Given the base and perpendicular, to find the hypothenuse.

Rule. - Add the square of the base to the square of the perpendicular, and extract the square root of the sum: the result is the hypothenuse.

Given one side and the hypothenuse, to find the other side.

Rule. - Subtract the square of the given side from the square of the hypotheiiuse, and extract the square root of the remainder: the result will be the other side.

Examples.

1. Measure off on the end sill 6 feet from the corner of the louse, and on the side sill 8 feet from the same corner: what must be the length of a pole that shall just reach the outside of the sills at those points, when the sills are square? Ans. - 10 feet.

The square of one side is 6 x 6 = 36; the square of the other side is 8 x 8 = 64 + 36 = 100, the square root of which is 10.

2. A brace has a run of 4 feet x 3 feet 6 inches. What is the length of the brace?

Reduce the feet and inches to inches, in this case; square the length of each run and extract the square root of their sum: the result will be the length of the brace in inches.

3. A square measures 6 feet on a side. What will be the diameter of a circle that shall just enclose it?

The diagonal of the square will be the diameter of the circle.

All circles are to each other as the squares of their radii, diameters, or circumferences.

To find the diameter or circumference of a circle which shall contain a certain number of times the area of a given circle: -

Rule. - Square the given diameter or circumference, and state the question as in proportion; and the fourth term is the square of the required answer, extracting the square root of which gives the answer.

Examples.

1. If a one-inch rope will sustain a weight of 500 lbs., how much will a two-inch rope sustain? lxl: 2x2:: 500 lbs.: (answer). Ans. - 2,000 lbs.

2. If a 3/4-inch pipe will empty a cistern in 1 hour 17 minutes, how long will it take a 1 1/2-inch pipe to do it? 3/2 x 3/2: 3/4 x 3/4 :: 77 minutes: (answer). Ans. - 19 1/4 minutes.

3. If a one-inch rope will sustain 500 lbs., what is the size of a rope to sustain 1,000 lbs.? 500 : 1,000 :: 1 x 1 : (the square of the answer) = 2, the square root of which is 1 2/5+. Axs. - l 2/5+ inches.

4. If a chain made of 1/4-inch round iron will sustain a weight of 1 1/2 tons, of what sized iron should a chain be made to sustain a weight of 3 tons? 1 1/2 : 3 :: 1/4 x 1/4: (the square of the answer) = 1/8, the square root of which is .353+ = almost 3/8 inch: therefore a chain made of 3/8-inch round iron is rather more than twice as strong as one made of 1/4 inch iron.

Table of Square Roots from 1 to 100, inclusive.

Cube Root.

The Cube Root is the root of a third power: it is called cube root because the cube or third power of any number represents the contents of a cubic body of which the cube root is the length or breadth of one of the sides.

Rule for extracting the Cube Root. - 1. Point off the given number into periods of three figures each, counting from units' place toward the left in whole numbers, and toward the right in decimals.

2. Find the greatest cube in the left-hand period, and write its root for the first figure in the required root; subtract the mbe from the left-hand period, and to the remainder bring lown the next period for a dividend.

3. At the left of the dividend write three times the square of the first figure of the root, and annex two ciphers for a tria] divisor; divide the dividend by the trial divisor, and write the quotient for a trial figure in the root.

4. Annex the trial figure to three times the former figure, and write the result in a column marked 1, one line below the trial divisor; multiply this term by the trial figure, and write the product on the same line in a column marked 2; add this term as a correction to the trial divisor, and the result will be the complete divisor.

5. Multiply the complete divisor by the trial figure, and subtract the product from the dividend; and to the remainder bring down the next period for a new dividend.

6. Add the square of the last figure of the root, the last term in column 2, and the complete divisor together, and annex two ciphers for a new trial divisor, with which obtain another trial figure in the root.

7. Multiply the unit figure of the last term in column 1 by 3, and annex the trial figure of the root, for the next term of column 2; add this term to the trial divisor for a complete divisor, with which proceed as before.

Note 1. - If at any time the product be greater than the dividend, diminish the trial figure of the root, and correct the erroneous work.

Note. 2. - If a cipher occur in the root, annex two more ciphers to the trial divisor, and another period to the dividend; then proceed as before with column 1, annexing both ciphers and trial figure.

Example. - What is the cube root of 79.112?

Operation

Application of the Cube Root.

I wish to make a box, the length, breadth, and depth of which are to be equal, to hold 50 bushels of grain. What is the length of one side of this box?

We first find the number of cubic inches in 50 bushels, then extract the cube root: the result is the length or depth of the box in inches.

Cubes are to each other as the cubes of their sides.

Spheres (round balls) are to each other as the cubes of their diameters or circumferences.

To find the side, diameter, circumference, or altitude of any solid which is similar to a given solid: -

Rule. - State the question as in proportion, and cube the given sides, diameters, circumferences, or altitudes: the cube root of the fourth term of the proportion is the required answer.