2. Required the area of an ellipse whose axes are 24 and 18. Ans. 339.2928. PROBLEM XV. To find the area of any portion of a parabola. RULE.-Multiply the base by the perpendicular height, and take two-thirds of the product for the required area. 1. To find the area of the parabola ACB, the base AB being 20 and the altitude CD, 18. Ans. 240. A 2 Required the area of a parabola, the base the altitude 30. MENSURATION OF SOLIDS. The mensuration of solids is divided into two parts. 1st. The mensuration of their surfaces; and, The following is a table of solid measures: 1728 cubic inches 27 cubic feet 4492 cubic feet 282 cubic inches 231 cubic inches 2150.42 cubic inches 2dly. The mensuration of their solidities. We have already seen, that the unit of measure for plane surfaces is a square whose side is the unit of length. A curved line which is expressed by numbers is also referred to a unit of length, and its numerical value is the number of times which the line contains its unit. If, then, we suppose the linear unit to be reduced to a right line, and a square constructed on this line, this square will be the unit of measure for curved surfaces. = The unit of solidity is a cube, the face of which is equal to the superficial unit in which the surface of the solid is estimated, and the edge is equal to the linear unit in which the linear dimensions of the solid are expressed (Book VII. Prop. XIII. Sch.). = = = C = B being 20 and Ans. 400. = 1 cubic foot. 1 cubic yard. 1 cubic rod. 1 ale gallon. 1 wine gallon. 1 bushel. OF POLYEDRONS, OR SURFACES BOUNDED BY PLANES. PROBLEM 1. To find the surface of a right prism. RULE.-Multiply the perimeter of the base by the altitude, and the product will be the convex surface (Book VII. Prop. I.). To this add the area of the two bases, when the entire surface is required. 1. To find the surface of a cube, the length of each side being 20 fect. Ans. 2400 sq. ft. 2. To find the whole surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and altitude 20 feet. Ans. 91.949. 3. What must be paid for lining a rectangular cistern with lead at 2d. a pound, the thickness of the lead being such as to require 7lbs. for each square foot of surface; the inner dimensions of the cistern being as follows, viz. the length 3 feet 2 inches, the breadth 2 feet 8 inches, and the depth 2 feet 6 inches? Ans. 21. 3s. 10 d. PROBLEM II. To find the surface of a regular pyramid. RULE.-Multiply the perimeter of the base by half_the_slant height, and the product will be the convex surface (Book VII. Prop. IV.): to this add the area of the base, when the entire surface is required. 1. To find the convex surface of a regular triangular pyra mid, the slant height being 20 feet, and each side of the base 3 feet. Ans. 90 sq. ft. 2. What is the entire surface of a regular pyramid, whose slant height is 15 feet, and the base a pentagon, of which each side is 25 feet? Ans. 2012.798. PROBLEM III. To find the convex surface of the frustum of a regular pyramid. RULE.-Multiply the half-sum of the perimeters of the two bases by the slant height of the frustum, and the product wili be the convex surface (Book VII. Prop. IV. Cor.). 1. How many square feet are there in the convex surface of the frustum of a square pyramid, whose slant height is 10 feet, each side of the lower base 3 feet 4 inches, and each side of the upper base 2 feet 2 inches? Ans. 110 sq. ft. 2. What is the convex surface of the frustum of an heptagonal pyramid whose slant height is 55 feet, each side of the lower base 8 feet, and each side of the upper base 4 feet? Ans. 2310 sq. ft. PROBLEM IV To find the solidity of a prism. RULE.-1. Find the area of the base. 2. Multiply the area of the base by the altitude, and the product will be the solidity of the prism (Book VII. Prop. XIV.). 1. What is the solid content of a cube whose side is 24 inches? Ans. 13824. 2. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? Ans. 21. 3. How many gallons of water, ale measure, will a cistern contain, whose dimensions are the same as in the last example? Ans. 12917. 4. Required the solidity of a triangular prism, whose height is 10 feet, and the three sides of its triangular base 3, 4, and 5 feet. Ans. 60. PROBLEM V. To find the solidity of a pyramid. RULE.-Multiply the area of the base by one-third of the altıtude, and the product will be the solidity (Book VII. Prop. XVII.). 1. Required the solidity of a square pyramid, each side of its base being 30, and the altitude 25. Ans. 7500. 2. To find the solidity of a triangular pyramid, whose altitude is 30, and each side of the base 3 feet. Ans. 38.9711. 3. To find the solidity of a triangular pyramid, its altitude being 14 feet 6 inches, and the three sides of its base 5, 6, and 7 feet. Ans. 71.0352. 4. What is the solidity of a pentagonal pyramid, its altitude being 12 feet, and each side of its base 2 feet? Ans. 27.5276. 5. What is the solidity of an hexagonal pyramid, whose altitude is 6.4 feet, and each side of its base 6 inches? Ans. 1.38564. PROBLEM VI. To find the solidity of the frustum of a pyramid. RULE.-Add together the areas of the two bases of the frustum and a mean proportional between them, and then multiply the sum by one-third of the altitude (Book VII. Prop. XVIII.). 1. To find the number of solid feet in a piece of timber, whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base 6 inches, the altitude being 24 feet. Ans. 19.5. 2. Required the solidity of a pentagonal frustum, whose altitude is 5 feet, each side of the lower base 18 inches, and each side of the upper base 6 inches. Ans. 9.31925. Definitions. 1. A wedge is a solid bounded by five planes viz. a rectangle ABCD, called the base of the wedge; two trapezoids ABHG, DCHG, which are called the sides of the wedge, and which intersect De each other in the edge GH; and the two triangles GDA, HCB, which are called the ends of the wedge. A When AB, the length of the base, is equal to GH, the trapezoids ABHG, DCHG, become parallelograms, and the wedge is then one-half the parallelopipedon described on the base ABCD, and having the same altitude with the wedge. The altitude of the wedge is the perpendicular let fall from any point of the line GH, on the base ABCD. PROBLEM VII. To find the solidity of a wedge. 2. A rectangular prismoid is a solid resembling the frustum of a quadrangular pyramid. The upper and lower bases are rectangles, having their corresponding sides parallel, and the convex surface is made up of four trapezoids. The altitude of the prismoid is the perpendicular distance between its bases. RULE. To twice the length of the base add the length of the edge. Multiply this sum by the breadth of the base, and then by the altitude of the wedge, and take one-sixth of the product for the solidity. Let L-AB, the length of the base. 7=GH, the length of the edge. b=BC, the breadth of the base. h=PG, the alutude of De the wedge. Then, L—l=AB—GH= AM. G A N P A M H B Suppose AB, the length of the base, to be equal to GH, the length of the edge, the solidity will then be equal to half the parallelopipedon having the same base and the same altitude (Book VII. Prop. VII.). Hence, the solidity will be equal blh (Book VII. Prop. XIV.). to If the length of the base is greater than that of the edge, let a section MNG be made parallel to the end BCH. The wedge will then be divided into the triangular prism BCH-M, and the quadrangular pyramid G-AMND. The solidity of the prism =1bhl, the solidity of the pyramid −3bh(L—l); and their sum, ibhl+}bh(L—1)=‡bh3l+¦bh2L -bh2l=bh(2L+I). If the length of the base is less than the length of the edge, the solidity of the wedge will be equal to the difference between the prism and pyramid, and we shall have for the solidity of the wedge, bhl—bh(l—L)=‡bh3l—¡bh21+‡bh2L=‡bh(2L+1). 1. If the base of a wedge is 40 by 20 feet, the edge 35 feet, and the altitude 10 feet, what is the solidity? Ans. 3833.33.` 2. The base of a wedge being 18 feet by 9, the edge 20 feet, and the altitude 6 feet, what is the solidity? Ans. 504. PROBLEM VIII. To find the solidity of a rectangular prismoid. RULE.-Add together the areas of the two bases and four times the area of a parallel section at equal distances from the bases: then multiply the sum by one-sixth of the altitude. T |