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TO MEASURE A CIRCLE.

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ART. 4.—The diameter of a circle being given, to find the circumference.

Rule.As 7 : is to 22 :: 80 is the given diameter : to the circum ference. Or, more exactly, as 113 : is to 355 : : &c. the diameter is found inversely,

Note. The diameter is a right line drawn across the circle through its centre.

1. The i and 6 inch No ir fit.

EXAMPLES,

1. What is the circumference of a wheel whose diameter is 4 feet?-as ay : 22 :: 4 : 12,57 the circumference.

2. What is the circumference of a circle whose diameter is 35?-As many : 22 : : 35 : 110 Ans.--and inversely as 22 : 7 :: 110 : 35, the diameter, &c.

16=1,3 Or, 18>

Supp every way;

feet of ear

Art. 5.-To find the area of a Circle.

ART. 7.

dimen: timber and de

Rule.product by

RULE.-Multiply half the diameter by half the circumference, and the product is the area; or if the diameter is given without the circumference, multiply the square of the diameter by ,7854, and the product will be the area.

EXAMPLES. 1. Required the area of a circle whose diameter is 12 inches, and circumference 37,7 inches.

18,85–half the circumference.

6=half the diameter.

1. AS 18 inches ng; ho

ift. 9 inch

113,10 area in

square

inches. 2. Required the area of a circular garden whose diameter is 11 rods?

,7854
By the second method, 11x11 121

Ans. 95,0334 rods
SECTION 2.-OF SOLIDS.

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Solids are estimated by the solid inch, solid foot, &c. 1728 of these inches, that is, 12 x 12 x 12 make 1 cubic vi solid foot

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ART. 6.-To measure a Cube. Definition. A cube is a solid of six equal sides, each of in which is an exact square.

RULE.--Multiply the side by itself, and that product by the same side, and this last product will be the solid content of the cube.

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1. The side of a cubic block being 18 inches, or 1 foot and 6 inches, how many solid inches doth it contain ? ft. in ft. 1 6=1,5 and 1,5 x 1,5 x 1,5=3,375 solid feet. Ans. Or, 18 x 18 x 18-5832 solid inches, and 1931-3,375.

2. Suppose a cellar to be dug that shall contain 12 feet e every way, in length, breadth and depth; how many solid I feet of earth must be taken out to complete the same?

12 x 12 x 12=1728 sold feet, the Ans. Art. 7.-To find the content of any regular solid of three

dimensions, length, breadth and thickness, as a piece of
timber squared, whose length is more than the breadth
and depth.
RULE.—Multiply the breadth by the depth, or thickness, and that

product by the length, which gives the solid content. E!

EXAMPLES,

1. A square piece of timber, being one foot 6 inches, or 18 inches broad, 9 inches thick, and 9 feet or 108 inches long;

how many solid feet doth it contain ?
1 ft. 6 in.=1,5 foot
9 inches ,75 foot.

Prod. 1,125 x 9-10,125 solid feet, the Ans.
in. in. in solid in.
Or 18x9x 108=17496-1728=10,125 feet.

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But, in measuring timber, you may multiply the breadth in inches, and the depth in inches, and that product by the length in fect, and divide the last product by 144, which will give the solid content in fcet, &c.

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2. A piece of timber being 16 inches broad, 13 inches thick, and 20 feet long, to find the content ?

Breadth 16 inches.
Depth 11
Prod. 176 x 2053520 then, 3520-144=24,4 feet. Ans.

3. A piece of timber 15 inches broad, 8 inches thick, and 25 feet long; how many solid feet doth it contain ?

Ans. 20,8+ feet. Art. 8.-When the breadth and thickness of a piece of

timber are given in inches, to find how much in length will make a solid foot.

Or, 18

18

324)

RULE.—Divide 1728 by the product of the breadth and depth, and the quotient will be the length making a solid foot.

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1. If a piece of timber be 11 inches broad and 8 inches ep, how many inches in length will make a solid foot ?

11 x8=88)1728(19,6 inches. Ans. 2. If a piece of timber be 18 inches broad and 14 inches deep, how many inches in length will make a solid foot? 18 x 14=252 divisor, then, 252) 1728(0,8 inches. Ans.

ART. 9.-To measure a Cylinder. Definition.—A Cylinder is a round body whose bases are circles, like a round column or stick of timber, of equal big. ness from end to end.

If the and its when he

llx

hewn s

Art. 1

of a

diar

ROLE.--Multiply the square of the diameter of the end by ,7854 which gives the area of the huse; thion multiply the area of tie base boy the length, and the product will be the solid content,

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EXAMPLE.

What is the solid content of a round stick of timber of equal bigness from end to end, whose diameter ia 18 inches, and length 20 feet?

Ho inclu and:

18 in.= 1,5 ft.

X1,5
Square 2,25,7854–1,76715 area of the base.

+20 length.
Ans. 35,34300 solid content,
Or, 18 inches.

18 inches.

324,7854=251,4696 inches, area of the base.

20 length in feet. 144)5089,3920(35,313 solid feet. Ans. ART. 10. To find how many solid feet a round stick of

timber, equally thick from end to end, will contain when

hewn square.

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RULE. Multiply twice the square of its semi-diameter in inches by the length in feet, then divide the product by 144, and thie quotient will be the answer.

EXAMPLE. If the diameter of a round stick of timber be 22 inches and its length 20 feet, how many solid feet will it contain when hewn square ?

11x11x2x20= 144=33,6+ feet, the solidity when hewn square ART. 11. To find how many feet of square edged boards

of a given thickness, can be sawn from a log of a given diameter.

RULE. Find the solid content of the log, when made square, by the last article-Then say, As the thickness of the board including the saw calf : is to the solid feet :: so is 12 (inches) to the number of feet of boards.

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EXAMPLE.

How many feet of square edged boards, 11 inch thick, including the saw calf, can be sawn from a log 20 feet long and 24 inches diameter?

12x12x2x20;144–40 feet, solid content

As 1 : 40 : : 12 : 384 feet, the Athe

beam, and the

product by 33

&wer.

EXAMPLE.

di Art. 12. The length, breadth and depth of any square brax Rule... being given, to find how many bushels it will contain,

RULE. Multiply the length by the breadth, and that product by the depth, divide the last product by 2150,425 the solid inches in a statute bushel, and the quotient will be the an

Suppose

beain and There is a square box, the length of its bottom is 50 inches, breadth of ditto 40 inches, and its depth is 60 inches; how many bushels of corn will it hold ? 50 X 40x60=2150,425=55,8-1 + or 55 bushels three

Multiply pecks. Ans. Art. 13. The dimensions of the walls of a brick building

being given, to find how many bricks are necessary to build it.

RULE. From the whole circumference of the wall measured Art. 15.round on the outside, subtract four times its thickness, then multiply the remainder by the height, and that product by the thickness of the wall, gives the solid content of the

Rule. whole wall; which multiplied by the number of bricka contained in a solid foot gives the answer.

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Therefore

EXAMPLE,

How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will it take to build a house 4.1 feet long, 40 feet wide, and 20 feet high, and the walls to be l foot thick?

8x4x2,5=80 solid inches in a brick, then 1723-80-21,6 bricks in a solid foot. 44+40+44+40=108 feet, whole length of wall.

- tines ilie thickness.

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3280 solid feet in the whole wall
21,6 bricks in a solid foot

Multiply by

Product,

As Art.

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70848 bricks. An

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