MISCELLANEOUS EXAMPLES.

Ex. 1. A gentleman desirous of making his kitchen gården, which is to contain 4 acres, a complete square, I demand what will be the length of the side of the garden?

Ex. 2. Six acres of ground are to be allotted to a square garden; but for the sake of more wall for fruit, there is to be a smaller square within the larger, which is to contain 3 acres, I demand the length of the sides of each square?

Ex. 3. What is the mean proportional between 12 and 75?*

Ex. 4. How long must a ladder be to reach a window 30 feet: high, when the bottom stands 12 feet from the house?†

To extract the cube root.

I. RULE. (1) Find, by trials, the nearest cube to the given number, and call it the assumed cube. (2) Say, as. twice the assumed cube added to the given number, is to twice the number added to the assumed cube, so is the root of the assumed cube to the root required nearly.

What is the cube root of 27455 ?

Here the nearest root that is a whole number is 30, the cube of which is 27000: therefore I say,

As 27000 X 2 + 27455: 27455 × 2 + 27000 :: 30

or 81455 81910: 30: 30.1675.

NOTES.

of 4 being 16, and the square of 5 being 25: it must therefore be worked thus:

20(4.4721 &c.

16

84) 400

336

887) 6400

6209

8942) 19100
17884

8944) 121600

By multiplying the root 4.4721 into itself, the answer will be 19.99996, &c., which is very nearly, though not quite, equal to 20; and by carrying the operation still further, greater accuracy would be obtained.

This is found by multiplying the given numbers together, and taking the square root of the product.

+ Square the given numbers, and take the square root of their

sum.

H

It is evident that the true root, omitting the last two figures, is somewhere between 30.16 and 30.17, the former being too little, the latter something too large. By taking the root thus found 30.16, as the assumed cube, and repeating the operation, the root will be had to a still greater degree of exactness.

Ex. 1. What is the cube root of 15625?

Ex. 2. Whrt is the cube root of 140608 ?

Ex. 3. What is the cube foot of 444194947 ?

Ex. 4. What is the cube root of the difference between 140608 and 14025?

II. RULE 1. Separate the given number into periods of three figures each, beginning from units place; then from the first period subtract the greatest cube it contains, put the root as a quotient, and to the remainder bring down the next period for a dividend. 2. Find a divisor by multiplying the square of the root by 300, see how often it is contained in the dividend, and the answer gives the next figure in the root. 3. Multiply the divisor by the last figure in the root. Multiply all the figures in the root by 30, except the last, and that product by the square of the last. Cube the last figure in the root. Add these three last found numbers together, and subtract this sum from the dividend; to the remainder bring down the next period for a new dividend, and proceed as before.

Ex. 5, What is the cube root of 444194947 ?

95976

95976

1732600 divisor

5218937

3

.......

5198400

2052076 X 30 X 9 27 3X 3X 3

5218947

Ex. 6. What is the cube root of 46656?

Ex. 7. What is the cube root of 65939264 ?

Ex. 8. What is the cube root of 3, carried to 2 places of decimals?

Ex. 9. What is the cube root of?

Ex. 10. What is the cube root of 12?

1728

Ex. 11. What is the cube root of .729?

Ex, 12. What is the cube root of .003375 ?

MISCELLANEOUS EXAMPLES.

Ex. 1. A corn-factor requires a cubical bin that shall hold 840 bushels of wheat: I demand the inside length of one of its sides. See Table, p. 49, note.

Ex. 2. In a cubical building that measures 2744 feet, what is the length of a side?

ARITHMETICAL PROGRESSION.

When a series of numbers increases or decreases by some common excess, or common difference, it is said to be in arithmetical progression, such as 1, 3, 5, 7, 9, &c., and 12, 10, 8, 6, 4, &c.

The numbers which form the series are called the terms of the progression; of these the first and last are called the

extremes.

Any three of these terms being given, the others may be easily found. I. When the first term a, and the last term z, and the number of terms n, are given, to find the sum of all the terms, s.

RULE. Multiply the sum of the extremes by the number of terms, and divide by 2, the quotient is the answer: or

Ex. 1. What is the sum of the terms of an arithmetical series, whose first term is 5, last term 29, and the number of terms 7 ?

* These symbols will be easily remembered; a and z being the first and last letters of the alphabet, may properly represent the first and last terms of any series; the reason of the other letters is sufficiently obvious,

Ex. 2. The first and last terms of a series are 3 and 111, and the number of terms 37: what is the sum ?

Ex. 3. How many strokes do the clocks of Venice strike in 24 hours, where they strike from 1 to 24?

Ex. 4. The first and last terms of a series are 1 and 1000, and the number of terms 100: required the sum.

Ex. 5. If 100 stones are placed in a right line, exactly a yard asunder, and the first one yard from a basket, what length of ground will a man go over, who gathers them up, one by one, returning with each to the basket?

Ex. 6. What must a man give for 54 timber trees, for which he pays 5 shillings for the first, and 20l. for the last, and the prices of the others being in arithmetical progression ?

Ex. 7. A butcher buys a drove of oxen, consisting of 32; for the first he pays 15s., and for the others he is to pay in arithmetical progression, so that for the last he is to pay 381.: what will they all come to?

Ex. 8. A horse-dealer sends to a fair 63 hoises, of various kinds and worth, which he is willing to dispose of according to the principles of arithmetical progression, demanding 37. only for the first, provided he had 531. for the last: how much did he receive for the whole, and what was the average value of each horse?

II. The first and last terms, a and z, and number of terms being given, to find the common difference, d.

RULE. The difference of the extreme terms divided by the number of terms, less 1, will be the common difference sought:

Ex. 1. What is the common difference of an arithmetical progression, whose extremes are 8 and 200, and the number of terms 17?

Ex. 2. When the extremes of an arithmetical progression are 6 and 37, and the number of terms 18, what is the common difference? Ex. 3. A gentleman gives at Christmas, among his 25 poor neighbours, a sum of money in arithmetical progression: to the least needy he gives 5 shillings, and to the poorest, with a very large family, he gives five guineas: what was the common difference?

NOTE.

*This mark,

is put for the difference of any two numbers, which ever of them is largest; thus, suppose a be 10, and ≈ 4, then a z 6, or it may be, a is 4, and z 15; then a

11.

Ex. 4. A traveller is out on his journey a month, of which he travels 25 days; on the first he rides 7 miles, and on the last, having little to do, he comes 43 miles: how much was the daily increase of his travelling, and how many miles did he ride in the whole?

III. The extreme terms a and z, and common difference d being given, to find the number of terms n.

RULE. Divide the difference of the extremes by the common difference, and the quotient increased by unity is

Ex. 1. When the extremes are 4 and 106, and the common difference is 3, what is the number of terms?

Ex. 2. If the least term be 6, the greatest 216, and the common difference 5, what is the number of terms?

Ex. 3. What debt can be paid, and in what time, supposing I agree to lay by 3s. the first weeks. the next, 11s. the third, and so on in arithmetical progression, till the last saving be four guineas?

Ex. 4. I set out for Hastings, which is 69 miles from this place, and I walk the first day 4 miles, the second 7, increasing every day by 3 miles, and on the last 19 miles: how many days will the journey take?

In addition to the above, the learner may commit to memory the following facts on the subject:

1. If three numbers are in arithmetical progression, the sum of the extremes is equal to double the mean term; as 6, 9, 12, where 6+ 12 = 2 × 9 = 18.

2. If four numbers be in arithmetical progression, the sum of the two extremes is equal to the sum of the means; as 5, 8, 11, 14, where 5 + 14 = 8 + 11 = 19.

3. When the number of terms is odd, the double of the middle term will be equal to the sum of the extremes; or of any other two means equally distant from the middle term; as 3, 8, 13, 18, 23, 28, 33, where 3 + 33 = 2 × 18 = 13+ 23=8+ 28.