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 being the sum of the present values of all the future payments, is also the sum which, being put out and improved at compound interest, will just suffice for the payment of the annuity (2).

7. Hence it follows, that when the annuity is L. 1, the number of years purchase and parts of a year, is the same as the number of pounds and parts of a pound in its present value.

And throughout this article, whenever the quantum of an annuity is not mentioned, it is to be understood to be L. 1.

8. The sum of which the simple interest for one year is L. 1, is evidently that which, being put out at interest, will just suffice for the payment of L. 1 at the end of every year, without any augmentation or diminution of the principal; and, being equivalent to the title to L. 1 per annum for ever, is called the value of the perpetuity, or the number of years purchase the perpetuity is worth.

But, while the rate remains the same, the annual interests produced by any two sums, are to each other as the principals which produce them; therefore, since $$5 : 1 :: 100 : \frac{100}{5} = 20$$, when the rate of interest is 5 per cent., the value of the perpetuity is 20 years purchase. In the same manner it appears, that according as the rate may be 3 or 6 per cent. the value of the perpetuity will be $$\frac{100}{3} = 33\tfrac{1}{3}$$, or $$\frac{100}{6} = 16\tfrac{2}{3}$$ years purchase; and may be found in every case, by dividing any sum by its interest for a year.

9. All the most common and useful questions in the doctrines of compound interest and annuities-certain, may be easily resolved by means of the first four tables at the end of this article. Their construction may be explained by the following specimen, rate of interest 5 per cent.

10. The calculation must begin with Table III., the first number in which should evidently be 1·05, the amount of L. 1 improved at interest during one year; which, being multiplied by 1·05, the product is 1·025, the second number; this second number being multiplied by 1·05, the product is 1·157625, the amount at the end of three years. And so the calculation proceeds throughout the whole of the column; each number after the first, being the product of the multiplication of the preceding number, by the amount of L. 1 in a year (1).

11. The number against any year in Table I. is found by dividing unity by the number against the same year in Table III. (2); thus, the number against the term of six years in Table I. is $$\frac{1}{1{\cdot}340096} = {\cdot}746215$$. All the numbers in that table after the first, may also be found, by multiplying that first number continually into itself (2).

12. The number against any year in Table II. being the sum of the numbers against that and all the preceding years in Table I.; is found by adding the number against that year in Table I. to the number against the preceding year in Table II.; thus, the number against 4 years in Table II., being

13. If each payment of an annuity of L. 1 be put out as it becomes due, and improved at compound interest during the remainder of the term, it is evident that at the expiration of the term, the payment then due will be but L. 1, having received no improvement at interest. That received one year before will be augmented to the amount of L. 1 in a-year; that received two years before will be augmented to the amount of L. 1 in two years; that received three years before to the amount of L. 1 in three years, and so on until the first payment, which will be augmented to the amount of L. 1 in a term one year less than that of the annuity.

Hence, it is manifest, that the number against any year in Table IV. will be unity added to the sum of all those against the preceding years in Table III.

And, therefore, that the number against any year in Table IV. is the sum of those in Tables III. and IV. against the next preceding year.

Thus, the number against seven years in Table IV., being

14. The method of construction is obviously the same at any other rate of interest.

15. All the amounts and values which are the objects of this inquiry, evidently depend upon the improvement of money at compound interest; it is, therefore, that the first, second, and fourth tables, all depend upon the third.

But every pound, and every part of a pound, when put out at interest, is improved in the same manner as any single pound considered separately. Whence, it is obvious, that while the term and the rate of interest remain the same, both the amount and the present value, either of any sum, or of any annuity, will be the same multiple, and part or parts of the amount or the present value found against the same term, and under the same rate of interest in these tables, as the sum or the annuity proposed is of L. 1.

So that to find the amount or the present value of