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The writers upon the fubjedt of annuities on lives 
have very juftly diftinguifhed them into two kinds : 
In the firfl, the annuitant is entituled to receive a pay- 
ment if he be alive on the day on which it becomes 
due ; but if he dies on the preceding day, or fooner, 
his heirs have no claim to any part of the payment, 
fo to have became due ; but in the fecond, if the an- 
nuitant dies at any intermediate time, between the 
days of payment, his heirs are to receive a part of 
the annuity, proportional to the time elapfed, between 
the preceding day of payment and the annuitant’s 
deceafe. 
This latter kind of annuities have been diftinguifhed 
from the former, by the words, fecured by a grant of 
lands ; becaufc, where lands are leafed for lives, the 
conditions are generally fuch as are above deferibed. 
The values of the firft kind of annuities have 
been inveftigated upon principles purely arithmetical; 
but, in order to perform the latter, fluxions have been 
ufed (as I humbly conceive) without any neceffity : 
But as the inveftigation of the former may be ufefully 
made a part of the latter, I fhall firft recite the me- 
thod of performing that, and then proceed to attempt 
the other, upon the fame principles. 
If, with the fiigacious mathematician Mr. Abra- 
ham De Moivre, we fuppofe the decrements of life 
to be equal {viz. that out of a number of perfons, 
alive at a given age, equal to the number of years 
that a perfon of that age hath a poflibility of living, 
there will die one in each year, till they are all ex- 
tindt) ; then, out of a number of chances equal to 
that number of perfons, which may, for infiance, 
be 
