14. What is the square root of

9

?

81

Ans. 3.

OBS. 3.—The square root of a fraction may be found by extracting the root of the numerator and denominator.

OBS. 4.-When the numerator and denominator are surd numbers, re duce the fraction to a decimal, and extract the root as above directed.

19. What is the square root of å 20. What is the square root of ?

Ans. .866+.

Ans. .9355+.

21. How many rows on one side of a square cornfield, containing 15376 hills?

Ans. 124.

22. An army of 242064 men are drawn up in a solid body, in the form of a square. What is the number of men in rank and file? Ans. 492.

row?

23. A man has 841 peach-trees, which he wishes to plant in the form of a square. How many must be planted in each Ans. 29. 24. There is a circular pond, containing 110889 square rods. What will be the length of a square field containing the same number of rods? Ans. 333 rods. 25. A number of men gave £22 1s. for a charitable purpose, each giving as many shillings as there were men. What was the number of men? Ans. 21. 26. What is the length of one side of a square acre of land? Ans. 12.64+.

27. The diameter of a circle is 6 inches. What is the diameter of a circle 4 times as large?

Ans. 12.

OBS. 5.-Circles are to one another as the squares of their diameter; therefore, to find the required diameter, square the given diameter, multiply the square by the given ratio, and the square root of the product will be the diameter required.

28. The diameter of a circle is 24 feet. eter of a circle one-fourth as large?

What is the diam

Ans. 12 feet.

29. In the right-angled triangle ABC, the side AC is 9 feet, and the side BC 12 feet. What is the length of the side AB?

30. What is the distance between the opposite corners of a room, 20 feet in length and 15 in width?

Ans. 25 feet.

31. If the distance between the opposite corners of a room be 25 feet, and the width of the room be 15 feet, what is the length? Ans. 20 feet. 32. If a room be 20 feet in length, and 25 feet between the opposite corners, what is the width ? Ans. 15 feet.

33. Two men owning a pasture 32 rods in width, and 50 rods between the opposite corners, agreed to divide said pasture into two equal parts by a wall running through it lengthwise. Suppose they pay 50 cents a rod for building the wall, what does it cost them? Ans. $19.209.

34. Suppose a ladder 50 feet long, to be so placed as to reach a window 30 feet from the ground on one side of the street, and without moving it at the foot, will reach a window 20 feet high, on the other side; what is the width of the street? Ans. 85.825+ feet.

35. Two men travel from the same place-one due east, the other due north. One travels 40 miles the first day, the other 30. What is the nearest distance between them at night? Ans. 50 miles.

36. A. and B. set out together, and travel in the same direction on parallel courses, which are 20 miles apart. A. travels 45 miles, and B. 25. What is the distance between them at night? Ans. 28+ miles.

37. Suppose a pine-tree to stand 25 feet from the end of a house 40 feet in length, the foot of the tree being on a level with the foundation of the chimney, which stands in the centre of the house, and a line reaching from the foot of the tree to the top of the chimney, be 75 feet, what is the height of the chimney? and if the height of the tree be of 3 of 4 of 14 of the height of the chimney, what will be the length of a line reaching from the top of the chimney to the top of the tree? (60 feet, height of the chimney. 75 feet, length of the line.

Ans.

{

Art. 229.—To find a mean proportional between two num bers.

RULE.

Multiply the given numbers together, and the square root of their product is the mean proportion sought.

1. What is the mean proportional between 3 and 12?

Operation.

3x12=36, and 36 6 Ans.

It is evident, that the ratio of 3 to 6 is the same as the ratio of 6 to 12; for 3, and 6.

12

2. What is the mean proportional between 12 and 48?

3. What is the mean proportional between 9 and 81?

Ans. 24

Ans. 27.

4. What is the mean proportional between 25 and 625 ? Ans. 125.

EXTRACTION OF THE CUBE ROOT.

Formation of the Cube, and Extraction of the Cube Root.

Art. 230.—The third power, or cube of any number, is the product of that number multiplied into its square; and the cube root is a number which, multiplied into its square, will produce the given number.

Roots and powers are correlative terms; that is, if 3 is the cube root of 27, then 27 is the third power, or cube, of 3.

There are but nine perfect cubes among numbers expressed by one, two, or three figures; each of the other numbers has for its cube root a whole number, plus a fraction. Thus 64 is the cube of 4, and 27 is the cube of 3; therefore, the cube root of each number between 27 and 64 must be 3 plus a frac tion.

What is the cube of 24?

tens. units.

24=2+ 4

2+4

8+16

4+ 8

4+16+16

2+ 4

16+64+64

8+32+32

8+48+96+64=13824

It will be perceived, from the above process, that the cube of a number composed of tens and units, is made up of four parts, viz: 1. The cube of the tens, (8 thousands.) 2. Three times the product of the square of the tens into the units, (48 hundreds.) 3. Three times the product of the tens into the square of the units, (96 tens.) 4. The cube of the units, (64 units.)

To extract the cube root is to find a number which, multiplied into its square, will produce the given number.

What is the cube root of 13824?

Operation.
13824(24

8

22×3 12)58 24

As this number is greater than 1000, which is the cube of 10, but less than 1,000,000, its root will consist of two figures, tens and units; but the cube of tens cannot be less than thousands; therefore, the three figures, 824, on the right, cannot form a part of it. Hence we separate these from 13 by a point, and look for the cube of tens in 13, the left-hand period. The root of the greatest cube contained in 13 is 2, which is the tens in the required root; for the cube of 20, which is 8000, is less, and the cube of 30, which is 27000, is greater than the given number;

therefore, the required root is composed of 2 tens, plus a certain number of units less than ten.

We now subtract 8, the cube of the tens, from 13, and bring down the next period, 824. We have now 5824, which contains the three remaining parts of the cube, viz: Three times the product of the square of the tens into the units, plus three times the product of the tens into the square of the units, plus the cube of the units. Now, as the square of tens gives hundreds, it follows, that three times the square of the tens into units must be contained in 58, which we separate from 24 by a line. If we now divide 58 by three times the square of the tens, we shall obtain the units of the required root. We may ascertain whether the unit figure be right, by cubing the quotient, or by applying the following principle: The difference between the cubes of two consecutive numbers is equal to three times the square of the least number, plus three times this number, plus 1. Thus, the difference between the cube of 3 and the cube of 4, is equal to 9×3+3×3+1=37, which is the difference between the cube of 3 and the cube of 4. Therefore, had we written 3 in the unit's place, the remainder would have been equal to 3 times the square of 23, plus three times 23, plus 1, which would show that the unit figure must be increased.

Thus far the illustration has been general,-applied to numbers merely—numbers in the abstract. We may now apply it to solid bodies. Numbers which represent, or stand for things, are called concrete, as question first below.

EXAMPLES.

Art. 231.-1. What is the length of one side of a solid block containing 13824 solid inches, or what is the cube root of 13824 ?

OBS.-The foregoing operation can be better understood by blocks prepared for the purpose. It is necessary to have one cubical block, of a convenient size, to represent the greatest cube in the left-hand period, and three other blocks, equal to the sides of the first block, but of indefinite thickness, to represent the additions upon the sides. Then three other blocks, equal in length to the sides of the cube, and their other dimensions equal to the thickness of the additions on the sides of the cube. Lastly: a small cubic block, of dimensions equal to the thickness of the additions, to fill the deficiency at the corner. By placing these blocks as above described, the several steps in the operation may be easily understood. It may be observed, however, that this illustration would serve only for concrete numbers, as in the above question.