The three heads, into which it has been found expedient to divide the general subject of the present paper, all relate to the consideration of a series of numerical quantities which are supposed to represent, either accurately or approximately, some known or implied law which necessarily has an influence in regulating their progression. Assuming that we only know the values of certain terms of the series of quantities, distributed at stated equidistant intervals, it has been shown that all the intermediate or absent terms are most accurately and expeditiously determined by the process of interpolation, already discussed and practically developed.

The author commences by stating that he does not propose to consider what sums should in practice be given for the surrender of policies, or what is the value to a purchaser of a policy offered for sale in the public market; but on this occasion confines himself to the examination and explanation of the theoretical value of a policy.

Interpolation, generally considered, is the process of determining numerical values intermediate to a series of other values that are already known, and it is of extraordinary utility in many calculations, and especially in the formation of extensive tables. If, in the direct calculation of a table, the values to be tabulated depend upon a complicated function, difficult of calculation, considerable labour and corresponding risk of error may usually be avoided by computing the values for a certain succession of equidistant intervals, and afterwards determining by interpolation all other values that may be requisite to complete the table and adapt it to the practical purposes for which it is constructed. In such case the method is made use of as a powerful auxiliary at the option of the computer.

Having read with much pleasure and profit the several articles upon this subject, chiefly in the Assurance Magazine, from such able contributors as Messrs. De Morgan, Jellicoe, Brown, Sprague, Pattison and others, whose names are known and honoured on this side the Atlantic as well as at home, I have thought that, in return, a brief account of the method adopted in the recent distribution of the surplus of our largest American Life Company might be acceptable to the members of the Institute of Actuaries.

The great changes which have lately taken place in India—the progress of railways, the extension of commerce, and the impetus which is likely to be given to the emigration of Europeans to those vast fields of enterprise—cannot but add peculiar interest to any inquiries tending to throw light on the causes which affect the health or life of Europeans in that country. The difficulties of arriving at any sound conclusions on these subjects have always been great. In dominions so widely spread, comprising all climates, from the most delightful temperatures of perpetual spring to the fever-haunted jungles and deadly swamp, every cause that affects the health of man may be found at work.

In six dialogues: the sixth contains a quadrature of the circle. But there is another edition of this work, without place or date on the title-page, in which the quadrature is omitted. This seems to be connected with the publication of another quadrature, without date, but about 1670, as may be judged from its professing to answer a tract of Wallis, printed in 1669. The title is Quadratura circuli, cubatio sphæræ, duplicatio cubi, 4to. Hobbes, who began in 1655, was very wrong in his quadrature; but, though not a Gregory St. Vincent, he was not the ignoramus in geometry that he is sometimes supposed.

If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavour to persuade me that he was larger than the elephant, I might by possibility be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be unable wholly to reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others—I should feel quite at my ease.

In the paper which I had recently the pleasure of submitting to the Institute, I gave an explanation of what was meant by the phrase “the value of a policy.” It was therein stated that, in one view, the value of a policy was the difference between the present value of the sum assured payable by the Office, and the present value of the premiums payable by the person assured; and in another view, that the value of a policy was the difference between the premiums received and the claims paid, accumulated at the rates of mortality and interest inherent in the calculation of the premiums.

The Friendly Societies, which are sometimes called Benefit Clubs, have for many hundred years been highly popular in England; not merely because they supplied one of the principal wants of the working classes—namely, relief in sickness or accidents which disable them from obtaining their usual wages—but because, by mutual association, they afforded some scope for the talents and sagacity of some of the members to be recognised and appreciated by others. But by the very nature of their composition they were deficient in the element of stability ; and notwithstanding the best intentions of the members, the Societies have been constantly broken up and reformed.

The paper commenced with a general view of the effect of emigration upon returns of deaths among the population of England and Wales, followed by an estimate of the extent to which any calculations based upon such returns, and the general census, must be thereby affected. This was followed up by some remarks on the uncertainty attending the statements of ages of the individuals enumerated in the census returns, and the ages returned as at death. A tabular view of the estimated effect of these causes of error upon the expectation of life, derived from the returns, was also given.

Five meetings of the International Statistical Congress have now been held, and it cannot be denied, that, with the vast increase in commerce, and in the social relations of different countries, they continue to grow in interest and importance. The first Congress, which may be truly said to owe its origin to the exertions of, and was presided over by, that eminent writer and thinker, M. Quetelet, was held in Brussels, in 1853; the second in Paris, in 1855 ; the third in Vienna, in 1857; the fourth in London, in 1860.

The assurance of a capital sum payable at the death of a husband is of much more recent adoption than the provision of an annuity during the life of his widow. Some of the earliest Assurance Companies were instituted with the main design of providing annuities to widows and sisters—now, however, forming a very small portion of their business—and very many funds and schemes, instituted at a very early period, are now in existence, throughout the various sections, professions and trades, of the community, whose only end is to secure annuities commencing on the death of the person upon whose income the annuitant may be supposed to be dependant.

The great advantages, social, commercial, and political, which would attend the use of one system of weights, measures, and coins throughout the world, have generally been admitted, but as generally considered impossible. Such a result is frequently deemed to be merely the dream of a visionary, or the speculation of a philosopher, who has no practical knowledge of the world, and is incapable of appreciating the difficulties which stand in the way of accomplishing so desirable an object.

The most important practical question arising in an actuary's practice is the following—“By what table of mortality, and according to what process, should the liabilities of a Life Assurance Company be estimated ?” Very wide differences of opinion appear to exist as to the proper answer to be given to this question. Thus, in the Assurance Magazine for January last, are contained two papers, written respectively by Mr. Jellicoe and Mr. Tucker, which bear more or less upon this subject ; and in which very conflicting opinions are expressed by those experienced actuaries.

Some years ago, before the enactment of the Ten Hours Factory Bill, I had the honour to be engaged by the Earl of Shaftesbury (then Lord Ashley) on an investigation of the average distances traversed daily by children and others employed as “piecers” in the cotton mills about Manchester and the surrounding districts. After taking the various dimensions of the mules or spinningjennies, and making the requisite observations as to the number of strokes of each machine per minute, and the average number of breakages of the threads at each stroke, I found that the necessary calculation involved a preliminary discussion of a somewhat curious mathematical problem.

If the series of the numbers living which constitute the law of mortality could be approximately derived, by collecting together the corresponding terms of a limited number of geometrical progressions, it is evident that the mathematical formula involving any combination of life contingencies and of the interest of money would expand into a certain number of geometrical series, the summation of each of which might be easily expressed. Such a result, as a general principle, would not appear to be attainable, so far as can be judged by an example I have been induced to test with the Carlisle table.

According to Mr. Locke, the circumference is three diameters, three-fourths the difference of the diameter and the side of the inscribed equilateral triangle, and three-fourths the difference between seven-eighths of the diameter and the side of the same triangle. This gives, he says, 3·18897. There is an addition to this tract, being an appendix to a book on the longitude.