The present paper was undertaken on happening to peruse very lately Messrs. Gray, Smith, and Orchard's laborious and useful work, entitled Assurance and Annuity Tables, according to the Carlisle rate of Mortality, at 3 per Cent. In the introductory section of that work, when referring to the larger Tables V. and VI., containing the single and annual premiums for survivorship assurances, it is there very justly remarked, “Considering the frequency with which occasion arises for the functions here tabulated, and the tedious nature of the operations requisite when their values have to be formed from other tables, it may, at first sight, seem surprising that complete tables of them had not been formed long ere now; but the wonder will cease when it is farther considered that, until recently, there had been pointed out no other method of forming them than the tedious operation just referred to.

The mode in which the Property and Income Tax of this country is levied, has created so much dissatisfaction, and is so universally acknowledged to be inequitable, that I am induced to call the attention of the Members of this Institute to the subject, with a view to arrive at some elucidation of the principles upon which direct taxation should be based: being satisfied that there is no tribunal before which the consideration of such a matter could be more properly brought, or where its peculiar features could be more thoroughly and efficiently investigated.

The origin and progress of the principles of insurance are subjects, which not only call for the endeavour to elicit and clear from seeming contradictions such records of the insurer’s calling as are blended with the remote history of mercantile customs and laws, but which likewise particularly suggest the necessity of some essay to connect those rude outlines with the more advanced stage of the theory, in its immediate alliance with the sciences of applied probabilities and political or social arithmetic.

We now pass to the traces of the early practice of LIFE INSURANCE, a subject which has in general been but very slightly noticed.

The estimation of the value of human life in its rudest approximation—that is, of the term of years which, on the average of a sufficient number of observations, would be observed to fall to the share of each of a given number of individuals born under certain circumstances, and continuing in certain climates or ranks—was to all appearance a subject the consideration of which, not only the speculations of the remoter ages had no impelling cause to enter upon, but one which, if pursued, would have led not only to scepticism, but even to persecution, for entering on a tacitly admitted forbidden ground.

To trace the succession of events which have given rise to the duplex system of assurance in this country, known as the Participating, and Non-participating, would not be altogether uninteresting or unprofitable. There is little doubt, that the immoderate premiums required at the outset by the few who were disposed to embark in the then novel speculation of life assurance, suggested the idea of mutual association, as a means of achieving the same objects on more advantageous terms; and thus the participating system owed, to a certain extent, its origin to the non-participating.

My object in the following paper is to give, in a few leading problems, a general and somewhat elementary view of the Theory of Probabilities, exhibiting the immediate dependence of the doctrine of Life Contingencies on the laws of probability and combination, and showing how apparently complicated problems, involving two and three lives, can be deduced directly from their very simplest and most elementary forms of expression.

Before I proceed to the more immediate object of this paper, viz., to the Consideration of some leading Problems in the Theory of Probabilities, I would premise a few observations with reference to the doctrine of Life Contingencies.

In my work on Notation, p. 13, I have asserted a fact which is of such essential importance in the elucidation of the doctrine, that it can scarcely be too often repeated, or too strongly impressed upon the mind of a student, viz,, that curtate expectations of life are the values of annuities which do not bear interest; and this is universally true, whether they be the expectations of single lives, of joint lives, of the longest of two or more lives, or of a life or lives after another or others,—in short, however remote or complicated the contingencies may be in which they are involved; and the values of annuities, or those series in which interest is involved, are invariably in the same form of expression as their corresponding curtate expectations, as thus employing the notation explained in the work already alluded to.

This subject has been so ably treated by Mr. Sang, in the paper transferred by Mr. Thomson’s permission to the pages of this Magazine, that it might seem almost superfluous to revert to it. Nevertheless, there are considerations connected with it which I believe to be of some importance, and which that gentleman has not adverted to; and I am therefore induced to submit the following remarks, by way of pendant to his observations :—

It has frequently been remarked, that however varied and uncertain may be the occurrence of the events to which the life of a single individual is exposed, the average return in a large mass of persons is so regular, as to be predicted with confidence within very small limits of error. On the uniform happening of these events under similar circumstances, or the discovery of the laws by which they are governed, the actuary depends for the application of the theories of probabilities to the many useful purposes for which they may be rendered available.

The Council of the Institute of Actuaries are making a laudable attempt to collect such data as the Assurance Offices possess, with a view to determine the mean duration of human life in different parts of the Globe; and thence to ascertain the proper rate of premium to be charged in the several localities to which persons whose lives are assured may be desirous of proceeding. The almost entire absence of any authentic information upon this subject renders such a proceeding of the greatest importance and interest; since it is only by such means we can hope to be able to substitute, for the present uncertain and arbitrary charges, rates which are based upon actual experience, and which are justified by the circumstances of each particular case.

It has frequently been remarked, that however varied and uncertain may be the occurrence of the events to which the life of a single individual is exposed, the average return in a large mass of persons is so regular, as to be predicted with confidence within very small limits of error. On the uniform happening of these events under similar circumstances, or the discovery of the laws by which they are governed, the actuary depends for the application of the theories of probabilities to the many useful purposes for which they may be rendered available.

The chief, perhaps the sole, applications of the Theory delivered in Part I., are to Church Livings and Copyhold Estates. Of these applications we shall treat in order.

The principle on which the valuation of the rights of presentation to Church livings is subjected to treatment by this theory is, that the worth of the living is greater than that of the services to be performed by the holder of it. It is conceived, therefore, that the patron, in presenting a clerk to a benefice, is entitled to consider that, in addition to affording the clerk an opportunity for the exercise of his vocation, he is conferring upon him also an annuity due for his life, whose amount is the excess of the value of the benefice over that of the clerk's services. The law sanctions the arrangements that arise from this mode of viewing the subject, and permits the open purchase and sale of the right of presentation, whether for ever or for any number of times; save and except during the "mortal sickness" of an incumbent, or during the period between his death and the appointment of his successor; the purchase of a right of presentation in these circumstances being held to be simony, and the guilty party consequently being amenable to law.

The interest which the Members of the Institute of Actuaries take in all matters relating to assurance, has induced me to present a few statements which exhibit the loss upon marine risks under certain classifications; and at the request of their valuable and talented secretary, I lay before them the data upon which tables for assuring the following benefits were constructed by me in the early part of last year for a projected company.