Or, in measuring boards, you may multiply the length in feet by the breadth in inches, and divide by 12, the, quotient will give the answer in square feet, &c. Thus, in the foregoing example, 21 x 18+12=31,5 as before. 4. If a board be 8 inches wide, how much in length will pake a square foot ? Rule.--Divide 144 by the breadth, thus, 8)144 Ans. 18 in. 5. If a piece of land be 5 rods wide, how many rods in length will make an acre ? RULE.—Divide 160 by the breadth, and the quotient will be the lepgth required, thus, 5)160 Ans. 32 rods in length. ART. 3.-To measure a triangle. Definition.— A triangle is any three cornered figure which is bounded by three right lines.* RULE. Multiply the base of the given triangle into half its perpendicular height, or half the base into the whole perpendicular, and the product will be the area. EXAMPLES. 1. Required the area of a triangle whose base or longest side is 32 inches, and the perpendicular height 14 inches. 32 7=224 square inches the Answer. 2. There is a triangular or three cornere: lot of land whose base or longest side is 51. rods; the perpendicular from the corner opposite the base measures 44 rods; how many acres doth it contain ? 51,5 x 22=1133 square rods,=7 acres, 13 rods. * A Triangle may be either right angled or oblique; in either case the teacher can easily give the scholar a right idea of the base and perpendiculo, by marking it down on the slate, paper, &c. 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 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. EXAMPLES. 1. What is the circumference of a wheel whose diameter is 4 feet? -as 7 : 22 :: 4 : 12,57 ihe circumference. 2. What is the circumference of a circle whose diameter js 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 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 circunference. 6-half the diameter. 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 SOLID8. Solids are estimated by the solid inch, solid foot, &c. 1728 of these inches, that is, 12x12x12 make I 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? fl. in fi. 1 6=1,5 and 1,5 x 1,5 < 1,5–3,375 solid feet. Ans. Or, 18 x 18 x 18=5832 solid inches, and 193=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? 12 - 12 - 12–1728 sold feet, the Ans. Art.7.-To find the content of any regular solid of three dinensions, length, hreadth and thickuess, 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 9 inches ,75 foot. Prod. 1,125 9-10,125 solid feet, the Ans. in. in. in. solid in. Or 18X9X 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 hy 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 fiud the content ? Breadth 16 inches. Prod. 176 x 20=3520 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. 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 inany inches in length will make a solid foot ? 11*8=82)1728(19,6 inches. Ans. 2. If a piece of tiinber be 18 inches broad and 14 inches deep, how many inches in length wilt nake a solid foot? 18 x 14=252 divisor, then, 252)172516,8 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. Role.—Mulliply 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 timior of equal biguess froni end to end, whose diameter is 18 inches, and length 20 feet? 18 in.=1,5 ft. x1,5 Square 2,25 x,7854=1,76715 area of the base. +20 length. Ans. 35,34300 solid content. Or, 18 inches. 18 inches. 324x,7851=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 inches and its length 20 feet, how many solid feet will it contain when hewn square ? 11X11 X220=144=33,6+ feet, the solidity when hewn square. Art. 11. To find how many feet of square edged boards of a given thickncss, 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. EXAMPLE. How many feet of square edged boards, 17 inch thick, including the saw calf, can be sawn from a log 20 feet long and 24 inches diameter? 12 x 12 y 2x20-144-40 feet, solid content. As į į: 10 : : 12 : 384 feet, the Ans. |