To find the value of an annuity to continue forever, at a given rate of compound interest.

Let p equal present worth, u equal annuity, r equal interest of £1, or $1, R equal 1+r equal principal and interest of £1, or $1, for one year.

Xu. But since t is infinite, Rt is infinitely

R

greater than 1, whence R'—1=R' and p= (x=)

TR

Suppose a freehold estate (or one which is bought to continue forever) of $100 per annum; what is it worth, allowing the buyer & per cent for his money?

6

If a freehold estate is bought for $16662, the allowance of 6 per cent made to the buyer; I demand the yearly

rent.

By theorem 2. 1666 x 1.06-1-$100. Ans.

CASE 3. EXAMPLE 3.

If a real estate of $100 per annum, be sold for $16663, Idemand the rate per cent.

By theorem 3.

1666+100

1.06 amount

16662

of £1, or $1, at 6 per cent, and 1.06-1.06-186 6 per cent. Ans.

NOTE. As the time is supposed to be infinite, in such questions as the preceding, no formula is necessary to shew the value of t.

Ef 2.

PURCHASING FREEHOLD ESTATES IN REVERSION.

From the doctrine advanced in case 1 of the purchas

If a freehold estate of $100 per annum, to commence 2 years hence, is to be sold, what is its worth, allowing the purchaser 6 per cent for his present payment?

By theorem 1.

Suppose a freehold estate, to commence 2 years hence, is sold for $1483.2384, allowing the purchaser 6 per cent for his present payment; what is the yearly income?

By theorem 2.

and 1666

1483.2384x 1.06% = 1666 x 1.06-16663-100. Ans.

LOGARITHMS.

As we have had occasion to use logarithmns, in solving several problems in the preceding part of this work, we shall briefly show how they may be constructed, and also, something of their uses.

Logarithms are a series of numbers, so contrived, that by them the work of multiplication may be performed by addition; and the operation of division may be done by subtraction. Or, logarithms are

the indices, or series of numbers in arithmetical progression, corresponding to another series of numbers in geometrical progression. Thus:

ƒ0, 1, 2, 3, 4, 5, 6, &c. Indices, or logarithms. 1, 2, 4, 8, 16, 32, 64, &c. Geom. progression.

Or,

0, 1, 2, 3, 4, 5, 6, &c. Indices, or log. 1, 3, 9, 27, 81, 243, 729, &c. Geometrical series.

Or, -0, 1, 29 3, 4, 5, 6, In.or L. 1, 10, 100, 1000, 10000, 100000, 1000000, &c. Geometrical series. Where the same indices serve equally for any geometrical series, or progression.

Hence it appears, that there may be as many kinds of indices, or logarithms, as there can be taken kinds of geometrical series. But the logarithms most convenient for common uses, are those adapted to a geometrical series increasing in a tenfold ratio, or progression, as in the last of the foregoing examples. In the geometrical series, 1, 10, 100, 1000, &c. if between the terms 1 and 10, the numbers 2, 3, 4, 5, 6, 7, 8, 9 were interposed, indices might also be adapted to them, in an arithmetical progression, suited to the terms interposed between 1 and 10, considered as a geometrical progression. Moreover, proper indices may be found to all the numbers that can be introduced between any two terms of the geometrical series.

But it is evident, that all the indices to the numbers under 10, must be less than 1; that is, they must be fractions. Those to the numbers between 10 and 100, must fall between 1 and 2; that is, they are mixed numbers, consisting of 1, and some fraction. Likewise, the indices to the numbers between 100 and 1000, will fall between 2 and 3; that is, they are mixed numbers, consisting of 2 and some fraction; and so of the other indices. Hereafter the integral part only of these indices, will be called the index; and the fractional part will be called the logarithm

The computation of these fractional parts, is called making logarithms; and the most troublesome part of this work, is to make the logarithms of prime numbers, or those which cannot be divided by any other numbers than themselves and unity.

RULE,

For computing the logarithms of numbers.

Let the sum of its proposed number be called A. Divide 0.8685889638* by A, and reserve the quotient; divide the reserved quotient by the square of A, and reserve this quotient; divide the last reserved quotient by the square of A, reserving the quotient still; and thus proceed as long as division can be made. Write the reserved quotients orderly under each other, the first being uppermost; divide these quotients respectively by the odd numbers 1, 3, 5, 7, 9, 11, &c.; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, &c., and let the quotients be written orderly under each other; add them together, and their sum will be a logarithm. To this logarithin add the logarithm of the next less number, and the sum will be the logarithm proposed.

*0.8685889638+ is the quotient of 2, by 2.50258509st which is the logarithm of 10; according to the first form of Lord Napier, the inventor of logarithms. The manner in which Napier's logarithm of 10 is found, may be seen in most books of algebra, and in systems of mathematics generally, where the analytical art is discussed.

+ The logarithm of 10 itself, is derived from the follow

ing theorem, viz. If 10-1+e e is found; but,

1-e

ell

+

e13

11 13

&c. =2.3025850896

Napier's Log of 10. Brigg's Log, of 10=1.

Here the next less number is 1, and 2+1=3 — A,

add the logarithm of 10.000000000

Their sum. 0.301029993 log. of

The manner in which the division is here carried on, may be readily perceived, by dividing, in the first place, the given decimal by A, and the succeeding quotients by A2; then letting these quotients remain in their situation, as seen in the example, divide them respectively by the odd numbers, and place the new quotients in a column by themselves. By employing this process, the operation is consi-derably abreviated.