for the Valuation of Life- annuities. 
1 19 
By the fame Heps it will appear, that the prefent value 
of an annuity certain for n years to be received in quar- 
terly payments, each a quarter of the annual payment, is 
1 q. And alfo, that the pre- 
1 
- r 
4 
I r 
-rxif- 
4 4 
4 » 
• — P 
r X I + - 
4 
4 « 
fent value of an annuity certain for n years, to be re- 
ceived in momently payments, each the fame propor- 
tional part of the yearly payment that the moment is of 
the year, mull be p 
r 
\ 1000, &c, n 
r x 1 H 
1- 
IOOO, & 
cJ 
But, by the 
binomial theorem, 1 + 
IOOO, &C. H 
= i +rn 
r*n* 
2x3. 
r 4 « 4 
+ 
IOOO, <xc. 
, See. which feries approximates indefinitely to 
2x3x4’ x x J 
that number of which rn is the hyperbolic logarithm, by 
prob. 1. fedhxi. vol.II. of Mr. simpson’s Fluxions; or by 
prop. 1 .p.40 . of his Treatife on Trigonometry. Therefore, 
- = p--L=m, as explained before* 
p- 
r x I -f 
IOOO, &c. 
] 000, &c* 
See p. no. 
If the value of an annuity of Jf. 1 for n years is. 
required payable half-yearly, and the half-yearly in- 
1 tereft of f. 1 , inftead of being half the yearly intereft 
(or - j, is fuppofed to be 1 + r' z - 1 ; the anfwer will be> 
°>5 
0,5 
°,5 °»5 
1 1 + r 1 -|- ^ i I 4 rl 
a ) 
&:c. continued to 2 n terms = 
