MENSURATION OF SOLIDS. A solid is a figure that has length, breadth, and thickness. The boundaries of solids are surfaces. A surface, no part of which is plane, is called a curved surface. Any solid bounded by planes, is called a polyhedron. When the solid is contained by four planes, it is called a tetrahedron; by six, a hexahedron; by eight, an octahedron; by twelve a dodecahedron; and by twenty, an icosahedron. The planes that bound or contain any polyhedron, are called its sides, or faces; and the lines bounding its sides, its edges; the edge, however, in common language, is called the side. A prism is a solid contained by planes, of which two are opposite, equal, similar, and parallel, and the other sides are parallelograms. According as the two opposite and parallel planes are triangles, rectangles, or polygons, the prism is said to be triangular, rectangular, square, or polygonal; and the perpendicular distance between these two sides is called the altitude of the prism. When the lateral edges are perpendicular to the end or base, it is called a right prism, and in other cases an oblique prism; and a line joining the centres of the terminating planes, or parallel sides, is called the axis of the prism. The unit of measure for solids is a cube, the length of whose edge is the lineal unit; and the number of cubic units in any solid, is called its volume, or solid contents, or cubic contents. A cube is a solid comprehended under six equal sides, each side being an exact square. Cube one of its edges, or the lineal side. To find the distance between the opposite corners of the cube: Extract the square root of three times the square of one of its edges. To cube any number by the sliding rule: Place the number on C over 1 on the line D; then over the number found on D, will be found its cube, or third power, on the line C. For the method of extracting the cube root by the sliding rule, see ¶ 11. EXAMPLES. 1. How many solid inches in a cube whose edge is 12 inches? in one whose edge is 20 inches? in one whose edge is 30 inches? and in one whose edge is 35 inches? Answers, 1728; 8000; 27000; and 42875. 2. What is the superficial area of a cube 20 inches on each side, or edge? and how many bushels will a cubic box of this size contain? Ans. 163 square feet; capacity, 3 bushels, 2 pecks, and 7 quarts. 3. How many solid feet, and how many bushels in a cubic box 24 inches on a side? Answers, 8 cubic feet; and the capacity is 6 bushels, 1 peck, and 6 quarts, nearly. The square root of 2150.42, the number of solid inches in a bushel, is 46.37, nearly. Therefore a cubic or square box 46.37 inches on a side and one inch in depth, will hold a bushel; and if the box be two inches deep it will hold two bushels, and if three inches in depth it will hold three bushels, &c. And the height or depth of a box (the bottom being square) being given, its capacity will be as the square of its side. Therefore, if we place the side of a cube in inches, or the depth of any square box in inches, found on C, over 46.37 on D, then over the side found on D, will be found its capacity in bushels on the line C. Thus, in example 2d, if we place 20 over 46.37 on D, over 20 found on D will be found 3.71 bushels on the line C; and, if we place 24 on C over 46.37 on D, over 24 on D will be found 6.45 bushels on C. 2150.42 is 13.387, which is the side of a The square root of square, which, if one foot in depth, will contain one bushel. 13.387 is therefore the gauge point for bushels, when the depth of a square box is given in feet. Thus, in example 3d, the side of the cube being 2 feet, place 2 on C over 13.387 on D, and over 24 on D will be found 6.45 bushels on C. As 13.387 is an important gauge point, a dot or mark should be made on the rule under 13.387. See 22, under example 14. 4. How many bushels, and how many ale, and how many wine gallons, will a cubic vessel hold, its lineal side being 3 feet? Answers, 21.701 bushels; 202 wine, and 165.45 ale gallons. To find the number of wine and ale gallons by the sliding rule: Place the altitude or depth of any square vessel in inches over the square roots of the number of cubic inches in an ale or in a wine gallon; and then, over the side in inches found on D, will be found the number of gallons on C. Thus, place 36 on C over 15.2 on D, and over 36 found on D will be found 202, the number of wine gallons; and place 36 on C over 16.79 on D, and over 36 found on C will be found the number of ale gallons, viz. 165.45. 5. How many wine barrels of 31 gallons will a cubic cistern hold, which is 50 inches deep? Ans. 17.21 barrels, nearly. 6. How many ale barrels and how many coal bushels of 40 quarts each, will a cubic vessel hold, whose side is 45 inches? Answers, 33.8 bushels, and 9.07 barrels. The gauge point for wine barrels (the side of the cubic, or depth of the square vessel, taken in inches, being placed over the gauge point) is the square root of 7276.5, viz. 85.3, nearly; and the gauge point for ale barrels of 36 gallons is 100.75, and for coal bushels of 40 quarts, 51.84. Consequently, to solve the above examples by the sliding rule:-Place the side of the cube over the respective gauge points on the line D, and over the given side in inches will be found the required contents. 7. The side of a cubic cistern is 4 feet; required its contents in feet, in wine and ale barrels, and in bushels of 32 and 40 quarts. Answers, 91.13 feet; 212 wine, and 15.62 ale barrels ; 73.45 and 58.76 bushels. 8. What is the distance between the opposite corners of a cube whose side is 15 feet? Ans. 26 feet, nearly. TIMBER MEASUREMENT. The measurement of timber will be found fully illustrated in this and the following sections. The methods here given are by far the most expeditious, whilst they are strictly accurate. The old method of measuring round and rectangular timber, by means of the girt, has been passed over as totally unworthy of notice, in consequence of its inaccuracy, as usually practised. To find the content of square timber:- Multiply the square of the side by the length. Or, if the length be taken in feet, and the side of the stick in inches :— Multiply the square of the side in inches by the length in feet, and divide the product by 144, and the quotient will be the content in cubic feet. The method of computing the content of square timber, by the sliding rule, will be found in the following paragraph; and the method of finding the content of round timber will be found fully illustrated in paragraphs 36 and 37. 34. PARALLELOPIPED. |