27. There is a wall 15 feet high, and in front of it is a pavement 24 feet wide. How long a ladder is required to reach from the outside of the pavement to the top of the wall? The hypoteneuse is required, therefore, 15 x 15-225; and 24x24 576; then, 225+576-801; and V801=28.3,+ Ans.

28. A certain tree is broken off 8 feet from the ground, and, resting on the stump, touches the ground at the distance of 12 feet. What is the length of the part broken off? Ans. 14.42+ feet.

29. There is a fort standing by the side of a river, 24 yards high, and a line 36 yards long will just reach from the top of the fort to the opposite side of the river. What is the width of the river? Ans. 26.832 yards.

30. Two ships sail from the same port, one due east, and the other due north. What is the distance between them, when one has sailed 100 miles, and the other 168 miles? Ans. 195.5+ miles.

31. A man shot a bird sitting on the top of a steeple 80 feet high, while standing at the distance of 60 feet from its base. How far did he shoot? Ans. 100 feet.

32. A rope 100 feet long attached to the top of a steeple, touches the ground when drawn perfectly straight, 20 feet from its base. How high is the steeple? Ans. 98 feet,

nearly.

33. Two boys were playing with a kite, the line of which was 520 feet in length. When the string was all out, one of them standing directly under the kite, and the other holding the string, the distance between them was 312 feet. What was the perpendicular height of the kite? Ans. 416 feet.

QUESTIONS. When is a number involved? What is a root? What is a power of any number? On what does the particular power produced depend? How does the power obtained compare with the number of multiplications in producing it? How is a required power expressed? What is the figure denoting the power called? How is a fraction involved? If the given quantity be a mixed number, what must be done? If a number be raised to two different powers, how is the power obtained by multiplying these two powers together, expressed? How is any power of a given number divided by another power of the same number? When the number to be raised to a power is in part a decimal, how is the number of decimals to be cut off from the required power ascertained? What is evolution? What is a root of a number? What is a power of any number? What are irrational powers? What are surds? How many methods are there of expressing roots? What is the first method? What root is expressed by the radical sign without any index or exponent? If

6

any other root is to be expressed, how is it done? How many times is the root to be taken as a factor in producing its corresponding power? What is the other mode of expressing roots? What is an advantage of this mode? When a fractional index is used, what does the denominator denote ? What does the numerator? When several numbers are to be added, and the root of the sum extracted, how is the operation expressed? What does the root of the product of several numbers equal? Give an illustration. What is the square root of any number? What is a square? How is the area of a square found? Give the illustration. How is the length of the sides of a square found? What is the rule for extracting the square root? What is Note 1st? Note 2d? Note 3d? Why do we take twice the root for a divisor? Why in dividing do we omit the right hand figure of the dividend? Why do we place the quotient figure on the right of the divisor? How is the square root of a vulgar fraction extracted? The vulgar fraction may first be reduced to a decimal and the root of the decimal extracted, if preferred. What is a parallelogram? How is its area found? How may a square equal in area to a given parallelogram be found? What is Note 5th? Note 6th? To what is the square of the hypoteneuse of a right angled triangle equal? If the base and perpendicular be given, how may the hypoteneuse be found? If the hypoteneuse and one of the legs of a triangle be given, how may the other leg be found? The base and perpendicular are called the legs of a triangle. How are the operations in Square Root proved? Ans. By multiplying the root into itself.

EXTRACTION OF THE CUBE ROOT.

A CUBE is bounded by six equal plane surfaces, each of which is a square; that is, the length, breadth, and depth of a cube are equal. (See fig. 1st.) The area of each of the six equal surfaces is found by squaring the measure of its side, as has already been explained in Square Root.

If the adjoining figure represent a cubic block measuring three feet in length, breadth, and thickness, the superficial area of each face is 3x 3=9 square feet. Now if the divisions marked by the dotted lines on each face of the block, were extended through it, in either direction, the whole would be divided into 9 parts, each one foot square and 3 feet long, and susceptible of being divided each

Breadth

Denth

FIG. 1.

Length

into three, and consequently the whole, into 27 blocks, each one

cubic foot. We have then the following general principle. The content of a solid or cubic body is found by multiplying its length, breadth, and thickness into each other; or, what is the same in effect, by cubing one of these dimensions.

Extracting the cube root is a process, the reverse of the preceding, that is, it is finding the length of one of the sides of a cubic body, the solid content of that body being given; or it is finding from a given number, another number, whose cube or third power shall equal that number.

RULE 1st. Separate the given number into periods of three figures each, by placing a point first over the unit figure, and advancing toward the left when the number consists of integers only; but to the right and left both, when it consists of integers and decimals, and make the last period of the decimal complete, by annexing cyphers, whenever necessary.

2d. Find by trial the greatest cube root of the left hand period, and place it as in square root; then subtract its cube from the same period, and bring down the next period of three figures to the remainder for a dividend.

3d. Square the root figure and multiply its square by 3 for a divisor, and see how many times it is contained in the dividend, omitting the first two right hand figures, and place the result as the second figure in the root.

4th. Multiply the divisor by the last quotient figure, and, placing two cyphers on the right of the product, place the result under the dividend. Also multiply the square of this same last quotient figure by the former figure or figures of the root, and also by 3; and, placing one cypher on the right of the product, write it under the preceding product. Lastly, under this, write the cube of the last quotient or root figure, and make the sum of these three numbers a subtrahend.

5th. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend.

6th. To obtain a new divisor proceed as before, and thus continue the operation, till all the periods of the given number have been brought down.

Note 1st. In obtaining each divisor, square the whole root obtained, and multiply that square by 3.

Ex. 1. What is the cube root of 10648?

OPERATION.

1 0 6 48 (22

23-8

Art. 3d, Rule, 2a×3=12 (Div.) 12)2 6 4 8 (See Art. 2d.)

In dividing, the two right hand figures of the dividend are omitted. (See rule, Art. 3.) Proof, 223-10648.

Explanation. We will suppose the number 10648, in the preceding sum, to be so many feet of timber, one foot square, and that it is required to find how large a cubic pile they will form; that is, what will be the length, breadth, and depth of a cubic pile containing that number of solid feet. To make each step as clear as possible, we will repeat the preceding operation.

20 feet.

FIG. 2.

20 feet

400 Sqr. feet

The number given when pointed, (see rule,) is divided into two periods, viz. 10 and 648. We therefore know that the required root will consist of two figures, and by trial we find the root of the first or left hand period to be 2, that is, 2 tens, or 20. Hence, 10648(20. The same as is seen in the above operation, excepting that a cypher is placed on the right of the root figure 2, to give it its true value. This gives the linear measure of a cubic block which the 10 (10.000) of the given number will make. Now, since 20 is the linear measure of the cube, (that is, the direct and not diagonal measure from corner

20 feet Superficies 400 Sqr. ft.

Solid Contents
8000 solid ft.

400 Sqr.feet

to corner,) it is also the linear measure of each of the six equal square faces of that cube. Therefore, 20×20=400, the area of each face; and 400×20=8000, the number of cubic feet required to make a cubic body, whose linear measure is 20 ft.

20 feet

(See fig 2.) By this step we have then disposed of 8000 of the 10648 solid feet. Hence,

1 0 6 4 8 (20
(

8000

2648

(This same effect is obviously produced in the first solution of this sum, by cubing the 2, and subtracting it from the left hand period, 10, and then bringing down the next period, 648, to the remainder.) There now remains 2648 feet to be so added to the block already formed, that the whole shall be a perfect cube. This is done by making equal additions on any three of the equal faces which lie contiguous to each other. The reason of this is obvious. A solid body has length, breadth, and thickness; and in a cubic body, these dimensions are all equal, and by making the additions as here directed, they are equally increased.

The scholar will now understand why three times the square of the root obtained is taken as a divisor. The square of the root is the superficial area of one of the sides or faces of the cubic block, and this multiplied by 3 gives the area of three faces or sides, which is the number of sides to which equal additions are to be made. Hence, dividing the quantity to be added to the cube now obtained by this area, determines the thickness of the addition. This, in the sum now under consideration, is 2 feet, as seen at fig. 3. But these additions are evidently limited in size to the original block; consequently, the corners E, E, E, (fig. 3,) remain to be filled before a perfect cube is produced.

Now to determine the quantity here added. Each face of the original cube contains 400 square feet, (see fig. 2,) and this multiplied by 2, the depth of the addition, gives 800 solid feet as the content of the addition made to each face; the whole addition therefore is 800 x3=2400 solid feet, and 2648-2400248 solid feet, the quantity yet remaining to be disposed of.

FIG. 3.