40 M E M O I R S of tfje 
du6l of the two numbers of perfbns of the two given a^es 5 and by 
"ivhat was faid before, after the term propoled, the rectangle H D 
will be as the number of perfbns of the younger age that lurvive 5 
and the rectangle A E as the number of thofc that die ; fb like- 
wife the rectangles AI, FD will be as the numbers living and 
dead of the other age 5 hence the rectangle HI will be as an 
equal number of both ages furviving^ the re6langle FE being 
the produfl: of the decealed, or Yjy, an equal number of both 
dead 5 the rectangle GD or Ky, a number living of the younger 
age, and dead of the elder 5 and the rectangle A G or r Y a num- 
ber living of the elder, but dead of the younger : This being un- 
derftood, it is obvious, that as the whole re61:anole A D or N ;; is 
to the gnomon FABDEGorN;; Y}', (b is the whole number 
of perfons or chances, to the number of chances that one of the two 
pcrfons is living; and as AD, or N;/, is to FE, or Yy, fo 
are all the chancevS, to the chances that both are dead ; whereby 
may be* computed the value of the reverfion after both lives ; and 
as A D to G D or R^, fo are the whole number of chances, to 
the chances that the younger is living, and the other dead ; 
whereby may be compared, what value ought to be paid for the 
revcrfion of one life after another, as in the cafe of providing for 
cleroymen's widows, and others by fuch reverfions j and as A D 
3s to A G, or r Y, fo are all the chances, to thofe that the elder 
furvives the younger'. 
Ufe feventh, if three lives are propofed, to find the value of 
an annuity, during the continuance of any of thole three lives ^ 
the rule is, as the produfi: of the continual multiplication of the 
three numbers in the table, anfwering to the ages propofed, is to 
the difference of that produ6t, and of the produft of the three 
numbers of the decealed of thole ages, in any given term of 
Tears • lb is the prefent value of a fum of money to be paid cer- 
tainly after fo many years, to the prefent value of the fame lum 
to be paid, provided one of thole three perfons be living at the 
expiration of that term; which proportion being yearly repeated, 
the fum of all thofe prefent values will be the value of an annuity, 
j^ranted for three fuch lives ; but to explain this, together with all 
the calcs of furvivance in three lives 3 let N be the number in the 
table for the younger age, ;; for the fecond, and v for the elder ; 
let Y be thole 'dead^of the younger -in the term propofed, y thofe 
dead of the fecond age, and v thofe dead of the elder; and let 
R be the remainder of the younger age, r that of the middle age, 
and f the remainder o( the elder ; then will R -}- Y be z=: N, 
r -]- r ^znandg-j- "J :^ v-, and the continual product of the 
ifcrec 
