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TO MEASURE A CIRCLE. ART. 4. The diameter of a Circle being giren, to find the circumference.
RULE. As 7. is to 22 : : so is the given diameter : to the circumference. 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. What is the circumference of a wheel whose diameter 1 4 feet ? -as 7 : 22 : ; 4: 12,57 the circumference.
2. What is the circumference of a circle whose diameter is 35 - As 7 : 22 : : 35 : 110 Ans. -and inversely as 22 : 7 :: 110 : 35, the diameter, &c. ART. 5. To find the area of a Circle.
RULE. Multiply half the diameter by half the circomference, 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.
113,10 area in square inches. 2. Required the area of a circular garden whose diame er is 11 rods?
97854 By she second method, 11X11 = 121
Ans. 95,0334 rods SECTION 2. OF SOLIDS. Solids are estimated by the solid inch, solid foot, &c. 1728 of these inches, that is 12x12x12 make 1 cubic or Xolid food
Art. 6. Te measure a Cube. Definition. A cube is a solid of six oqual sides, each of 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.
1. The side of a cubic block being 18 inches, or 1 root and 6 inches, how many solid inches doch it contain ? ft. in. ft. 16=1,5 and 1,5X1,5X1,5=3,375 solid feet. Ins. Or, 18x18x18=5832 solid inches, and 9=3,375.
2. Suppose a cellar to be dug that shall contain 12 feet every way, in length, breadth and depth ; how many solid feet of earth must be taken out to complete the same ?
12x12x12=1728 solid feet, the Ans. ART. 6. 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.
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 ?
i ft. 6 in=1,5 foot. 0 inches
Prod. 1,125x9=10,125 solid feet, the Ans.
in. in. in. solid in. Or, 18X9X108=174964-1728=10,125 feet.
But, in measuring timber, you may wultiply the breadth in inches, and the depth in inches, and that product by the length in feet, and divide the last product by 144, which will give the solid content in feet, &c.
2. A piece of timber being 16 inches broad, 11 inches thick, and 20 feet long, to find the content ?
Breadth 16 inches.
Prod, 176x20=3520 then, 3520-144=24,4 feet,
the Answer. 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.
RULE. Divide 1728 by the product of the breadth and depth, and the quotient will be the length making a solid foot.
1. If a piece of timber be 11 inches broad and 8 inches deep, 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 in ches deep, how many inches in leng? I will make a solid foot ? 18X14-252 divisor, then 252,1728(6,8 inches. Ans.
ART. 9. To measure a Cylinder. Definition.--A Cylinder is a round budy whose bases are circles, like a round column or stick of timber, of equal bigness from end to end.
RULE. Multiply the square of the diameter of the end by ,7854 which gives the area of the base ; then multiply the area of the base by the length, and the product will be the solid content.
What is the solid content of a round stick of timber of equal bigness froin end to end, those diameter 1o 18 de ches, and length 20 fcet ?
18 in.1,5 ft.
Square 2,25X,7854=1,76715 area of the base
x 20 length.
Ans. 35,34300 solid conteni'.
324,7854=254,4696 inches area of the base,
20 length in feet.
144)5089,3920(35,343 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.
RULE. Multiply twice the square of its semi-diameter in inches by the length in feet, then divide the product by 144, and the quotient will be the answer.
EXAMPLE If the diameter of a'round stick of timber be 22 inchies and its length 20 feet, how many solid feet will it contain i when hewn square ?
11x11x2X20--144-33,6+ feet, the solidity when
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 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.
EXAMPLE How many feet of square edged boards, 14 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. K2 As 1 : 40
$9% fpet, the Ani.
Art. 12. The length, breadth and depth of any square box 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 in. ches in a statute bushcl, anu the quotient will be the answer
There is a square box, the length of its bottom is 50 inches, breadth of ditto 40 inches, and its depth is 60 in. ches how
bushels of corn will it hold ? 50X40X60-2150,425= 55,84+ or 55 bushels three pecks. Ans. ART. 13. The dimensions of the walls of a brick build
ing being given, to find how many bricks are necessary to build it.
RULE From the whole circumference of the wall measured round on the outside, subtra.'t 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 whole wall ; which multiplied by the number of bricks contained in a solid foot gives the answer.
many bricks 8 inches long, 4 inches wide, and 21 inches thick, will it take to build a house 44 feet long, 40 feet wide, and 20 feet high, and the walls to be one foot thick ?
8X4X2,5–80 solid inches in a brick, then 1728+80 21,6 bricks in a solid foot. 44440+44+40=168 feet, whole length of wall.
---4 four times the thickness,
3280 solid feet in the whole wals. Multiply by 21,6 bricks in a solid foot.
Product, - 70848 bricks. Ans.