A right-angled triangle contains a right angle.

The side opposite the right angle is called the hypotenuse. The other two sides are called the base and perpendicular.

It is an established theorem that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

The annexed figure illustrates this theorem and the following rules.

RULE 1.-Extract the square root of the SUM of the square of the base and the square of the perpendicular; the result will be the HYPOTENUSE.

RULE 2.-Extract the square root of the DIFFERENCE between the square of the hypotenuse and the square of the given side; the result will be the other side required.

Examples.

1. What is the hypotenuse of a right-angled triangle whose base is 36 ft. and perpendicular 45 ft.? Ans. 57.6 ft. 2. If the hypotenuse of a right-angled triangle is 65 feet, and the base 52 feet, what is the perpendicular ?

Ans. 39 feet. 3. The hypotenuse of a right-angled triangle is 80 feet, and the perpendicular 48 feet, what is the base ?

Ans. 64 feet.

4. Two ships start from the same point at the same time. In six days, one has sailed 500 miles due east, and the other 400 miles due north. What is their distance apart ?

5. How far from the base of a building must a ladder 100 feet in length be placed so as to reach a window 60 feet from the ground? Ans. 80 feet.

6. A room is 32 feet long and 24 feet wide; what is the distance betweon the opposite corners ? Ans. 40 ft.

7. A boy in flying his kite let out 500 feet of string and then found that the distance from where he stood to a point directly under the kite was 400 feet; how high was the kite? Ans. 300 feet.

CUBE ROOT.

ART. 170. The Cube Root of a number is a number which multiplied by itself twice, will produce the given number. Thus, the cube root of 64 is 4, since 4x4x4-64.

The process of finding the cube root of a number is best understood, as in square root, by involving a number, and thus ascertaining the law of the formation of the power.

The first nine numbers are,

1, 2, 3, 4, 5, 6, 7, 8, 9, and their cubes,

1, 8, 27, 64, 125, 216, 343, 512, 729.

From which it is seen that the cube of any number composed of one order of figures may contain one, two, or three orders.

Conversely, the cube root of any number composed of one, two, or three orders, is composed of but one order.

The numbers in the second line above are the only perfect cubes below 1000.

Again, 10' 1000 and 90°-729000. From which it is seen that the cube of tens gives no order below thousands, or above hundreds of thousands. In the same manner it may be shown that the cube of any number must contain at least three times as many orders, less two, as the number cubed. Thus the cube of any number composed of four orders must contain either ten, eleven, or twelve figures.

Let us now involve a number composed of two orderstens and units-to the third power, and observe the law of formation.

54°50'+3 x 50 x 4+3 x 50 x 4+4= 125000+30000+2400 +64-157464.

By using algebraic symbols, it may be rigidly shown that what is true of the above number, is true of any number composed of tens and units; that is,

The cube of any number composed of tens and units is equal to the cube of the tens, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. Let us now proceed to determine a process by which the cube root of a number may be found.

Ex. 1. What is the cube root of 157465 ?

Explanation.-Since 157464 is composed of six orders, the root will be composed of two, and since the cube of tens give no order below thousands, we separate the number into periods of three figures each by placing a dot over units, and another over thousands. Now, according to principles above explained, 157 must contain the cube of the ten's figure of the root. The greatest cube in 157 is 125, the cube root of which is 5. Place 5 for the ten's figure of the root. Subtract the cube of 5 from 157, and annex 4 of the next period to the remainder, giving 324. Now three times the product of the square of the tens by the units must be found in 324, since the square of tens gives no order below hundreds.

Square 5 tens and multiply the result by 3 for a trial divisor to find the next root figure. Place the quotient below the order in the root. It may be too large, since three times the product of the tens by the square of the units may give orders above tens, thus forming a part of 324, cube 54, and since the result is not greater than 157464, place 4 for the unit's figure of the root.

Ex. 2. What is the cube root of 34328125 ?

1. Separate the given numbers into periods of three figures each, commencing at units.

2. Find the greatest perfect cube in the left hand period, and place its root on the right as the highest order of the root. 3. Subtract the cube of the root figure from the left hand period, and to the remainder annex the first figure of the next period for a dividend.

4. Take three times the square of the root figure now found for a trial divisor, and place the number of times it is contained in the dividend, for the next figure of the root. Cube the root now found, and if the result is less than the first two periods of the given number, bring down the first figure of the next period for a new dividend; if, however, the cube is greater than the first two periods, diminish the last root figure by 1.

5. Take three times the square of the root now found for a new trial divisor, and place the number of times it is contained in the new dividend for the third figure of the root. Cube the three figures of the root, and subtract the result from the first three periods of the given number. Continue the operation in a similar manner until all the periods are used.

Notes.-1. When any dividend is not large enough to contain its trial divisor, place a cipher for the next figure of the root, and take three times the square of the root thus formed for a new trial divisor. Form a new dividend by bringing down the remaining two figures of the period, and the first figure of the next period.

2. When there is a remainder after all the periods are used, annex periods of ciphers and continue the operation until the requisite number of decimal places is obtained.

3. Extract the cube root of both terms of a common fraction, when they are perfect powers; otherwise multiply the numerator by the square of the denominator, and divide the root of the product by the denominator. The result will be the root with an error, less than one divided by the denominator.

4. In extracting the cube root of decimals or mixed decimals, ciphers must be added, to fill the periods.