The valuation of reversions, absolute and contingent, is a matter of great and growing importance. The greater part of the landed property in the United Kingdom being settled in strict entail, proposals are being constantly made by the expectant heirs for loans upon their reversionary estates; and so long as the present law and practice of settlement prevail, so long will the making advances on the security of reversionary interests in landed property continue to be an important part of the business of such Life Insurance Companies as are wise enough to cultivate it. Although the amount of the landed property in the country remains unaltered, yet the rentals are steadily increasing, and consequently the amount of the money which can be employed in the way above described may also be expected to increase. Again, the vast and rapid increase of wealth among the middle classes of the country has led to a great increase in the number of life interests and reversions under marriage settlements and wills; and in consequence of the many vicissitudes in trade, such reversionary interests in personal property are constantly offered for sale in increasing numbers. There is also, perhaps, a growing disposition, when the owners of reversionary interests in either real or personal estate become involved, to press their interests for public sale, instead of arranging privately a sale to some member of the family. On the other hand, as it is becoming better known how ready Offices of good standing are to make advances on contingent reversionary interests, applications are now made to them, which would formerly have been made to private money lenders, whose charges were, to say the least, excessive. From these combined causes, the number of reversions in the market is increasing, and seems likely to increase.

Those of our readers who have studied the paper of M. Maurice on Interpolation, of which a translation appeared in our last number, will no doubt be glad to compare with it Briggs's own description of his method of Interpolation. His original work however appears to be very scarce; and the chapters in which he describes his method—the 12th and 13th—are omitted, even in the Edition published by Vlacq in Briggs's lifetime. We believe, therefore, that this translation by Mr. Williams of those parts of Briggs's Preface in which he describes his method of Interpolation, will prove very acceptable to our readers.—ED. J. I. A.

The studies of an actuary depend so much for their success on the accurate and full collection of statistics, that we cannot but feel a deep interest in every effort that is made to improve the method of and to lay down sound principles for obtaining them. Hitherto we have been more especially concerned with the laws of population, vitality, and disease, because they form the basis of the vast extension of life assurance in this and other countries. But in the course of the last few years the progress of social science has brought forward many questions tq. which the doctrine of probability as to the recurrence of events may be applied, which, though not strictly within the limits of our professional pursuits, afford many opportunities for the use and extension of our science.

The calculation of extensive tables of logarithms could not fail to give rise to the method of interpolation; and such indeed was its origin. After Napier had made his memorable discovery, and he and Briggs had recognized the advantages that would result from the adoption of 10 as the base of a system of logarithms, it was Briggs who courageously undertook the immense labour required for calculating those famous tables, published in London in 1624, which have not been surpassed in extent and accuracy by any subsequent work. It is well known, however, that Briggs left a great hiatus in those tables, and that the logarithms of numbers between 20,000 and 90,000 are not to be found in them; but scarcely were the first tables printed, when Briggs undertook with fresh energy the computation of the Logarithms of the Trigonometrical Lines, and he was on the point of terminating this vast enterprise when arrested by death. It was his friend Gellibrand who completed the work, and gave it to the world in 1633.

Life Assurance is an institution possessing such an important influence in the welfare and well-being of Society that it is most essential the principles upon which it is based and conducted should be such as to ensure its lasting stability and prosperity.

After a Life Assurance Office has been once established upon a firm basis, there is no part of its management so important as that relating to the periodical valuation of the liabilities under its policies. It is my object in this essay to inquire into the various methods adopted for that purpose in the present day; and to consider the various results they produce when different tables of mortality and rates of interest are employed in the calculations; and also to ascertain the effects which the employment of different data may have upon the present and future condition of an Office.

No life has been written of the original projector of the Equitable Society, except in a column of the Biographie Universelle by M. Nicollet. Dodson's name was, and even still is, so familiar to the actuary, chiefly through the Mathematical Repository, and the impulse he gave to life-contingency problems, that this Journal is the proper place of deposit for what can be collected concerning him. The article above mentioned tells very little. He succeeded Hodgson [which should have been Robertson] in the chair of mathematics at Christchurch Hospital in 1756 [1755] and died November 23, 1757. He published the Antilogarithmic Canon, which others had contemplated [and executed too, but the manuscript was lost] and which he had the courage to execute up to a certain point [his table is the counterpart of Vlacq's largest direct table: five figures of argument and eleven of tabular result]. He could not balance the success of the ordinary tables: the writer doubts whether the table was ever used on the continent [he might have added, England: who uses either Vlacq or Dodson? Their tables are for help to other table-makers, and always were, though both of them intended more]. He published, the Calculator in 1747, a collection of tables at the end of which [say in the proper place in the middle] is an abridgment of the antilogarithmic table. But he is best known in England by his Mathematical Repository, and by his zeal for benevolent institutions [say his determination to found an assurance office to which himself should be admissible].

Having thus found the required formula for the value of , viz.:—

we are now in a position to processed with the transformation of the expression found in the Number of this Journal

Let there be letters x, y, z,…each of which has values, choices, or drawings. Let their number be σ, and let, ξ, η, ζ,…be their several numbers of drawings : x1, x2,…xξ the drawings of x; and so on. Let Σx, Σy,…be the sums of the drawings of x, y, &c.; and Σ:x, Σ:y, the average drawings: so that , and so on.

The application of bonus to the hastening of the time fixed by the original contract for the payment of a life assurance policy, and the using of it to extinguish the premiums payable during the latter years of life, are systems, which, partly from their growing favor with the public, and consequent adoption by many offices, and still more from the fact, that the subject is by no means exhausted, notwithstanding various able articles that have appeared in this Journal, seem to me to render apology unnecessary for again treating of it.

It is stated in the notes to Cousin's éloge, and has been repeated, that Fourier was placed at the head of a bureau de statistique by the prefect of the Seine, the Comte de Chabrol. But there is hardly any authentication. Anything done in the subject by Joseph Fourier, the most powerful working member of the Egyptian commission, the originator of mathematical results of the first order of genius, novelty, and utility, and an experienced public man, must be worthy of attention. But so little is his statistical career known, that the Biographie Universelle (Michaud, 1856) only ventures its mention of the connexion of Fourier with the volumes presently described as suivant plusieurs personnes bien instruits.

An Insurance Office, which, in consideration of fixed premiums undertakes the payment of a sum on the death of the assured, strictly speaking lays a wager with each of the assured, in which the stakes are proportional to the probabilities of the happening of the two contrary events—his living and his dying. Every assurance of a sum payable at death admits of being divided into partial assurances for a fixed period, as a year. Let the sum assured be S and the probability of death within a year p, then pS will be the stake of the assured, or the premium; and (1−p)S the stake of the office, i.e. the amount which the office would lose in case of the death of the assured within the year. The two stakes together therefore amount to the sum assured.

There is no part of the subject of Life Insurance which has occasioned so much difficulty, and given rise to so much diversity of opinion and of practice, as that of the distribution of profits. The methods which have been adopted are very various, and very few of them seem founded upon any intelligible principle.

Many attempts have been made of late years to form an exact theory on this subject, and to deduce systematic rules; but it cannot, I think, be yet said that there is any method which is generally recognised as theoretically correct, and capable of application in all cases.

There must at the present time be a vast amount of British capital embarked in loans. Foreign states are constantly holding out their lures with more or less of success to our monied men, and financial and other associations are always ready to take charge of the funds of such of them as prefer investing at home. In this state of matters it is somewhat remarkable that there is nothing to be found in our books on interest on the subject of loans. These are usually—and perhaps intentionally—so complicated with conditions in regard to premiums, discounts, times and modes of repayment, &c., as to render it almost always a matter of extreme nicety to determine the rate paid by the borrower for the accommodation, and that realized by the lenders on their investments. And yet, as I have just said, in no English work that I am aware of, is there anything to be found having special reference to the subject.

My second example is the Austrian Loan of 1865. The conditions are as follows:—

734,694 Bonds, each of £19. 17s. 0d., issued at £13. 14s. 4d., that is, at a discount of £6. 2s. 8d. each.

Subscribers will be at liberty to pay their Scrip in full on any one of the above dates, under discount at 6 per cent per annum.

In the number of the Journal for April, 1868, Herr Wilhelm Lazarus, of Hamburg, reported the recent establishment in Berlin of a German Life Assurance Institute, a society having generally the same object as the Institute of Actuaries. As the proceedings of such an association cannot fail to interest the members of the Institute of Actuaries and the readers of its Journal, we purpose laying before them some of the more interesting papers read before the German Institute; and as a fit introduction to these we now give the following translation of the laws of that Society, for which we are indebted to Mr. J. Hill Williams.—ED. J. I. A.

Irrational Functions, strictly speaking, are functions whose numerical values cannot be expressed by finite fractions. In the present connexion the term irrational has a wider signification. It is used to denote any function whose numerical value cannot, in general, be expressed exactly within the limits as to decimal places, to which we restrict ourselves in the table under formation. As here employed therefore the term designates not only transcendental functions, (as exponential, logarithmic, circular, &c.,) and algebraical irrational functions, (which are such as contain fractional powers of the variable in either numerator or denominator, or both,) but also algebraical fractional functions, which are such as contain integer powers of the variable in both numerator and denominator, or in the latter only. And it is to the formation of tables of the values of functions such as these that our attention is now to be directed.

On the 1st February, 1868, this Company, “now the largest moneyed Corporation “in America, celebrated its “silver wedding,” as it was termed by the Governor of the State of New York on the occasion of the 25th annual meeting. The Report then presented and the general Prospectus of the Company have been forwarded to us by the “President” of the Company, who appears to be virtually what in this country we should call “Managing Director.” These documents are so instructive and interesting in themselves, and illustrate so remarkably the application to our pursuits of that originality of the American character, which is fast becoming as proverbial as its energy, that we propose to notice their contents in some detail.

The interest with which I have perused Mr. Gray's very able paper in the last number of the Journal, has led me to give some consideration to the general theory of varying Annuities; and as the results of the investigation appear to me somewhat striking, I am induced to think that a brief paper on the subject may not be unacceptable to that section of the scientific world to which this Journal more particularly addresses itself.