Timber and some kinds of stone are bought and sold by the solid (or cubic) foot, the contents being found by multiplying the length, breadth, and thickness together; timber is sometimes bought and sold by the ton.

2. A stone is 5 feet 3 inches long, 3 feet 7 inches wide, and 2 feet 9 inches thick; required the number of cubic feet it contains. Ans. 51 F. 8' 9" 9". 3. How many cubic feet in a piece of timber 37 feet long, 1 foot wide, and 1 foot 3 inches thick ? Ans. 69 F. 4′ 6′′. 4. What will 9 pieces of timber come to, at $2,50 per ton, each piece being 294 feet long, I foot 9 inches, by 1 foot 7 inches, the side widths. Ans. $36,78+.

Diggers and fillers up of docks, wharves, &c. work by the square of 6 feet long, wide and deep, making 216 cubic feet to a square.

5. How many squares have been dug out of a cellar, 41 feet long, 21 feet wide, and 8 feet 9 inches deep? Ans. 35 squares, 153 F. 1' 6". 6. A dock has been filled, which was 126 feet 7 inches long, 50 feet 4 inches wide, and 12 feet deep; how much does it come to at $2,75 per square?

Ans. $1013,96146.

The freight of bales, boxes, chests, trunks, &c. is commonly paid so much per cubic foot, or so much per

ton.

7. How many cubic feet are in a box 7 feet long, 4 feet 2 inches wide, and 3 feet 4 inches deep?

Ans. 104F. 2'. 8. How many tons (40 cubic feet to the ton) are in 7 bales, each 5 feet, 9 inches long, 4 feet wide, and 3 feet 5 inches thick ? Ans. 13 tons, 30 F. 1'.

9 How much must be paid for the freight of 11 trunks, each being 4 feet long, 1 foot 7 inches wide, and 1 foot 4 inches deep, at $15,25, per ton?

Ans. $35,41,3,15

Grindstones are sold by the cubic foot, called a stone.

To find the number of stones (or cubic feet) contained in a grindstone, having the diameter and thickness given in inches.

RULE. Multiply the diameter by itself, then multiply this product by 11, divide the second product by 14, the quotient will be the superficial area of its side; then multiply this area by the thickness of the stone, in inches, and divide the product by 1728, the quotient will be the number of stones.

EXAMPLES.

1. How many stones are contained in a grindstone, its diameter being 36 inches, and 4 inches thick, and what is it worth at $1,25 per stone?

2. What is the value of a grindstone 25 inches diameter, and 6 inches thick, at $1,75 ?

Ans. $2,98,3+ 3. Required the value of a grindstone 7 inches thick, and 41 diameter, at $2,10 per stone. Ans. $11,23,5,+ To find the tonage of double decked vessels.

RULE Multiply the length of the keel by the breadth on the main beam, and that product by the depth, (all in feet) and divide the last product by 95 for tons.

NOTE. Half the breadth on the main beam is called the depth of double decked vessels.

United States tonage is thus found :

RULE. Take the length from the fore part of the stem, to the hind part of the stern post, above the upper decks, the breadth at the broadest part above the main wales; half this breadth is called the depth; then deduct from the length of the breadth, the remainder multiply by the breadth, and that product by the depth, and divide this last product by 95, for tons.

For single decked vessels.

RULE. Take the length and breath as above, and subtract of the breadth from the length the remainder multiply by the breadth, and that product by the depth from the under side of the deck plank, to the ceiling at bottom; divide this last product by 94 for

tons.

EXAMPLES.

1. What is the tonage of a ship, 87 feet from the fore part of the stem, to the hind part of the stern post above her upper deck, and 25 feet wide, United States tonage ? Ans. 23688. 2. What is the tonage of a schooner (single deck) 69 feet from the fore part of the stem, to the hind part of the stern post, 17 feet wide, and 74 feet deep?

Ans.

VUL

79 71

94.

VULGAR FRACTIONS.

Vulgar Fractions are parts of an unit, (or whole number) and are expressed by two numbers with a line drawn between them, the number over the line is called the numerator, and that under it the denominator.

The denominator is the number of parts the unit is supposed to be divided into, and the numerator is the number of those parts the fraction contains; is one half, i. e. an unit divided into halves, or two parts, this is one half, or one of those parts; is three fourths, 25 is twenty five twenty sixths, i. e. if an unit be divided into twenty six parts, it is twenty five of those parts.

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NOTE. A remainder after division is q numerator of a fraction, of which the divisor is its denominator.

Vulgar Fractions are of four kinds, viz. pro per, improper, compound, and mixed.

A fraction is said to be proper when the numerator

is less than the denominator; thus, 3 4 풀, 1, 3, 7, 15, &c. Improper fractions have their numerators greater than their denominators; thus,, . 19, 27, &c.

A compound fraction is a fraction of a fraction, thus, of of of, &c. i. e, one half of three fourths, two thirds of one half, three eighths of nine twelfths, &c.

A mixed fraction consists of an integer, or whole number, with a fraction annexed; thus 73, 127, are expressed seven and three ninths, twelve and seven eighths, &c.

NOTE. Any whole number may be reduced to an improper fraction, by considering it as a numerator, and placing under it for a denominator, with a line drawn between them, thus, &, 15 127 i. e. eight ones, fifteen ones, one hundred and twenty seven ones; here the 8, the 15, and the 127, are reduced to improper fractions.

8

REDUCTION OF VULGAR FRACTIONS.

CASE I. To reduce vulgar fractions to their lowest terms.*

RULE. Divide the numerator and denominator by such a number as will divide them both without a remainder; then divide these quotients (if need be) by the same, or any other number which will divide both without a remainder, &c. till no number except 1 will divide so: Or, find a common measure for dividing them at once; thus,

Divide the denominator by the numerator, then divide the numerator by the remainder, and then divide

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* Fractions are said to be in their lowest terms when they are expressed by the least numbers possible, thus, &c. when reduced to their lowest terms, will be 1

48 1296) 5769 1. &c.