Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/92

82 however, improved and extended by the addition of the columns (M and R) for finding the values of assurances. Davies s treatise on annuities, as issued by his executors in 1855, with the explanation that it is an uncompleted work, but that the completed portion had been in print since 1825, contains several other tables of the same kind. In the pre face to this work it is stated that " the most important dis tinction between the two methods is, that Mr Davies s method is much simpler in principle than that of Mr Barrett, as the columnar numbers given by the latter must be con sidered more as the numerical results of algebraical expres sions ; whereas in Davies s arrangement it will be found, on reference to age 0, that the number in column D represents the number of children just born, and those opposite ages 1, 2, 3, 4, &c., to the end of life, the present sums which would be required for the payment of 1 to each survivor of such children at the end of 1, 2, 3, 4, &c., years to the extremity of life ; and the sum thereof inserted in column N, opposite age 0, represents the present fund required to provide the payment of annuities of 1 each for life to all the children given in column D at age ; and from this method very considerable amount of labour is avoided by multiplying the number living at each age by a fraction less than a unit ; but by Barrett s method, the number living at each age has to be multiplied by the amount of 1 improved for as many years as are equal to the difference between that age and the greatest tabular duration, as already stated, which makes each product a large multiple of the number living." This passage, we are informed, correctly represents Mr Davies s own views on the subject. It may be noticed that Davies does not employ the notation used above, D x, N z , &c., but omits the subscript x. Thus, instead of the formula r&eq ^- he would write N. In some respects this notation is perhaps preferable to tha t now used, as it is certainly better, when there is no risk of confusion, to omit the subscript x. But Davies s notation cannot be adopted without alteration, as N x might be mistaken for the number in the column N" oppo site the age 1. We may, however, consistently with the principles of the notation adopted by the Institute of Actuaries, write the formula _rj^ s&eqrnj. The notation at present commonly used is due to David Jones, whose work (mentioned below) was the first that contained an extensive series of commutation tables. On a general review of the whole evidence, we cannot help thinking that Barrett s merits in the matter have been somewhat exaggerated. The first idea of a commutation table was not due to him, but (leaving Tetens out of view) to Dale and Morgan ; and it is certain that he was familiar with the latter s treatise. The change he introduced into the arrangement of the table, namely, multiplying by a power of (1 +i) instead of by a power of v, is the reverse of an improvement ; and accordingly, his form of table has never been in practical use by any person but himself, excepting only Babbage. It is, of course, not to be denied that great credit is due to him as a self-educated man, for perceiving more clearly than his predecessors the great usefulness of the commutation table ; but in our opinion he does not stand sufficiently out from those who preceded and followed him, to justify the attempt to attach his name to the columnar method of calculating the values of annuities and assurances. Those who desire to .obtain further information on the matter, and to learn the views of other writers, can refer to the appendix to Baily s Life Annuities and Assurances, De Morgan s paper " On the Calculation of Single Life Contingencies," Assurance Magazine, xii. 348-9 ; Gray s Tables and Formulae, chap. viii. ; the preface to Davies s Treatise on Annuities ; also Hend- riks s papers in the Assurance Magazine, No. 1, p. 1, and No. 2, p. 12 ; and in particular De Morgan s " Account of a Correspondence between Mr George Barrett and Mr Francis Baily," in the Assurance Magazine, vol. iv. p. 185. The principal D and N tables published in this country are contained in the following works:— 1em 1em 1em 1em 1em 1em 1em 1em 1em 1em 1em 1em The explanations of the tables in the last four works are by Dr William Fair, F.R.S. Very unfortunately, these tables are not all arranged upon the same principle, but those contained in the Reports of the Registrar-General, in the English Life Table, in Chisholm s and in Henry s tables, are so arranged that the column N is shifted down one year, so that in them the ratio N&eqp gives, not the value of the ordinary annuity, but the -L a; value of the annuity increased by unity, or the annuity-due. It is very needful to bear this in mind for the prevention of error ; and the existence of a difference of this kind is ex tremely perplexing. For information upon the subject of this confusing change, see De Morgan s paper " On the Forms under which Barrett s Method is represented, and on Changes of words and symbols," Ass. Mag., x. 302. All the preceding methods require a considerable amount of calculation in order to obtain the value of an annuity on a life of any particular age. We will now explain some methods of approximation, by means of which we can calculate with much less labour the value of an annuity at