A B=225=15 feet, Answer.

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

Answer, 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? Answer, 20 feet. 32. If a room be 20 feet in length, and 25 feet between the opposite corners, what is the width?

Answer, 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?

Answer, $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? Answer, 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? Answer, 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? Answer, 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 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. Answer, 75 feet, length of the line. {

To find a mean proportional between two numbers.

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.

3×12=36, and /36-6, Answer.

It is evident, that the ratio of 3 to 6 is the same as the ratio of 6 to 12, for §-2, and 12-2.

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

Answer, 24.

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

Answer, 27.

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

EXTRACTION OF THE CUBE ROOT.

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

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 fraction.

What is the cube of 24?

tens. units.

24-2 +4

2 +4

8+16

4 +8

4+16+16

24

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.: 1st, the cube of the tens, (8 thousands.) 2d. Three times the product of the square of the tens into the units, (48 hundreds.) 3d. Three times the product of the tens into the square of the units, (96 tens.) 4th. 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?

2

Operation.

13824(24

8

2X3=12)58124

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 square 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 9x3+3x3+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 three 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.

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

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.

Having distinguished the given number into periods of three figures each, denoted by the index of the root, we perceive, by the number of periods, that the root will consist of two figures. As the cube of ten cannot be less than a thousand, 10×10×10=1000, we look for the cube of tens in the second, or left hand period. We find, by trial, the greatest cube in 13, or 13000, to be 8, or 8000, and its

root, 2 or 2 tens, (the length of one side of the cube, Fig. I.) which we place in the quotient, as the first figure of the Operation. root, and its cube,

2

3

20X20×20=8000, 13324(24, root. under that period;

2=2X2X2=8

2X300+60=1268) 5824

1200X4=4800 60X4X4 960 4X4X4= 64

5824

and, subtracting it, we have a remainder of 5, or 5000-to which we bring down the next period. Had the cube contained but 8000 solid inches, we should now have found its root, or the length of one

side, But we have 5824 inches to be added to the cube,

20

20

made.

FIG. I.

20

20

FIG. II.

20

and in such a manner that its cubic form shall not be altered. It is obvious, that an equal addition must be made on three sides. As each side is 20 inches square, we have 20×20 X3=1200; or, which is the same thing, multiply the square of the quotient by 300. 2x2 X300 1200 inches surface, to which the additions are to be

It will be seen (Fig. II.) that there are three deficiences along the sides, a a a, where the additions meet, 20 inches in length, 20×3=60, or multiply the quotient by 30; 2X30-60. We have, then, 1200+60=1260, which 30 may be considered the points where the additions are to be made. Then 5824-1260-4 inches, the thickness of the addition, or the second figure. of the root. The area of the sides multiplied by the thickness, 1200X4=4800 inches, the amount of the addition upon the sides. Then the number of inches necessary to fill the deficiencies where

20

20

a