be the Number in the Table for the Younger Age, n for the Second, and for the Elder Age; let Y be those dead of the Younger Age in the term proposed, y those dead of the Second Age, and v those of the Elder Age; and let R be the Remainder of the younger Age, r that of the middle Age, and p the Remainder of the Elder Age. Then shall R+Y be equal to N, r+y to n, and p+v to v, and the continual Product of the three Numbers Nnv shall be equal to the continual Product of R+Yxr+Yxp+v, which being the whole number of Chances for three Lives is compounded of the eight Products following. (1) Rrp, which is the number of Chances that all three of the Persons are living. (2) rpY, which is the number of Chances that the two Elder Persons are living, and the younger dead. (3) Rpy the number of Chances that the middle Age is dead, and the younger and Elder living. (4) Rru being the Chances that the two younger are living, and the elder dead. (5) pYy the Chances that the two younger are dead, and the elder living. (6) rYu the Chances that the younger and elder are dead, and the middle Age living. (7) Ryu, which are the Chances that the younger is living, and the two other dead. And Lastly and Eightly, Yyu, which are the Chances that all three are dead. Which latter substracted from the whole number of Chances Nnv, leaves Nnv- Yyu the Sum of all the other Seven Products; in all of which one or more of the three Persons are surviving. To make this yet more evident, I have added Fig. 8. wherein these Eight several Products are at one view exhibited. Let the rectangled Parallelepipedon ABCDEFGH be constituted of the sides AB, GH, &c. proportional to N the number of the younger Age; AC, BD, &c. proportional to n; and AG, CE, &c. proportional to the number of the Elder, or v. And the whole Parallelepipedon shall be as the Product Nnv, or our whole number of Chances. Let BP be as R, and AP as Y; let CL be as r, and Ln as y; and GN as p, and NA as v; and let the Plain PRea be made parallel to the plain ACGE; the plain NVbY parallel to ABCD; and the plain LXTQ parallel to the plain ABGH. And our first Product Rrp shall be as the Solid STWIFZeb. The Second, or rpY will be as the Solid EYZeQSMI. The Third, Rpy, as the Solid RHOVWIST. And the Fourth, Rrv, as the Solid ZabDWXIK. Fifthly, pYy, as the Solid GQRSIMNO. Sixthly, rYu, as IKLMGYZA. Seventhly, Ryv, as the Solid IKPOBXVW. And Lastly, AIKLMNOP will be as the Product of the 3 numbers of persons dead, or Yyv. I shall not apply this in all the cases thereof for brevity sake; only to shew in one how all the rest may be performed, let it be demanded what is the value of the Reversion of the younger Life after the two elder proposed. The proportion is as the whole number of Chances, or Nnv to the Product Ryv, so is the certain present value of the Sum payable after any term proposed, to the value due to such Chances as the younger person has to bury both the elder, by the term proposed; which therefore he is to pay for. Here it is to be noted, that the first term of all these Proportions is the same throughout, viz. Nnv. The Second changing yearly according to the Decrease of R, r, p, and Encrease of Y, y, v. And the third are successively the present values of Money payable after one, two, three, &c. years, according to the rate of Interest agreed on. These numbers, which are in all cases of Annuities of necessary use, I have put into the following Table, they being the Decimal values of one Pound payable after the number of years in the Margent, at the rate of 6 per Cent. It were needless to advertise, that the great trouble of working so many Proportions will be very much alleviated by using Logarithms; and that instead of using Nnv-Yyu for the Second Term of the Proportion in finding the value of Three Lives, it may suffice to use only Yyu, and then deducting the Fourth Term so found out of the Third, the Remainder shall be the present value sought; or all these Fourth Terms being added together, and deducted out of the value of the certain Annuity for so many Years, will leave the value of the contingent Annuity upon the Chance of Mortality of all those three Lives. For Example; Let there be Three Lives of 10, 30, and 40 years of Age proposed, and the Proportions will be thus: As 661 in 531 in 445 or 156190995, or Nnv to 8 in 8 in 9, or to 33 in 41 in 48, to 39 in 50 in 58, 576, or Fyv for the first year, So 0,9434. to 0,00000348 for the second year, so 0,8900. to 0,00002462 for the third year, so 0,8396. to 0,00008128 for the fourth year, so 0,7921. to 0,00016650 for the fifth year, so 0,7473. to 0,00031071 for the sixth year, so 0,7050. to 0,00051051 And so forth to the 60th year, when we suppose the elder Life of Forty certainly to be expired; from whence till Seventy we must compute for the First and Second only, and from thence to Ninety for the single youngest Life. Then the Sum Total of VOL. XVIII, T all these Fourth Proportionals being taken out of the value of a certain Annuity for 90 Years, being 16, 58 years Purchase, shall leave the just value to be paid for an Annuity during the whole term of the Lives of three Persons of the Ages proposed. And note, that it will not be necessary to compute for every year singly, but that in most cases every 4th or 5th year may suffice, interpoling for the intermediate years secundum artem. It may be objected, that the different Salubrity of places does hinder this Proposal from being universal; nor can it be denied. But by the number that die, being 1174. per Annum in 34000, it does appear that about a 30th part die yearly, as Sir William Petty has computed for London; and the number that die in Infancy, is a good Argument that the Air is but indifferently salubrious. So that by what I can learn, there cannot perhaps be one better place proposed for a Standard. At least 'tis desired that in imitation hereof the Curious in other Cities would attempt something of the same nature, than which nothing perhaps can be more useful. Some further Considerations on the Breslaw Bills of Mortality. By the same Hand with the former. Phil. Trans. vol. xvii. p. 653 (No. 198, for March 1693). SIR,—What I gave you in my former Discourse on these Bills, was chiefly designed for the Computation of the Values of Annuities on Lives, wherein I believe I have performed what the short Period of my Observations would permit, in relation to exactness, but at the same time do earnestly desire, that their Learned Author Dr. Newman of Breslaw would please to continue them after the same manner for yet some years further, that so the casual Irregularities and apparent Discordance in the Table p. 599. may by a certain number of Chances be rectified and ascertain❜d. Were this Calculus founded on the Experience of a very great number of Years, it would be very well worth the while to think of Methods for facilitating the Computation of the Value of two, three, or more Lives; which as proposed in my former, seems (as I am inform'd) a Work of too much Difficulty for the ordinary Arithmetician to undertake. I have sought, if it were possible, to find a Theorem that might be more concise than the Rules there laid down, but in vain; for all that can be done to expedite it, is by Tables of Logarithms ready computed, to exhibit the Rationes of N to Y in each single Life, for every third, fourth or fifth Year of Age, as occasion shall require; and these Logarithms being added to the Logarithms of the present Value of Money payable after so many Years, will give a Series of Numbers, the Sum of which will shew the Value of the Annuity sought. However for each Number of this Series two Logarithms for a single Life, three for two Lives, and four for three Lives, must necessarily be added together. If you think the matter, under the uncertainties I have mentioned, to deserve it, I shall shortly give you such a Table of Logarithms as I speak of, and an Example or two of the use thereof: but by Vulgar Arithmetick the labour of these Numbers were immense; and nothing will more recommend the useful Invention of Logarithms to all Lovers of Numbers, than the advantage of Dispatch in this and such like Computations. Besides the uses mentioned in my former, it may perhaps not be an unacceptable thing to infer from the same Tables, how unjustly we repine at the shortness of our Lives, and think ourselves wronged if we attain not Old Age; whereas it appears hereby, that the one half of those that are born are dead in Seventeen years time, 1238 being in that time reduced to 616. So that instead of murmuring at what we call an untimely Death, we ought with Patience and unconcern to submit to that Dissolution which is the necessary Condition of our perishable Materials, and of our nice and frail Structure and Composition: And to account it as a Blessing that we have survived, perhaps by many Years, that Period of Life, whereat the one half of the whole Race of Mankind does not arrive. A second Observation I make on the said Table, is that the Growth and Encrease of Mankind is not so much stinted by anything in the Nature of the Species, as it is from the cautious difficulty most People make to adventure on the state of Marriage, from the prospect of the Trouble and Charge of providing for a Family. Nor are the poorer sort of People herein to be blamed, since their difficulty of subsisting is occasion'd by the unequal Distribution of Possessions, all being necessarily fed from the Earth, of which yet so few are Masters. So that besides themselves and Families, they are yet to work for those who own the Ground that feeds them: And of such does by very much the greater part of Mankind consist; otherwise it is plain, that there might well be four times as many Births as we now find. For by computation from the Table, I find that there are nearly 15000 |