RULE.

I. Reduce the vulgar fraction, or mixed number, to its simplest fractional form.

II. Then extract the square root of the numerator and denominator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal, and proceed as in Case II.

4. What is the square root of of 1 of 4 of 7 ?

5. What is the square root of 4?

Ans. .

Ans. 2.027 nearly.

Ans. 0.8044 nearly.

Ans. 0.052 nearly.

CASE IV.

When there are many decimal places required in the root, we may, after obtaining one more than half the number required, find the rest by dividing the remainder by the last TRUE DIVISOR, deprived of its right-hand figure.

EXAMPLES.

1. What is the square root of 11 to 10 decimals?

After obtaining five decimals in the root, by the usual method, we had 317756 for a remainder; the last true divisor was 663322, which, when deprived of its righthand figure, becomes 66332. Hence, we divided the remainder 317756 by 66332, and found 47903 for a quotient; which, being written immediately after the figures already found, gives the root sought to 10 decimals.

2. What is the square root of 3 to 10 decimals?

Ans. 1.7320508075.

3. What is the quare root of 0.00008876684 to 10 places of decimals?

Ans. 0.0094216155.

4. What is the square root of 0.8867081113724 to 10 places of decimals? Ans. 0.9416517994.

EXAMPLES INVOLVING THE PRINCIPLES OF THE SQUARE

ROOT.

119. A triangle is a figure having three sides, and consequently three angles.

When one of the angles is right, like the corner of a square, the triangle is called a right-angled triangle. In this case the side opposite the right angle is called the hypotenuse.

It is an established proposition of geometry, that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

From the above proposition, it follows that the square of the hypotenuse, diminished by the square of one of the sides, equals the square of the other side.

By means of these properties, it follows that two sides of a right-angled triangle being given, the third side can be found.

EXAMPLES.

1. How long must a ladder be, to reach to the top of a house 40 feet high, when the foot of it is 30 feet from the house?

In this example, it is obvious that the ladder forms the hypotenuse of a right-angled triangle, whose sides are 30 and 40 feet respectively. Therefore the square of the length of the ladder must equal the sum of the squares of 30 and 40.

2. Suppose a ladder 100 feet long, to be placed 60 feet from the roots of a tree; how far up the tree will the top of the ladder reach? Ans. 80 feet.

3. Two persons start from the same place, and go, the one due north, 50 miles, the other due west, 80 miles. How far apart are they? Ans. 94.34 miles, nearly.

4. What is the distance through the opposite corners of a square yard? Ans. 4.24264 feet, nearly.

5. The distance between the lower ends of two equal rafters, in the different sides of a roof, is 32 feet, and the height of the ridge above the foot of the rafters is 12 feet. What is the length of a rafter? Ans. 20 feet.

6. What is the distance measured through the centre of a cube, from one corner to its opposite corner, the cube being 3 feet, or one yard, on a side?

Ans. 5.196 feet, nearly.

We know, from the principles of geometry, that all similar surfaces, or areas, are to each other as the squares of their like dimensions.

7. Suppose we have two circular pieces of land, the one 100 feet in diameter, the other 20 feet in diameter. How much more land is there in the larger than in the smaller?

By the above principle of geometry it follows, that the quantity of land in the two circles, must be as the squares of the diameters, that is, 1002 to 202, or as 25 to 1. Hence, there is 25 times as much in the one piece, as there is in the other.

8. Suppose by observation, it is found that 4 gallons of water flow through a circular orifice of 1 inch in diameter in 1 minute. How many gallons would, under similar circumstances, be discharged through an orifice of 3 inches in diameter, in the same length of time? Ans. 36 gallons.

EXTRACTION OF THE CUBE ROOT.

207

9. What length of thread is required to wind spirally around a cylinder, 2 feet in circumference and 3 feet in length, so as to go but once around?

It is evident that if the cylinder be developed, or placed upon a plane, and caused to roll once over, that the convex surface of the cylinder will give a rectangle, whose width is 2 feet, and length 3 feet; at the same time the thread will form its diagonal. Hence, the length of the thread is √4+9=√13=3.60555 feet, nearly.

EXTRACTION OF THE CUBE ROOT.

120. If we cube 45 by the usual process, we find 45391125.

If, instead of 45, we take its equal 40+5, and then cube it by actual multiplication, we shall have this