3oo Dr. Young’s formula for expressing 
annuity of £100. a year, and its value were £1000., upon 
the ordinary supposition of the payments commencing after 
the end of a year ; supposing that we desired to have the 
first payment made at the end of nine months, and the sub- 
sequent payments at annual intervals as ’ usual, we should 
have to add £25. to the purchase money, making it £1025. 
at whatever rate of interest the value might have been com- 
puted. If we began at six months, £50., and if at three 
months, £ 75. must be added to the purchase : it being ob- 
vious that an additional £100. would be equivalent to an 
anticipation of twelve months, or to an immediate payment 
of a year’s annuity. 
From this simple and incontestable principle, in which the 
second differences only are neglected, it is very easy to de- 
duce the values of annuities, payable at intervals shorter than 
a year. A11 annuity of 1, payable half yearly, is equal to two 
annuities of ■§•, the one beginning as usual at the end of the 
year, the other anticipated by half a year ; and the value of 
this portion is greater than the other by half of one of the 
payments, that is, by ^ : so that “ We may always find the 
value of a life annuity payable half yearly , by adding a quarter of 
a year to the tabular value of the same annuity 
In a similar manner it is very easily shown, that “ for 
quarterly payments , we must add J- of a year’s value to the com- 
putation made on the supposition of annual payments and 
“ the continual bisection of the interval would at last afford 
us the addition of half a yearly payment for the value of a daily 
or hourly payment of a proportional part of the given annuity 
“ It may also be observed, that when we reckon at 3 per 
