two given numbers as extremes; take also the number of terms in the series greater by two than the required number of means, and find the ratio by Rule III. (2.) Then the product of the ratio and the first extreme will be one of the means: the product of this mean and the ratio will be another, and so on.

When only one mean is required, it is very easily found by extracting the second root of the product of the extremes.

Exam. 10. Find three mean proportionals between 5 and 1280. Here, the series would consist of 5 terms, and the extremes are 5 and 1280; and hence the ratio is found, by the last rule, to be 4; and by multiplying the first term by this, the product by the same, &c., the means are found to be 20, 80, and 320.

Exam. 11. Find a mean proportional between 5 and 10.

Here, 5 × 1050; the square root of which is 7.0710678, &c., the mean required.

Exer. 14. Find two mean proportionals between 1 and 2. Answ. 1.259921, and 1.587401.

15. Find a mean proportional between and 100. Answ. 3.16227766,

Exer. 16. If a thrasher agree to work 18 days for a farmer, on condition of receiving two grains of wheat for the first day's work, 6 grains for the second, 18 for the third, &c.: what would be the value of all he would be entitled to receive, supposing 7680 grains to fill a pint, and the wheat to be worth 7 shillings a bushel? Answ. £275-17 - 5.

17. Suppose a house, having 20 windows, to be sold at the rate of 4s. for the first of these windows, 6s. for the second, 9s. for the third, and so on; the value of each being increased by one half of itself, to find the value of the next; for how much would it be sold? Answ. £1329-14 - 01.

18. It is said that an Indian invented the game of chess, and showed it to his sovereign, who was so much pleased with it, that he desired the inventor to ask any reward he chose. The inventor then asked one grain of wheat for the first square of the chess table, two for the second, four for the third, and so on; doubling continually to 64, the number of squares. Now, suppose it had been possible for the prince to pay this reward, what would have been the value of the whole at 12s. 6d. per cwt., 10,000 grains being supposed to weigh a pound avoirdupois? Answ. £10,293,942,005,418 - 5 - 61. 19. Find the value of the interminate decimal 463. Answ. 51. 20. Required the finite value of the decimal ·5185. Answ. 14. 21. Find the sum of the infinite series, 1, 1,

1, &c. Answ. 4' 8' 16' 22. Find the sum of the infinite series,,,,, &c. Answ.

27.

COMPOUND INTEREST.*

The method that naturally presents itself for finding the amount of a sum at compound interest, is to find its amount at simple interest at the end of the first year; then to take this amount as a new principal, and find its amount in like manner, which would be the amount at compound interest at the end of the second year, and the principal for the third year: the amount of which must be found in like manner. Continuing the process, we should thus find the amount at the end of the proposed time. This will be illustrated in the following example.

Exam. 1. Required the amount of £2500 at the end of 4 years, at 6 per cent. per annum, compound interest.

Here, the amount for one year is £2650; the amount of which for one year also is £2809, the amount at compound interest for two years. The amount of this, again, for 1 year, or the amount of the given sum at the end of the third year, is £2977 - 10 - 9; the amount of which for another year is £3156 - 310, the amount of £2500 for four years. The amount at simple interest would have been £3100, which is less than the foregoing amount, by £56 - 3 - 10.

When the time is short, this method may be practised without much trouble; but when it is long, the labour becomes very great. In this case, the methods that follow should be employed.

RULE I. To find the amount of one pound sterling for any number of years, at compound interest: (1.) Divide the amount of £100 for 1 year by 100, and the quotient will be the amount of one pound for 1 year. (2.) This amount involved to the power denoted by the number of years, will be the amount of one pound for that time.

The contracted mode of multiplication of decimals is peculiarly useful in this rule, and in computations in compound interest and annuities in general. So also is the contracted method of dividing decimals. (See pages 195 and 198.)

Exam. 2. Required the amount of one pound sterling for 20 years, at 4 per cent. per annum, compound interest.

Here, the amount of £100 for one year is £104.5; the hundredth part of which is £1.045, the amount of £1 for a year. The second power of this is £1.092025, the amount of £1 at the end of the second year. The product of this by itself, found by the contracted method of multiplication, is £1.192518, the amount at the end of the fourth year. The square of this, again, is £1.422099, the amount for eight years; the square of which is £2.022366, the

*For the definition of compound interest, see page 224.

amount at the end of the sixteenth year. Finally, the product of this by £1.192518, the amount for four years, is £2-411708, or £28-23, nearly, the amount of £1 for 20 years. At simple interest, the amount would have been only £1 - 18.

Had the products here been found at full length, the labour would have been immense. In the last multiplication, one of the factors would have contained 49 figures, the other 13, and the product 61. It should be carefully remarked, however, that the decimal part of the amount found as above, will rarely be true in all its places. (See page 195.) A trifling error in rejecting or overestimating a figure at the end of a decimal may accumulate, and render the accuracy of the last figure or the last two figures doubtful. Thus, in the preceding result, the last two figures should have been 14 instead of 08; a difference, however, which would occasion only the trifling error of rather more than a penny in the amount of £1000. When great accuracy is required, the amounts should be brought out to a greater number of places, and the last figure or two of the final amounts rejected, or not depended on. The larger the given sum also, and the longer the time, this is the more necessary, as the effect of the error is the more perceptible.

Exam. 3. What is the amount, true to six places of decimals, of £1 for 6 years, payable half-yearly, at 5 per cent. per annum, compound interest?

Here, the payments being half-yearly, the amount of £100 for half a year is £102 - 10, or £102.5; and, consequently, that of £1 for the same time is £1.025: the square of this is 1050625, or 1.0506250, the amount for two half-years, or one year. Multiplying this by 1.050625, by the contracted method, we obtain 1·1038129, the fourth power of £1.025, or the amount of £1 for two years; the third power of which is 1.3448888, the twelfth power of £1.025, or the amount of £1 for six years; or, if only six figures of decimals be retained, 1.344889.

Exercises. Find the amounts of £1 in the following exercises, at the given rates per cent. per annum :—

RULE II. To find the amount, or the interest, of any sum, at compound interest, for a given time, and at a given rate: Find the amount of £1 for the given time, by Rule I., and multiply it by the given sum.

If the principal be subtracted from the amount, the remainder will be the interest.

Exam. 4. Required the amount of £760 - 14 - 4 for 12 years, at 5 per cent. per annum, compound interest.

1.795856

760-14

- 4

107751360

12570992

897928 for 10s.
359171

By Rule I., the amount of £1 is found to be £1.795856. This being multiplied by 760, and aliquot parts being taken for 14s. 4d., as in the margin, the result is £1366-2-9, the amount; and the principal being subtracted from this, there remains, for the interest, £605 - 8 - 5. The simple interest would have been £456-8 -71. The same result would have been obtained by using, instead of 14s. 4d., the equivalent decimal, and the answer would in that case have been found by multiplication alone. Exercises. Find the amounts of the following sums, at the given rates per cent. per annum :

29931

4s.

4d.

£1366 137590, or
£136629, the amount

Exercises.

Answers.

7. 251 16

6 for 9 years, at 5 per cent., &c.

390 13 3

Exer. 15. If a boy, 12 years old, have a legacy of £1396-16 - 8 left to him, how much will he have to receive at the age of 21, the legacy being improved by compound interest, at 5 per cent. per annum? Answ. £2166 - 18 - 114.

16. Find the amount of £648 from the 6th till the 21st year of a boy's life, at 4 per cent. per annum, compound interest. Answ. £1299-16-612.

17. If a merchant commence trade with a capital of £1200, and each year, after paying all expenses, increase the capital of the former year by a fifth part of itself; how much will he be worth at the end of 30 years? Answ. £284,851 - 11 - 61

RULE III. To find the principal, which, at a given rate, and in a given time, would amount to a given sum; Or, to find the present worth of a sum at compound interest,

for a given time, and at a given rate: Divide the given sum by the amount of £1, found by Rule I.

The present worth of £1 may be found by dividing it by its amount for the given time.

Exam. 5. What sum must be lent at compound interest, at 5 per cent. per annum, at the birth of a child, so that the amount may be £300068 at the end of 21 years?

Here, the amount of £1 for 21 years being 2-785962, we have for answer £3000-3÷2·785962 = £1076·9469 = £1076 - 18 - 11.

Exercises. Required the present worths of the following sums, or the principals that would produce them, at compound interest, at the given rates per cent. per annum :

Exercises.

£ s. d.

18. 324 18 6 for 9 years, at 5 per cent., &c.

Exer. 22. A brother is to pay his sister a portion of £4500 at the end of 11 years: how much will discharge the debt at the end of 4 years, compound interest being allowed at 4 per cent. per annum on the sum he pays? Answ. £3306 - 14-63.

23. With what sum must a merchant commence trade, so as to be worth £15,000 at the end of 12 years, if he may be expected to clear annually an eighth of his capital? Answ. £3649 - 14 - 73.

24. Whether is it better to sell a farm for £1000 payable at present, £1000 payable at the end of 5 years, and £1000 payable at the end of 10 years; or to sell it at £3000 payable at the end of 5 years, compound interest being allowed at 4 per cent. per annum? Answ. Better at three payments by £31 - 14 - 21.

The reason of the first rule will appear from the following considerations. The amount of £1 for a year will evidently be a hundredth part of the amount of £100 and as £1 is to its amount for a year, so is any other principal to its amount for the same time. Hence, to take a particular instance, the amount of £1 for a year at 5 per cent. will be 1.05; and by the nature of compound interest, this will be the principal for the second year. Then, as the principal, £1 £105, its amount: the principal, £1.05 £1.052, the amount at the end of the second year, and the principal for the third. Again, as £1: £105, its amount: the principal, £1·052 : £1053, the amount at the end of the third year, and the principal for the fourth. It will thus appear, that the amount of one pound for any number of years, will be equal to £1.05 raised to the power denoted by the number of years. The amount of £1 being thus determined, it is plain that the amount of any other principal will be had by multiplying the amount of £1 by that principal, since the amount will evidently be proportional to the principal; which proves