Hence, this rule for extracting the third or cube root of any given number:-Commencing at the unit figure, cut off periods of three figures each till all the figures of the given number are exhausted. Then find the greatest cube number contained in the first period, and place the cube root of it in the quotient. Subtract its cube from the first period and bring down the next three figures; divide the number thus brought down by 300 times the square of the first figure of the root, and it will give the second figure; add 300 times the square of the first figure, 30 times the product of the first and second figures, and the square of the second figure together, for a divisor; then multiply this divisor by the second figure, and subtract the result from the dividend, and then bring down the next period, and so proceed till all the periods are brought down.

For instance, in finding the cube root of 48228544, the operation will stand thus:

48'228'544(364 root.

27

3276)21228

19656

393136)1572544

1572544

Divided by 300×32=2700 Divided by 363×300=288800

If any thing remains, add three ciphers, and proceed as before; but for every three ciphers that are added, one decimal figure must be cut off in the root. And if the cube root of a fraction or a mixed number be required, reduce the fraction to a decimal, and proceed as in whole numbers: the decimal part however must consist of periods of three figures each, if not, ciphers must be added.

Examples in extracting the Cube Root. Ex. 1. Required the cube root of 512000000.

Answer, 800.

Ex. 2. Required the cube root of 447697125.

Ex. 3. Required the cube root of 2.

Ans. 765.

Ans. 1.259921.

Ex. 4. Required the cube root of 44361864.

Ex. 5. Required the cube root of .0001357.

Ans. 354.

Ans. .05138, &c.

Ex. 6. Required the cube root of or .018115942.

Ex. 7. Required the cube root of 133.

DUODECIMALS.

Ans. .262 nearly.

Ans. 2.3908.

Fractions whose denominators are 12, 144, 1728, &c. are called duodecimals; and the division and subdivision of the integers are understood without being expressed as in decimals. The method of operating by this class of fractions, is principally in use among artificers, in computing the contents of work, of which the dimensions were taken in feet, inches, and twelfths of an inch.

RULE.-Set down the two dimensions to be multiplied together, one under the other, so that feet shall stand under feet, inches under inches, &c. Multiply each term of the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each immediately under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place removed to the right hand when the multiplier is inches, and two places when the parts become the multiplier. The sum of these partial products will be the answer.

Or, instead of multiplying by the inches, &c. take such parts of the multiplicand as these are of a foot.

Or, reduce the inches and parts to the decimal of a foot, and proceed as in the multiplication of decimals.

For example, multiply 2 feet 6 inches by 2 feet 3 inches

Here, the 7, which stands in the second place, does not denote square inches, but rectangles of an inch broad and a foot long, which are to be added to the square inches in the third place; so that, 7 × 12 + 6 - 90 are the square inches, and the product is 5 square feet, 90 square inches. And this manner of estimating the inches must be observed in all cases where two dimensions in feet and inches are thus multiplied together.

Or, the product may be found by reducing the inches to the decimal of a foot: thus 6 inches .5 of a foot; hence, 2.5×2.25 5.625 square feet, but .625 of a square foot is equal to .625×144-90 square inches, the same as before.

Examples in Duodecimals.

Ex. 1. Multiply 35 feet 4 inches into 12 feet 3 inches. Ans, 434 square feet 47 square inches.

Ex. 2. Multiply 7 feet 9 inches by 3 feet 6 inches.

Ans. 27 square feet 18 square inches. Ex. 3. Multiply 7 feet 5 inches 9 parts by 3 feet 5 inches Ans. 25 square feet 1023 square inches. Ex. 4. Multiply 75 feet 9 inches by 17 feet 7 inches. Ans. 1331 square feet 135 square inches.

3 parts.

Ex. 5. Multiply 97 feet 8 inches by 8 feet 9 inches. Ans. 854 square feet 84 square inches.

PRACTICAL GEOMETRY

DEFINITIONS.

1. GEOMETRY is that science which treats of the descrip tions and properties of magnitudes in general.

2. A point is that which has position, but not magni

tude.

3. A line is length without breadth; and its bounds or extremes are points.

4. A right line is that which lies evenly between its extreme points.

5. A superficies is that which has length and breadth only and its bounds or extremes are lines.

6. A plane superficies is that which touches in every part any right line that can be drawn in that superficies. 7. A solid is that which has length, breadth, and thickness; and its bounds or extremes are superficies.

8. A plane rectilineal angle is the inclination or opening of two right lines which meet in a point.

9. One line is said to be perpendicular to another, when t makes the angles on both sides of it equal to each other.

10. A right angle is that which is formed by two lines that are perpendicular to each other.*

il. An acute angle is that which is less than a right angle.

12. An obtuse angle is that which is greater than a right angle.

Any angle differing from a right one, is, by some writers, called an oblique angle.