remedy. It is not necessary to explain here the reasons for the delay which has taken place in the fulfilment of that intention; but it is satisfactory to know that the postponement has not been without its use. Many valuable hints have been offered; and although some marked differences of opinion exist as to any proposed remedies, the explanations given of the existing rules are, without any exception, admitted to be correct.

The system of fire insurance has been practised in Great Britain, either by private societies, public companies, or corporate bodies, for more than 180 years. In an article on fire insurance in Vol. XII., Part II., of the Encyclopædia Britannica, 8th edition, recently published, Mr. F. G. Smith draws attention to the various attempts which were made to interest the Corporation of the City of London in the subject, and to the proposals which for this purpose were submitted to the Court of Common Council between the years 1669 and 1680, and more especially to one from Mr. Deputy Newbold, which appears to have been the most acceptable. The delay which occurred before the report of the Committee was presented, enabled other private individuals to originate a scheme for fire insurance; and by the advertisements of the day it appears that they offered to insure against damage by fire, brick houses at 6d. in the pound, and timber houses at Is. in the pound—rates which showed the little knowledge which at that time prevailed upon the subject, provided sufficient business could be reckoned upon to obtain an average of the risks. On the 13th October, 1681, the Court of Common Council decided to effect fire insurances on houses within the city and liberties, and engaged a sufficient fund and undoubted security by the Chamber of London, on lands and ground rents, to provide for the fulfilment of their contracts. The war that ensued between those private Societies which had been the first in the field, and the city insurers, gave rise to much amusing pamphleteering and advertisements in the Gazette. In 1681, 1682, and 1683, the journals of the Court of Common Council record the signing of many policies, and refer to the discussions and arguments of their opponents; but the city authorities appear to have been soon weary of the scheme, for, by a resolution of the 13th November, 1682, the Court decided to relinquish the business, to cancel the existing contracts, and to return the money accepted for them.

In a paper which was read before the Institute of Actuaries on the 31st May, 1852, “On the Uniform Action of the Human Will, as exhibited by its Mean Results in Social Statistics,” I drew attention to the remarkable regularity with which marriages are contracted in any country, and the very small limits of difference from the average number which appear from year to year. The observations made by M. Quetelet in Belgium, from 1825 to 1845, showed that the extreme difference in the total number of marriages was little more than half the difference of the extremes in the number of deaths in the same period. Such a conclusion seemed to imply that the subject was worthy of more research. If the law of mortality can be so accurately defined at different ages, that pecuniary interests, amounting to some hundreds of millions sterling, can be valued and adjusted with the greatest nicety, it is reasonable to conclude that the labour of a statistical inquiry into the proportion of marriages at different ages would be rewarded with the discovery of some equally defined law, since the variations from year to year in a given number of facts appear, from a large number of observations, to be even less than in the former case. It is true that, as human life must fail at some time, from the natural decay of the powers of life, every interval of age after man has once attained maturity may be expected, under ordinary circumstances, to show a steady and progressive increase in the liability to disease and death. On the other hand, it may be, argued that marriage is the exercise of the free will of man—that consequently, it does not depend on the age or period of life, but on the arbitrary exertion of those feelings or mental and moral qualities which are not subject to natural laws, or at least not to such laws as we are able to express numerically in the same manner we can the law of mortality in any given population. If we consider, however, marriage as, in one sense, the natural provision for the preservation or increase of the species, and the counteraction to the law of mortality by which the species would perish, we should not be surprised to find that, however imperceptibly to individuals, there is a tendency to obey some unknown law of nature which at the period of maturity would lead to the maximum of marriages, and gradually diminish with age in the same manner as the tendency to disease and death increases with age. The motives and caprices of individuals would only have the same effect on the general results which the different habits of individuals may have in increasing or diminishing the rate of mortality. Accordingly, M. Quetelet, in a comparison of the number of marriages in Belgium for each five years of age after 21, for both sexes, for a period of five years consecutively, showed that the average results in each period scarcely differed at all from year to year. The table is so remarkable, that I have reduced the proportion to 100 of the total marriages in each year, and present it to show the small differences which will then be seen to prevail.

The calculus of finite differences was created by Taylor, in his celebrated work entitled Methodus Incrementorum, and it consists, essentially, in the consideration of the finite increments which functions receive as a consequence of analogous increments on the part of the corresponding variables. These increments or differences, which take the characteristic Δ to distinguish them from differentials, or infinitely small increments, may be in their turn regarded as new functions, and become the subject of a second similar consideration, and so on; from which results the notion of differences of various successive orders, analogous at least in appearance to the consecutive orders of differentials. Suc h a calculus evidently presents, like the calculus of indirect functions, two general classes of questions:—

1. To determine the successive differences of all the various analytical functions of one or more variables, as the result of a definite manner of increase of the independent variables, which are generally supposed to augment in arithmetical progression.

2. Reciprocally to start from these differences, or, more generally, from any equation established between them, and go back to the primitive functions themselves or to their corresponding relations

The present paper may be considered a sequel to that upon the “Mortality arising from Naval Operations,” which I had the honour to lay before the Society in their last session.

Although military affairs are in general looked upon in England with less interest than those of the navy, they derive at this time increased importance from the fact that nearly all the great operations of the present war have been upon land; and have, moreover, from a combination of unfortunate circumstances, been subjected to much criticism and discussion.

Since the establishment of the Statistical Society of London, so great an impetus has been given in this country to the study of statistics, that probably in no branch of knowledge have more important results been produced in the same time. This arises in a great measure from the very nature of statistics, and from the universality of their application. Every science, the facts of which can be expressed in a tabular form, comes within their range; and the great object of those who desire to elevate the character of statistics in the mind of the philosopher or the man of science is, to show how, by a correct and methodical classification of facts of every kind, the boundaries of knowledge are enlarged, new laws developed, and new theories suggested.