299 
the decrement of human life. 
The present values of 10 shillings are therefore assumed ; 
I, at 1 year .48077 ; II, .48058 
2 years .46228 .46192 
3 years .44450 .44398 
4 years .42740 .42674 
5 years .41096 .41018 
The latter column exhibiting obviously a larger deduction 
for discount than the former ; so that the rate of interest in 
the two calculations is by no means the same : although in 
the case of r=. 05, they would respectively represent the 
highest rate of interest allowed by our laws to be received 
without a new investment or engagement : but this arbitrary 
restraint ought certainly not to affect the mathematical con- 
sideration of the question. 
The difficulty, if any person thinks it such, may be avoided 
by a mode of investigation which I have lately had occasion 
to point out. “ An annuity, of which a payment is due on a 
given day, is more valuable than an annuity purchased on 
that day, and to commence a year after, by the amount of 
a year’s payment : and the value of a life annuity , becoming 
payable at any intermediate time between the day of purchase and 
its first anniversary , will be greater than the simple tabular 
value of the annuity by a sum proportional to the anticipation of 
the payment the increase of the value being very nearly 
uniform, when we suppose the anticipation to be gradually 
increased : this increase of the value comprehending ob- 
viously the greater probability as well as the greater proximity 
of each payment, and proceeding from day to day by very 
nearly equal increments. Thus, if we wished to purchase an 
