To find the solidity of a frustum of a direct wedge : Multiply the area of the lateral surface abnc, by the breadth of the wedge nr. Or, if the wedge be not of equal thickness throughout, multiply half the sum of the lateral surfaces by the breadth of the wedge, and the product will be the solidity, nearly. To find the solidity of an indirect wedge : To the length of the edge add twice the length of the back, or base, and reserve this sum; then multiply the altitude of the wedge by the thickness of the back, and this product by the reserved sum, and one-sixth of this product will be the solidity. To find the solidity of a prismoid : Add together the areas of the two ends, and four times the rea of the middle section parallel to the two ends, and one-sixth of the sum will be the mean area, which, multiplied by the altitude of the prismoid, gives the solidity. The prismoid may be regarded as the frustum of an indirect wedge. The length of the middle section of the prismoid is equal to half the sum of the lengths of the two ends, and its breadth is equal to half the sum of the breadths of the two ends. EXAMPLES. 1. What is the solidity of a direct wedge, whose back is 27 inches long and 8 inches thick, and its perpendicular height 40 inches? Ans. 4320 inches. 2. What is the solidity of the frustum of a direct wedge, the length of the base being 40 and its breadth 10 inches, the breadth of the other end 4 inches, and its altitude 30 inches? Ans. 8400 inches. 3. How many bushels will a box contain, its form being that of a frustum of a direct wedge, whose base is 3 feet in length, the breadths of the two ends 16 and 10 inches, and the height 18 inches? Ans. 3.9174 bushels. 4. What is the solidity of an indirect wedge, whose altitude is 14 inches, its edge 21, the length of the back 32, and its thickness 4 inches? Ans. 892 cubic inches. 5. Find the contents of a wedge, whose base is 16 inches long and 2 broad, its edge 10 inches, and its altitude 7 inches. Ans. 111.56 inches. 6. The length and breadth of the base of a wedge are 70 and 30 inches, the length of the edge 9 feet 2 inches, and the altitude 34.29016 inches; what is its solidity? Ans. 24.8048 feet. 7. Find the solidity of a prismoid, the length and breadth of its base being 10 and 8, those of the top 6 and 5, and the height 40 feet. Ans. 2120 cubic feet. 8. What is the solidity of a stick of timber, the length and breadth of one end being 2 feet 4 inches, and 2 feet, and those of the other end 1 foot and 8 inches, and its altitude or length 61 feet? Ans. 144.592 feet. 9. Find the capacity of a trough of the form of a prismoid, its bottom being 48 inches long and 40 inches broad, its top 5 feet long and 4 feet broad, and its depth 3 feet. Ans. 493 feet. 10. How many coal bushels, of 38 quarts each, will a box hold, its bottom being a rectangle 4 feet by 8, and the top a rectangle 6 feet by 10, and its depth 5 feet? Ans. 153 bushels. To gauge a prismoid:-Find the side of a square whose area is equal to the mean area of the prismoid, and then proceed in the same manner as with the square prism. A sphere or globe is a solid, comprehended under a convex surface, every point in which is equally distant from a certain point within, called the centre. Any line drawn from the centre to the surface is a radius; and any line drawn through the centre, and terminating at the surface at both extremities, is a diameter. The sphere may be generated by the revolution of a semicircle about its axis; and it may be considered as composed of an infinite number of cones or pyramids, whose bases form the convex surface, and whose vertexes all meet in the centre. The sphere is equal to a cone whose base is equal to the convex surface of the sphere, and its height equal to the radius of the sphere; it is likewise equal to two-thirds of its least circumscribing cylinder. Therefore, To find the solidity of a sphere : Multiply the area of its convex surface by one-third of the radius of the sphere, or by one-sixth of the diameter. Or, Find the solidity of a cylinder, whose diameter and length are equal to the diameter of the sphere, and two-thirds of it will be the solidity of the sphere. Or, Multiply the cube of the diameter of the sphere by .5236; or, when great accuracy is required, by .5235938. To find the greatest cube that can be cut from a sphere, its diameter being given : Extract the square root of one-third of the square of the diameter for the side of the cube. Or, Multiply the diameter by .57735. To find the diameter of a sphere, its solidity being given : Divide the solidity by .5236, and the quotient will be the solidity of the least circumscribing cube, the cube root of which will be the diameter. Το gauge a sphere:-Place two-thirds of the diameter over the gauge points for the cylinder, and over the diameter found on D will be found its solidity on the line C. The surface of a sphere is equal to the convex surface of its least circumscribing cylinder: it is also equal to four times the area of a great circle of the sphere. Therefore, To find the surface of a sphere :— Multiply its circumference by its diameter. Or, Multiply the square of its diameter by 3.1416. EXAMPLES. 1. How many cubic inches are contained in a sphere 25 inches in diameter ? Ans. 8181.25. 2. Find the solidity of a sphere, the diameter of which is 8 inches. Ans. 321.55585 cubic inches. 3. How many cubic feet of gas will a balloon of a spherical form contain, its diameter being 50 feet? Ans. 65449.85 feet. 4. What is the solidity, and what the surface of a sphere, whose diameter is 1? Answers, 0.5235988, and 3.1415926. 5. What is the diameter of a sphere whose capacity is 4 wine gallons? Ans. 12.08 inches. 6. How many solid feet, how many bushels, and how many ale gallons in a sphere whose diameter is 6 feet? Answers, 113.1 feet; 91 bushels; and 695 gallons, nearly. The gauge points for the last example may be found under the cylinder, 36; or the gauge points for the sphere, placing the whole diameter over the gauge point, may be found thus:-Divide the given number of inches (as 144 for feet, or 231 for wine, and 282 for ale gallons, &c.) by .5236, and extract the square root of the quotient. 7. What is the gauge point for solid feet in a sphere, if we place the diameter in feet over the gauge point? Ans. 16.58 inches. 8. What is the gauge point for bushels in a sphere ? Ans. 18.5 inches. 9. What is the gauge point for wine gallons in a sphere, the diameter in inches being placed over the gauge point? Ans. 21 inches, nearly. 10. How many wine gallons will a sphere contain, whose diameter is 12.1 inches? Ans. 4 gallons, nearly. diameter will it take to 11. How many spheres 4 feet in equal the contents of a sphere 8 feet in diameter ? Ans. 8 spheres. |