From which, any one of the quantities may be found, when the rest are given.

As to the whole interest, it is found by barely subtracting the principal p from the amount a.

Example. Suppose it be required to find, in how many years any principal sum will double itself, at any proposed rate of compound interest.

In this case the 4th theorem must be employed, making a=2p; and then it is

So, if the rate of interest be 5 per cent. per annum; then R=1+05 = 1.05; and hence

t=

log. 2 log. 1.05

=

•301030
⚫021189

14.2067 nearly;

that is, any sum doubles itself in 14 years nearly, at the rate of 5 per cent. per annum compound interest.

Hence, and from the like question in simple interest, above given, are deduced the times in which any sum doubles itself at several rates of interest, both simple and compound; viz.

The following Table will very much facilitate calculations of compound interest on any sum, for any number of years, at various rates of interest.

COMPOUND INTEREST.

The Amounts of 17 in any Number of Years.

1

1.0300 1.0350 1.0400 1.0450 1.0500 1.0600 21.0609 1.0712 1.0816 1.0920 1.1025 1.1236 31.0927 1.1087 1.1249 1.1412 1.1576 1.1910 41.1255 1.1475 1.1699 1.1925 1.2155 1.2625 1.1593 1.1877 1-2167 1.2462 1.2763 1.3382 6 1.1948 1.2293 1.2653 1.3023 1.3401 | 1.4185

5

7

8

1.4071 1.5036

1.2299 1.2723 1.3159 1.3609 1.2668 1.3168 1.3686 1.4221 1.4775 1.5939 9 1.3048 1.3629 1.4233 1.4861 1.5513 1.6895 10 1.3439 1.4106 1.4802 1.5530 1.6289 1.7909 11 1.3842 1.4600 1.5895 1.6229 1.7103 1.8983 12 1.4258 1.5111 1.6010 1.6959 1.7959 2.0122 13 1.4685 1-5640 1.6651 1-7722 1.8856 2.1329 14 1.5126 1-6187 1-7317 1.8519 1.9799 2.2609 15 1.5580 1.6753 1.8009 1.9353 2-0789 2.3966 16 1-6047 1-7340 1.8730 2.0224 2.1829 2.5404 17 1-6528 1.7947 1.9479 2-1134 2.2920 2-6928 18 1-7024 1.8575 2.0258 2.2085 2.40662-8543 19 1.7535 1.9225 2.1068 2.3079 2.5270 3.0256 20 1-8061 1-9828 2.1911 2-4117 | 2·6533 | 3.2071

The use of this Table, which contains all the powers, R2, to the 20th power, or the amounts of 17, is chiefly to calculate the interest, or the amount of any principal sum, time, not more than 20 years.

for any

annum

For example, let it be required to find, to how much 5231 will amount in 15 years, at the rate of 5 compound interest.

cent. per per

In the table, on the line 15, and in the column 5 per cent.

is the amount of 11, viz.
this multiplied by the principal

2.0789

523

1087-2647
10871 5s 31d.

5641 5s 31d.

Note 1. When the rate of interest is to be determined to any other time than a year; as suppose to a year, or a year, &c.: the rules are still the same; but then t will express that time, and R must be taken the amount for that time also.

Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year; it may be determined in the following manner :

1st, For any time which is some aliquot part of a year :Find the amount of 17 for 1 year, as before; then that root of it which is denoted by the aliquot part, will be the amount of 11. This amount being multiplied by the principal sum, will produce the amount of the given sum as required.

2d, When the time is not an aliquot part of a year :Reduce the time into days, and take the 365th root of the amount of 1l for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before. And in these calculations, the operation by loga. rithms will be very useful.

OF ANNUITIES.

ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual pay

ments.

Annuities are divided into those that are in Possession, and those in Reversion: the former meaning such as have commenced; and the latter such as will not begin till some particular event has happened, or till after some certain time has elapsed.

When an annuity is forborn for some years, or the payments not made for that time, the annuity is said to be in Arrears.

An annuity may also be for a certain number of years; or it may be without any limit, and then it is called a Perpetuity.

The Amount of an annuity, forborn for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due.

ANNUITIES.

The Present Worth or Value of an annuity, is the price. or sum which ought to be given for it, supposing it to be bought off, or paid all at once.

Let a the annuity, pension, or yearly rent;

n = the number of years forborn, or lent for;
R = the amount of 17 for 1 year;

m = the amount of the annuity;
v = its value, or its present worth.

Now, 1 being the present value of the sum R, by proportion the present value of any other sum a, is thus found:

year hence.

is the present value of a due 2 years

be the present values of a, due at the years respectively. Consequently the

α

+

R

+ +

R2 R3

R4

1 1

R

α a a

R3 R4

R5

&c. will

end of 3, 4, 5, &c. sum of all these, or

+ + + &c.) X

R2

R4
R3

a continued to n terms, will be the present value of all the n years' annuities. And the value of the perpetuity, is the sum of the series to infinity.

But this series, it is evident, is a geometrical progression,

1

but for its first term and common ratio, and the having R number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be

When the annuity is a perpetuity; n being infinite, R*

; that is, any annuity divided by

the interest of 17 for 1 year, gives the value of the perpetuipcr cent. ty. So, if the rate of interest be 5

=

Then 100a 5 = 20a is the value of the perpetuity at 5 per cent. Also 100a ÷ 4 25a is the value of the perpetuity at 4 per cent. And 100a ÷ 3 = of the perpetuity at 3 per cent. : and so on.

---

33a is the value

Again, because the amount of 17 in n years, is R", its increase in that time will be R". 1; but its interest for one single year, or the annuity answering to that increase, is 1; therefore, as R - 1 is to R1, so is a to m; that X a. Hence, the several cases relating to

R

is, m =

Rn 1

R

Annuities in Arrear, will be resolved by the following equations:

In this last theorem, r denotes the present value of an annuity in reversion, after p years, or not commencing till after the first p years, being found by taking the difference Rn 1 a

between the two values

years and p years.

R

RP 1 a

X and

= for n

RP

But the amount and present value of any annuity for any number of years, up to 21, will be most readily found by the two following tables.