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.”

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.

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.

The celebrated interminable fraction 3·14159…, which the mathematician calls π, is the ratio of the circumference to the diameter. But it is thousands of things besides. It is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, π must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention. As it is, our trigonometry being founded on the circle, π first appears as the ratio stated. If, for instance, a deep study of probable fluctuation from the average had preceded geometry, π might have emerged as a number perfectly indispensable in such problems as—What is the chance of the number of aces lying between a million +x and a million −x, when six million of throws are made with a die? I have not gone into any detail of all those cases in which the paradoxer finds out, by his unassisted acumen, that results of mathematical investigation cannot be: in fact, this discovery is only an accompaniment, though a necessary one, of his paradoxical statement of that which must be. Logicians are beginning to see that the notion of horse is inseparably connected with that of non-horse: that the first without the second would be no notion at all. And it is clear that the positive affirmation of that which contradicts mathematical demonstration cannot but be accompanied by a declaration, mostly overtly made, that demonstration is false. If the mathematicians were interested in punishing this indiscretion, he could make his denier ridiculous by inventing asserted results which would completely take him in.

Mr. Baron Maseres, in the preface to his cumbrous work on the Principles of the Doctrine of Life Annuities (1783, quarto), at p. xi., after referring to and commending for its utility the method for finding the value of an annuity on (x) in terms of that of an annuity on (x + 1), has the following:—

“This method was first communicated to me by Dr. Price, but it was published in the year 1779, by Mr. William Morgan, the actuary to the Society for Equitable Assurances, near Blackfriars Bridge, in his Treatise on the Doctrine of Annuities and Assurances on Lives, pp. 56, 57; and it had been published before by Dr Price himself, in his Treatise on Reversionary Payments, Note 0 of the Appendix, and likewise by Mr. Thomas Simpson, in his book on Life Annuities, Prob. 1, Coroll. 7; which last book was published so long ago as the year 1742.