TO MEASURE A CIRCLE. ART. 4. The diameter of a Circle being given, 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 555 :: &c. the diameter is found inversely. NOTE. The diameter is a right line drawn across the circle through its centre. EXAMPLES. 1. What is the circumference of a wheel whose diameter is 4 feet?-As 7: 22: 4: 12,57 the circumfe rence. 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. 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. : 113,10 area in square inches. 2. Required the area of a circular garden whose diame ter is 11 rods? ,7854 By the 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 12×12×12 make 1 cubic or solid foot. ART. 6. To measure a Cube. Definition. A cube is a solid of six equal 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. EXAMPLES. 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 1,5×1,5×1,5—3,375 solid feet. Ans. Or,. 18x18x18-5832 solid inches, and $33,375 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? 12×12×12=1728 solid 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. 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. =,75 foot. Prod. 1,125x9-10,125 solid feet, the Ans. in. in. in. solid in. Or, 18×9×108=17496÷1728-10,125 feet. But, in measuring timber, you may multiply 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. Depth 11 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. EXAMPLES. 1. If a piece of timber be 11 inches broad and 8 inches deep, how many inches in length will make a solid foot? 11x8=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? 18×14=252 divisor, then 252) 1728(6,8 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 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. EXAMPLE. What is the solid content of a round stick of timber of equal bigness from end to end, whose diameter is 18 inches, and length 20 feet? 18 in. 1,5 ft. Square 2,25x,7854-1,76715 area of the base, Ans. 35,84300 solid content. Or, 18 inches. 324x,7854-254,4696 inches, area of the base, 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 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 1.2 (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 ? 12×12×2×20-144-40 feet, solid content. As 14: 40: 12: 384 feet, the Ans. 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 inches in a statute bushel, and the quotient will be the answer. EXAMPLE. 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? 50×40×60÷2150,425=55,844 or 55 bushels, three pecks. Ans. 贯 ART. 13. The dimensions of the walls of a brick building being given, to find how many bricks are neces sary to build it. RULE. From the whole circumference of the wall measured 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 whole wall; which multiplied by the number of bricks contained in a solid foot, gives the answer. EXAMPLE. How many bricks 8 inches long, 4 inches wide, and 2 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? 8×4×2,5=80 solid inches in a brick, then 1728÷80 =21,6 bricks in a solid foot. 44+40+44+40=168 feet, whole length of wall. -4 four times the thickness. 164 remains. Multiply by 20 height. S280 solid feet in the whole wall. Product, 70848 bricks, Ans. D |