The Duke of Wellington, as Master-General of the Ordnance, having given orders in 1825 that practical architecture should form part of the course of instruction of the junior officers attending the Royal Engineer Establishment at Chatham, under my direction, which had hitherto been confined to their duties in the field, and a Professor in every respect competent having been selected for that duty, I was induced to enter into the details of measuring and estimating buildings and other works myself—which I had never done before, because, like other officers of my corps, who studied the other branches of practical architecture and engineering with pleasure, I had, when employed in garrison duties, left those details to the clerks of works, and to the overseers and foremen of the department.

To ascertain with precision the loss of life occasioned by war is an object of so much importance in every point of view, whether national or social, that whatever apology may be required for the manner in which the present subject is treated, none can be necessary for bringing it forward.

That so little should have been done towards the elucidation of the question, is no doubt partly attributable to the difficulty of obtaining accurate information with respect to it, but in a greater degree, perhaps, to the reluctance to dwell upon it felt by the public.

The object of this paper is to give a brief exposition of the different possible methods or classes of methods of equitably dividing the net profits of a Mutual Life Assurance Office among the members. By an equitable division must be understood one which is fair not only between persons of different ages entering the Office at the same time, but also between persons entering the Office at different epochs of its existence,–one from which the members of different ages will, relatively to one another, receive the same benefit from the profits during the earlier and growing years of the Office, and during years when the amount of its business is declining, as during its stationary period. I assume the premiums actually paid to be composed of the net premium calculated from a correct life table, and of a charge of 25 per cent. on the net premium added to guard against unfavourable contingencies, and that this additional charge is really the source of all the profit divisible among the members; or at least, that whatever profits arise from any other source may be fairly apportioned in the same proportions as appear equitable on the supposition that the charge is the sole source of profits. The addition to the premiums for defraying the expenses of the Office, for the sake of simplicity, I leave out of account. In consideration of the payments of the successive net premiums, the assured is guaranteed a sum at death such that the present prospective value of that sum is equal to the present value of the successive net premiums. Similarly, in consideration of the successive charges on the premiums, he should have the prospect of receiving a sum or sums either at death or at any times before that; not guaranteed, it is true, to any fixed amount, but variable with the circumstances of the Society, such that the present value of such sums or bonuses shall be equal to the present value of the successive charges. That this equation between the present values ought to subsist is as clear in the case of the charges and bonuses as in that of the net premiums and sums assured. If the system of division of profits be such that this equation always subsist, it must be fair as respects each member; and any Office adopting a system in which it does not subsist is committing as great an injustice towards its members as one which calculates the premiums on false life tables.

It is well known to those who have studied the theory of chances. that where the magnitude of a risk may be divided into an indefinite number of parts, it is possible, by this multiplication of smaller parts, to confine the probability of the occurrence of any given excess or deficiency in the result, above or below its mean value, within any given limits; and it is of great practical importance to ascertain both how far the subdivision ought to be carried with regard to insurances, and what are the inconveniences to be apprehended from the admission of occasional deviations from the general rule.

Suppose that the portion of surplus allotted to a particular policy at the division of profits, is applied so as to relieve the assured from the annual payments after arriving at an age to he determined by the amount of that portion or cash bonus, the other conditions of the policy remaining the same; or again, suppose that the cash bonus is so applied as to convert the assurance into an endowment assurance, payable at an age to be determined:–it is proposed to investigate formulæ to apply to these two cases.

In the first case, the cash bonus must be equal to the present value at the time of division of profits of a deferred annuity, of which the annual payments are equal to the premium for the assurance, the number of years for which the annuity is deferred being unknown; and the problem, in fact, is to find the number of years.