On a former occasion I had the honour of submitting to the consideration of the members of the Institute of Actuaries some statistics respecting marriages amongst the families of the peerage, from which were deduced tables showing the probabilities of marriage, whether for the first or second time, according to the age of the husband. The subject would, perhaps, have been better fitted for a discussion at a meeting of the Statistical Society, had it not been brought within the range of assurance topics by the avowed object for which the statistics were collected, viz., to throw some additional light on a class of risks which were every day becoming more numerous and important—the class of assurances to protect contingent reversions which would pass from presumptive heirs in the event of issue being born to a present tenant for life. They are now more familiarly known as “issue risks.”

The following collection of problems is intended to serve two purposes—first, as a continuation of a former article, published in vol. x. of this Magazine, under the title of “Solutions of the Compound-Survivorship Assurance Problems,” which treats of cases involving three lives only; and secondly, as an introduction, or preliminary step, to a complete exposition of the writer's method of constructing mortality and annuity tables upon Mr. Gompertz's celebrated hypothesis, a brief account of which method will be found in the eighth and ninth volumes.

In the last Number I gave the most elementary view I could arrive at of Arbogast's method of development. In the communication following I saw that Mr. Peter Gray had referred to Stirling's theorem; and this suggested that it might be useful to give, by means of common algebra only, an account of the two most important cases of summation of many terms of a divergent series.

In the last number of the Journal I explained and exemplified a method for the formation of logarithms and anti-logarithms to twelve places, and gave the tables requisite for its application. I am now to analyse and farther exemplify the method in question, and demonstrate the rules laid down in the previous paper.

It has become the practice of late years for Assurance Companies, when applied to for a policy of very large amount, to issue such an one as is required and to reduce the risk within the limits they have prescribed for themselves by effecting reassurances of portions of it with other Companies. The only alternative proceeding would of course be to issue a policy for the amount to which the Company limits itself, and to leave the applicant to effect the remainder of his proposed assurance where he may. Mr. Newbatt's paper, which, as having reference to a matter of official practice merely, is not inserted here, advocates the latter mode of procedure in preference to the former, and adduces a variety of arguments in support of that view.

The remarks which I have to offer for the consideration of this meeting have reference not to the deduction of the probabilities of living and dying from the facts observed, but to the mode of dealing with those probabilities, in their rough state, with the view of rendering them fit for the purpose for which they may ultimately be required.

So far as I am aware, the earliest method for the formation of logarithms, in which it was proposed to resolve the number whose logarithm is required into factors by a continuous process, is one which was published by Mr. Manning in the Philosophical Transactions for 1806. My acquaintance with Mr. Manning's method is derived, not from the original paper, which I have not seen, but from a reprint of it, “nearly as it stands,” in Young's Elementary Essay on the Computation of Logarithms. Mr. Manning applies his method to the formation of Naperian logarithms only; but it is equally applicable to the formation of common logarithms. I repeat, in accordance with this method, an example I have already more than once given.

In the investigation of a little curiosity in the matter of annuities, I had occasion to want four or five terms of log (a + bx + cx2 + …), and, by Arbogast's rules I accordingly wrote down the following, without a stroke of the pen more than is here given:—

This I might have continued to any number of terms for little beyond the trouble of writing. The method is easy to demonstrate and easy to practice; and it struck me that I might, in a few pages, give an account of it sufficient for use and detached from anything else. I therefore append it to the problem.

With the exception of the publication on a more limited scale (and which will be hereafter referred to), of that which is now to be developed, there exists at present no ready and practical method of forming to more than seven places the logarithms of numbers of more than seven or eight figures. The great extent of the tables requisite, if formed on the plan of our present seven-figure tables, is not only likely ever to prove a bar to their construction, but it would also render them too cumbrous, if constructed, to be easily and readily used. In the present method, the end in view is sought to be attained in another way. The principle of the method is the resolution of the number whose logarithm is required, by a direct and easy process, into factors of a peculiar form, the logarithms of which, to the requisite extent, admit of easy tabulation. The logarithms of the factors, then, being taken from the table, their sum is the logarithm of the given number.

The measure by which the Chancellor of the Exchequer proposes to extend the benefits of life assurance and annuities to the nation at large has now been before the public some time. Few Bills have of late attracted so much attention, and have been so earnestly discussed by all classes, as this. But beyond what was elicited at an interview that took place between the Chancellor of the Exchequer and the Actuaries of several Offices, and some passing remarks on the subject made at the last meeting of the Institute of Actuaries, when we had the advantage of hearing Mr. Samuel Brown's excellent paper on Friendly Societies, the public have as yet had no opportunity of hearing the opinions of those, who after all are best able to judge. It now appears, that the Chancellor of the Exchequer is opposed to a general inquiry and the calling for “persons, papers and records” by the Select Committee, to whom the Bill has been referred, and it therefore becomes all the more desirable that the merits of the proposition of the Chancellor should be calmly and impartially discussed by the members of this Institute.

When a series of quantities which depend on a fixed law, whether known or implied, follow each other in a due order of succession, the general accuracy of their numerical values may be satisfactorily tested by observing the regularity of the progression of a suitable order of differences. If the tabular quantities are the results of calculation from a given formula, with equidistant arguments, by differencing them up to a certain order the existence of an isolated error, if one should exist, is thus prominently exhibited and therefore speedily detected. Moreover the precise locality of the error and the fact of its individuality being indicated by a characteristic interruption of the law of progression, the proper correction is ascertained by carefully noting the central position of the disturbance, and revising the corresponding portion of the original computation. This work of revision may, however, be dispensed with, if necessary or desirable, as the magnitude as well as the locality of the indicated error can in general be readily deduced, with sufficient accuracy, directly from the differences, by the method of adjustment here given. The method, being so far independent of fundamental origin, is therefore equally applicable to the adjustment of a series of quantities derived from observation, or from statistical data; only in this latter case, as almost every quantity must be more or less affected by imperfections arising from defective information and other known or unknown incidental causes, the influence of which may sometimes be augmented by paucity of numbers, the progression of the differences is disturbed by a more complicated combination of the effects of an irregular series of errors, and it becomes more difficult to eliminate the value of each separate correction.

It is the object of the present article to put together a number of formulae which it may be useful to the actuary to find in one place. At the same time it may show all persons who possess an elementary knowledge of algebra, that they may, with no great amount of tables, and processes of very easy application, learn to compute the value of any benefit in which the duration of one life only is concerned. The same principles, with more extensive tables, apply to cases in which two or more lives are involved.

In briefly commenting upon the papers that have been presented to Parliament on the subject of life assurances and annuities now granted by Government, it is less my object to criticize the measures that have been adopted, than to explain the mode of construction of the several tables of rates that are being acted upon. In consequence of several assumptions and suppositions, these have not been computed in the mode usually adopted by actuaries.

In the month of June, 1862, after the meeting of the second International General Average Congress held in London, a committee was constituted, “for the purpose of establishing one uniform system of general average throughout the mercantile world,” The meeting of the council of the National Association for the Promotion of Social Science, held in York in the autumn of 1864, set apart three days for the consideration of this branch of jurisprudence; and the 26th of September and two following days were occupied with the discussion of the various disputed points connected with the subject, under the presidencies of Sir James Wilde and Sir Fitzroy Kelly. The last-named gentleman, in closing the sitting, in the course of his speech gave his opinion as to the course to be pursued in order to give the force of law to the amendments which had been proposed, with the view to promote the uniformity which is so desirable in connection with the adjustment of claims for general average. He considered that “in order to obtain a legislative sanction to the code which had just been completed, it would be advisable to obtain the distinct approval of the leading commercial bodies, particularly the Chambers of Commerce in the great towns; and to obtain, if possible, assurances on the part of the foreign Governments that they would be prepared to adopt the code upon its adoption in this country. …If possible, the code or rules should be made a Government measure; failing this, it should be entrusted to at least two independent members, one of whom must be a mercantile man, representing a mercantile constituency, and the other a lawyer of eminence; and that it would be desirable to go to work at once, while the public interest was alive to the measure.”

The Mythological Astronomy of the ancients; part the second: or the key of Urania, the wards of which will unlock all the mysteries of antiquity. Norwich, 1823, 12mo.

A Companion to the Mythological Astronomy, &c., containing remarks on recent publications. … Norwich, 1824, 12mo.

A new Theory of the Earth and of planetary motion; in which it is demonstrated that the Sun is vicegerent of his own system. Norwich, 1825, 12mo.

The analyzation of the writings of the Jews, so far as they are found to have any connexion with the sublime science of astronomy. [This is pp. 97–180 of some other work, being all I have seen.]

Every branch of exact science has its paradoxer. The world at large cannot tell with certainty who is right in such questions as squaring the circle, &c. Mr. Weyman Lee was the assailant of what all who had studied called demonstration in the question of annuities; and he can be exposed to the world: for his error arose out of his not being able to see that the whole is the sum of all its parts.

The twelve-figure table which accompanied Part I. was extracted from a table extending to twenty-four places, which is still in manuscript. Hence the details I am about to give of the methods of construction and verification made use of, will necessarily have reference to the more extensive table.

In a paper read before this Society on the 29th of April, 1861, and contained in the ninth vol. of the Journal, there is given a table showing the probability of living a year at each age, as evidenced by the males of the families of the peerage existing within the last half century; and it appeared to me that it would be worth while, if only as a matter of curiosity, to put the information there given into a form better adapted to ordinary calculation.