VII. If Three Lives are proposed, to find the Value of an Annuity during the continuance of any of those three Lives; the Rule is, As the Product of the continual Multiplication of the Three Numbers, in the Table, answering to the Ages proposed, is to the difference of that Product, and of the Product of the Three Numbers of the Deceased of those Ages, in any given Term of Years: So is the present Value of a Sum of Money, to be paid certainly after so many Years, to the present Value of the same Sum to be paid, provided one of those Three Persons be living at the Expiration of that Term. Which Proportion being yearly repeated, the Sum of all those present Values will be the Value of an Annuity granted for three such Lives. But to explain this, together with all the Cases of Survivance in Three Lives: Let N be the Number in the Table for the younger Age, n for the second, and ν 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 υ those of the elder Age; and let R be the Remainder of the younger Age, r that of the middle Age, and ρ the Remainder of the elder Age. Then shall R + Y be equal to N, r + y to n, and ρ + υ to ν, and the continual Product of the three Numbers N, n, ν, shall be equal to the continual Product of R + Y × r + y × ρ + υ, which being the whole Number of Chances for three Lives, is compounded of the eight Products following. (1) Rrρ, which is the Number of Chances that all three of the Persons are living. (2) rρY, which is the Number of Chances that the two elder Persons are living, and the younger dead. (3) Rρy the Number of Chances that the middle Age is dead, and the younger and elder living. (4) Rrυ being the Chances that the two younger are living, and the elder dead. (5) ρYy the Chances that the two younger are dead, and the elder living. (6) rYυ the Chances that the younger and elder are dead, and the middle Age living. (7) Ryυ, which are the Chances that the younger is living, and the two other dead. And Lastly and Eighthly, Yyυ, which are the Chances that all three are dead. Which latter subtracted from the whole Number of Chances Nnν, leaves Nnν - Yyυ 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 ν. And the whole Parallelepipedon shall be as the Product Nnν, 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 ρ, and NA as υ; 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 Rrρ shall be as the Solid STWIFZeb. The Second, or rρY will be as the Solid EYZeQSMI. The Third, Rρy, as the Solid RHOVWIST. And the Fourth, Rrυ, as the Solid ZabDWXIK. Fifthly, ρYy, as the Solid GQRSIMNO. Sixthly, rYυ, as IKLMGYZA. Seventhly, Ryυ, as the Solid IKPOBXVW. And Lastly, AIKLMNOP will be as the Product of the 3 Numbers of Persons dead, or Yyυ. 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 Nnν to the Product Ryυ; 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. Nnν. The second changing yearly according to the Decrease of R, r, ρ, and Increase of Y, y, υ. 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 Decimal Values of one Pound payable after the Number of Years in the Margent, at the Rate of 6 per Cent.

Years.Pres. Value
of 1 l.		Years.Pres. Value
of 1 l.		Years.Pres. Value
of 1 l.
10,9434190,3305370,1158
20,8900200,3118380,1092
30,8396210,2941390,1031
40,7921220,2775400,0972
50,7473230,2618450,0726
60,7050240,2470500,0543
70,6650250,2330550,0406
80,6274260,2198600,0303
90,5919270,2074650,0227
100,5584280,1956700,0169
110,5268290,1845750,0126
120,4970300,1741800,0094
130,4688310,1643850,0071
140,4423320,1550900,0053
150,4173330,1462950,0039
160,3936340,13791000,0029
170,3714350,1301
180,3503360,1227

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 Nnν - Yyυ for the second Term of the Proportion in finding the Value of Three Lives, it may suffice to use only Yyυ, 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 Nnν to 8 in 8 in 9, or 576, or Yyυ for the first Year, so 0,9434. to 0,00000348.

To 15 in 16 in 18, or 4320, for the second Year, so 0,8900. to 0,00002462.

To 21 in 24 in 28, or 14112 for the third Year, so 0,8396. to 0,00008128.

To 27 in 32 in 38, for the fourth Year, so 0,7921. to 0,00016650.

To 33 in 41 in 48, for the fifth Year, so 0,7473. to 0,00031071.

To 39 in 50 in 58, for the sixth Year, so 0,7050. to 0,00051051.