4. A ladder 75 feet long being placed with its foot in the street reaches a window on one side 45 feet high, and on the other side 60 feet high.

How wide is the street?

Ans. Street, 105.

Angle 60

Angle 45

When a number is both integral and decimal, as 455.742, the integers must be pointed off as if there were no decimals, and the decimals as if there were no integers; thus, 4'55.74'20'. Add more ciphers if necessary.

To extract the square root of a fraction whose denominator is not a perfect square, multiply both terms of the fraction by the denominator; thus, to extract the root of 3.x 5 = 18.

The root of the denominator is now 5; of the numerator, 15.0000 (3.873

The Cube is a solid having for its base a square and every side equal to the base all equal squares; it has therefore three dimensions, all equal; and as 1 foot is 12

inches, let the base be a square, each side 12 inches; the square will contain 144 square inches, and for every inch in height 144 solid inches, and 12 in height 1728 solid inches; that is

12 x 12 x 12 = 1728.

The cube of 1 is 1, and of 2 it is 8.

1 x 1 x 1= 1, or 2 × 2 × 2 = = 8,

and the cube of 12 is 1728 increased three figures.

9 × 9 × 9 = 729, and 99 × 99 × 99 = 970299. The cube of the largest digit has three places of figures, and the cube of two places of the largest digits has six places of figures; therefore, the increase for one figure in the root is three in the cube; hence the pointing off in periods of three figures.

Extracting the cube root consists in having given the cube or solid contents to find a side.

REM.-As the cube is the product of three dimensions, and as the root is one of those dimensions, the divisor must necessarily have two dimensions.

As 11 x 11 x 11 = 1331 = (10 + 1)3 = 1331.

a + b + c

a+b+c

a2+ab+ac

ab+b2+bc

+ ac+be+c2

a2+2ab+b2+2ac+2bc+c2

a + b + c

a3+2ab+ab2+2a2c+2abc+ac2

a2b+2ab2+b3 +2abc+bc2+2b2c

a2c+2abc+b2c+2ac2+2bc2+c3

a3+3a2b+3ab2+b3+3a2c+6abc+3ac2+3bc2+3b2c+c3

a+b+c

a3+3a2b+3ab2+b3+3a2c × 6abc+3b2c+3ac2 + 3bc2+c3 (

a3

3a2+3ab+b2) 3a2b+3ab2 +b3

3a2b+3ab2+b3

3a2+6ab+3b2+3ac+3bc+c2 ) +3a2c+6abc+3b2c+3ac2 +3bc2 + c3 3a2c+6abc+3b2c+3ac2 + 3bc2 + c3

In the first exemplification, it is evident that the first term of the root is a, the cube of which is a3; the second term of the root is b, and the divisor must be contained in what remains after deducting as exactly b times; hence it must be 3a2+3ab+b2 =, which is composed of 3 times the square of the first term of the root, three times the product of the first and second terms of the root, and the

square of the second term. In the second exemplification, the first and second terms of the root are obtained as in the first exemplification, and the third term must be c; the divisor must therefore be contained c times in what remains, hence it must be 3a2+ 6ab + 3b2 + 3ac + 3bcc; and how has it been obtained? Ans. In the same way that the second divisor was obtained, viz., first by taking 3 times the square of the root already obtained; secondly, 3 times this root multiplied by c, the next term, and the square of c, the last term of the root.

REM.-Always use the root already found as tens, as it is so relatively to the next figure of the root, so that the first 1 must be regarded as ten, and its square 100 × 3 30; then in multiplying

=

the first figure of the root it must again be regarded as 10.

2. Extract the cube root of 970299.

970,299 ( 99

729

24300) 241 299

2430 241 299

81

4. What are the dimensions of a cube containing 1728 cu. in. ? Ans. 12 in.

It is demonstrated in Geometry that similar cubical bodies are to each other as the cubes of their like dimensions; as, if a cube of 1 inch weigh 2 lbs., one of three inches would weigh 27 x 2 = 54 lbs.

5. How many cubes whose edges measure inch, would be contained in a cubical block whose edges are 2 inches?

1 × 1 × 1 = 14.

2 × 2 × 2 = 8 x 4 = 512, Ans.

6. How many shot, inch in diameter, can be made of a globe of lead 4 inches in diameter ?

= 4096.

4 x 4 x 4 = 64 × 4096

=252144, Ans.

7. How many cubes whose edges measure in. is contained in a cubical block whose edges are 3 inches?

Ans. 629 cubes.

8. How many square feet in the surface of a cube whose volume is 3375 cubic feet? Ans. 1350 sq. ft.

9. What is the length of an edge of a cubical bin which contains 500 bushels of wheat?

Ans.. 8 feet 6 inches.