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very moment of the life’s expiring ; and this by a 
proper, accurate, and geometrical calculation. 
I have been induced to take this method, for the 
following reafons j firft, by this fuppofition, the 
value of lives would receive but an inconfiderable 
increafe * fecondly, by this means, the feveral inter- 
vals of life, which, in the tables of obfervations, 
are found to have uniform decrements, may be the 
better conne&ed together. It is with this view that 
I have framed the two following problems, with 
their folutions. 
Problem I. 
To find the 'value of an annuity , fio c ircumfi an- 
ti ate d , that it fhall be on a life of a given age ; 
and that , upon the failing of that life , fiuch a 
part of the rent fhall be paid to the heirs of the 
late poffeffor of an annuity , as may be exactly 
proportioned to the time intercepted between that 
of the lafl payment , and the very moment of the 
life’s failing. 
Solution. 
L et n reprefent the complement of life, that is, 
the interval of time between the given age, 
and the extremity ofold-age, fuppos’d at 86. 
r the amount of i /. for one year. 
a. the logarithm of r. 
SP the prefent value of an annuity of i /. for 
the given time. 
Gfi the value of the life fought. 
Then — — = Q. 
r — i an 
I 2 
De- 
