309. To find the diameter and circumference of a circle, either from the other. RULE 1. As 7 is to 22, so is the diameter to the circumference, and as 22 is to 7, ɛo is the circumference to the diameter. RULE 2. As 113 is to 355, so is the diameter to the circumference, and as 355 is to 113, so is the circumference to the diameter. RULE 3. A3 1 is to 3.1416, so is the diameter to the circumference, and as 3.1416 is to 1, so is the circumference to the diameter. 310. To find the area of a circle. RULE.-Multiply half the circumference by half the diame ter, or the square of the diameter by .7854,- —or the square of the circumference by .07958,-the product will be the area, 311. The area of a circle given to find the diameter and circumference. RULE.-1. Divide the area by .7854, and the square root of the quotient will be the diameter. 2. Divide the area by .07958, and the square root of the quotient will be the circumference. 312. To find the area of an oval, or ellipsis. RULE-Multiply the longest and shortest diameters together, and the product by .7854; the last product will be the area. 1. What is the area of an oval whose longest diameter is 5 feet, and shortest 4 feet? 5x4x.7854-15.708ft. Ans. 2. What is the area of an oval whose longest diameter is 21, and shortest 17? Ans. 280.3878. 313. To find the area of a globe or sphere. RULE.-Multiply the circumference by the diameter, and the product will be the area. 1. How many square feet in the surface of a globe whose diameter is 14 inches and circumference 44? 44x14-616 Ans. 2. How many square miles in the earth's surface, its circumference being 25000, and its diameter 7957 miles ? Ans. 198943750. 3. What is the area of the surface of a cannon shot, whose diameter is one inch? Ans. 3.1416 inches. 4. How many square inches in the surface of an 18 inch artificial globe ? Ans. 1017.8784. 2. Mensuration of Solids. 314. Mensuration of Solids teaches to determine the spaces included by contiguous surfaces, and the sum of the measures of these including surfaces is the whole surface of the body. The measure of a solid is called its solidity, capacity, content, or volume. The content is estimated by the number of cubes, whose sides are inches, or feet, or yards, &c. contained in the body. 315. To find the solidity of a cube. RULE.-Cube one of its sides, that is, multiply the side by itself, and that product by the side again, and the last product will be the answer. 1. If the length of the side of a cube be 22 feet, what is its solidity? 22×22×22-10648 Ans. 2. How many cubic inches in a cube whose side is 24 in-. ches? Ans. 13824. 316. To find the solidity of a parallelopipedon. RULE.-Multiply the length by the breadth, and that product by the depth, the last product will be the answer. 1. What is the content of a parallelopipedon whose length is 6 feet, its breadth 2 feet, and its depth 1g feet? 6 x 2.5 x 1.75-26.25, or 264 feet. 2. How many feet in a stick of hewn timber 30 feet long, 9 inches broad, and 6 inches thick? Ans. 11 feet. 317. To find the side of the largest stick of timber that can be hewn from a round log. RULE.--Extract the square root of twice the square of the semidiameter at the smallest end for the side of the stick when squared. 1. The diameter of a round log at its smallest end is 16 inches'; what will be the side of the largest squared stick of timber that can be hewed from it? 18x8x2=11.31 in. Ans. 2. The diameter at the smallest end being 24 inches, how large square will the stick of timber hew? Ans. 16.97 in. 318. To find the solidity of a prism, or cylinder. RULE.--Multiply the area of the end by the length of the prism, for the content. 1. What is the content of a triangular prism, the area of whose end is 2.7 feet, and whose length is 12 feet? 2.7x12=32.4 ft. Ans. 2. What number of cubic feet in around stick of timber whose diameter is 18 inches, and length 20 feet? Ans. 35.343. 319. To find the solidity of a pyramid or cone. RULE.-Multiply the area of the base by the height, and one third of the product will be the content. 1. What is the content of a cone whose height is 12 feet and the diameter of the base 24 feet? 2. What is the content of a triangular pyramid, its height being 141 feet, and the sides of its base being 5, 21×24=3×3=35-63, 6 and 7 feet? and 6X.7854 × 12÷3= 20.453125, Ans. Ans. 71.035+ 320. To find the solidity of a sphere.* RULE.-Multiply the cube of the diameter by .5236, or multiply the square of the diameter by one 6th of the circumference. 1. What is the content of a sphere whose diameter is 12 inches ? 12×12x12.5236 =094.7808, Ans. 2. What is the solid content of the earth, its circumference being 25000 miles? Ans. 26385814912 miles. 3. Guaging. YOLS 321. Guaging teaches to measure all kinds of vessels, as pipes, hogsheads, barrels, &c. RULE. To the square of the bung diameter add the square of the head diameter; multiply the sum by the length, and the product by .0014 for ale gallons, or by .0017 for wine gallons. 1. What is the content of a cask whose length is 40 inches, and its diameters 24 and 32 inches? 32×32+24×24×40=64000A 64000><.0014-89.6 a. gal. A. 64000><.0017108.8 w.gal. A. 2. What is the content of a cask whose length is 20 inches, and diameters 12 and 16? 11.2 a. gal. Ans. 13.6 w. gal SECTION III. PHILOSOPHICAL MATTERS, 1. Of the fall of Heavy Bodies. 322. Heavy Bodies near the surface of the earth, fall one foot the first quarter of a second, three feet the second quarter, five feet the third quarter, and seven feet the fourth quarter, equal to 16 feet the first secund. The velocities acquired by falling bodies, are in proportion to the squares of the times in which they fall; that is, if 3 bullets be dropped at the same time, and the first be stopped at the end of the first second, the second at the end of the second, and the third at the end of the third, the first will have fallen 16 feet, the second, (2×2=4) four times 16, equal to 64; and the third (3×3=9) nine times 16, equal to 144 feet, and so *The surface of a sphere is found by multiplying its diameter by its circumference. |