PRISM when each of its bases is perpendicular to its other sides; and it is TRIANGULAR, QUADRANGULAR, &c., according as its base is a triangle, quadrangle, &c. Thus, P is a triangular prism. 3. A PAR-AL-LEL-O-PI'-PED is a prism whose bases and also its other sides are parallelograms. Thus, B is a parallelopiped. 4. A parallelopiped is RIGHT when each of its faces is a rectangle. A common chest, a bar of iron, brick, &c., are instances of right parallelopipeds. When each face of a right parallelopiped, as A, is a square, it is termed a cube. A cube has 6 equal square faces. 5. A CYLINDER is a round prism, having circles for its ends. Thus, C is a cylinder, of which the line E F passing through the centers of both ends, is called the axis. 6. A PYRAMID is a solid having any plane figure for a base, and its sides triangles, whose vertices meet in a point at the top, called the VERTEX of the pyramid. A pyramid is TRIANGULAR, QUADRANGULAR, &c., according as its base is a triangle, quadrangle, &c. Thus, A is a triangular pyramid. 7. A body which has a circular base, and tapers uniformly to a point named the VERTEX, is called a CONE. The axis of a cone is a line passing through the vertex and the center of the base. Thus, C is a cone of which B V is the axis. B V B 8. A FRUSTUM of any body, as a pyramid or cone, is what remains when the top is cut off by a plane parallel to the base. 9. A GLOBE, or SPHERE, is a body of such a figure, that all points of the surface are equally distant from a point within, called the center. The diameter of a sphere, is a line passing through the center, and terminated both ways by the surface. The radius of a sphere is a line drawn from the center to the surface. Thus, A B is a diameter: CA a radius; C being the center of the sphere. 10. The HEIGHT or altitude of a solid, is a line drawn from its vertex or top, perpendicular to its base. 11. The CONTENTS or SOLIDITY of a body, is the space within it. The magnitude of this space is expressed by the number of times it contains a given space called the measuring unit. 12. The MEASURING UNIT for solids, is a cube whose base is the measuring unit for surfaces; as a cu. in., cu. ft., &c. ART. 320. To find the SOLID CONTENTS of a PARALLELOPIPED. Rule.-Multiply the length, breadth, and depth together: the product will be the solid contents. 1. Find the solid contents of a parallelopiped: the length, 12 ft.; breadth, 3 ft. 3 in.; depth, 4 ft. 4 in. Ans. 169 cu. ft. 2. The solid contents of a rectangular stone: the length, 6 ft.; breadth, 2 ft. 6 in.; depth, 1 ft. 9 in. Ans. 26 cu. ft. 3. A block of marble, in the form of a parallelopiped, is in length, 3 ft. 2 in., breadth, 2 ft. 8 in.; depth, 2 ft. 6 in.: what its cost, at 81 cts. per cu. ft.? Ans. $17.10 4. How many solid feet in a box 4 ft. 10 in. long, 2 ft. 11 in. broad, and 2 ft. 2 in. deep? Ans. 30 cu. ft. 6'6" 4". ART. 321. The principles of the preceding rule are applied to the measurement of MASONS' AND BRICKLAYERS' WORK. Masons' work is measured by the solid foot, or by the perch, which is 161⁄2 ft. long, 18 in. broad, 1 ft. deep; And multiplying these numbers together, shows that a perch contains 24, or 24.75 cu. ft. To find the number of PERCHES in any WALL, or solid body. Rule. Find the contents in cubic feet, by multiplying together the length, breadth, and depth; then divide by 24.75 to obtain the contents in perches. 1. How many perches in a wall 97 ft. 5 in. long, 18 ft. 3 in. high, 2 ft. 3 in. thick? Ans. 161.6+ P. 2. In a wall 53 ft. 6 in. long, 12 ft. 3 in. high, 2 ft. thick, how many perches? Ans. 52.95+ P. 3. What cost a wall 53 ft. 6 in. long, 12 ft. 6 in. high, 2 ft. thick, at $2.25 a perch? Ans. $121.59+ 4. How many bricks in a wall 48 ft. 4 in. long, 16 ft. 6 in. high, 1 ft. 6 in. thick, 20 bricks to the cu. ft.? Ans. 23925, 5. At $5.875 per thousand bricks, allowing 20 bricks to wall the solid foot, what will it cost to build a wall 320 ft. long, 6 ft. high, 15 in. thick? Ans. $282. 6. How many bricks each 8 in. long, 4in. wide, 2.25 in. thick, will be required for a wall 120 ft. long, 8 ft. high, 1 ft. 6 in. thick? Ans. 34560. 7. What the cost of building a wall 240 ft. long, 6 ft. high, 3 ft. thick, at $3.25 per 1000 bricks, each brick 9 in. long, 4 in. wide, 2 in. thick? Ans. $336.96 ART. 322. TO FIND THE SOLID CONTENTS OF A PRISM, OR OF A CYLINDER. Rule. Find the area of the base, and multiply it by the perpendicular height, the product will be the solid contents. 1. Each side of the base of a triangular prism is 2 in.; its length 14 in.: find the contents. A. v/588-24.2487+cu. in. 2. Find the contents of a cylinder 12 ft. long, the diameter of each end, 4 ft. Ans. 150.7968 cu. ft. 3. The cu. in. in a bu., each end 18 in. in diameter, depth 8 in. Ans. 2150.4252 cu. in. 4. If the bu. contain 2150.4 cu. in., what are the solid contents of a cylindrical tub 6ft. in diameter and 8 ft. deep? Ans. 181.764 bu. 5. How many bu. in a box 15 ft. long, 5 ft. wide, 4 ft. deep? Ans. 241+ bu. 6. How many bu, in a box 12 ft. long, 3 ft. wide, 5 ft. deep? Ans. 144.6+ bu. ART. 323. TO FIND THE SOLID CONTENTS OF A Rule.-Multiply the area of the base by the perpendicular height, and take one-third of the product. 1. Find the solid contents of a square pyramid, the base, 5 ft. each side; perpendicular height, 21 ft. 2. The solid contents of a cone, base perpendicular height, 15 ft. Ans. 175 cu.ft. 10 ft. in diameter; 3. The diameter of the base of a conical glass-house is 37 ft. 8 in.; the altitude, 79 ft. 9 in.: what space is inclosed? Ans. 29622.0227+cu. ft. 4. A sq. pyramid is 477 ft. high; each side of its base 720 ft. find the contents in cu. yd. Ans. 3052800 cu. yd. : 5. How often can a conical cup 9 in. deep, 1 in. diameter, be filled from a gal. or 231 cu. in. ? Ans. 43.57+times. ART. 324. TO FIND THE SOLID CONTENTS OF THE FRUSTUM Rule.-1st. Find the area or surface of each end by the preceding rules. 2d. Find the area of the mean base by multiplying the areas of the upper and lower bases together, extracting the square root of the product. 3d. Add together the areas of the upper, the lower, and the mean base; multiply their sum by one-third of the altitude, the product will be the solid contents. 1. Find the solid contents of a block with square ends, each side of the lower base 3 ft.; and of the upper base 2 ft.: the altitude, 12 ft. Ans. 76 cu. ft. 2. The length of the frustum of a sq. pyramid is 18 ft. 8 in.; the side of its greater base 27 in.; in. what the contents? : that of its lesser base, 16 Ans. 61.2283950+cu. ft. 3. Find the contents of a glass in the form of the frustum of a cone; the diameter at the mouth 21 in.; at the bottom, 1 in.; the depth 5 in. Ans. 12.76275 cu. in. ART. 325. TO FIND THE SOLID CONTENTS Of a Globe. Rule.-Multiply the cube of the diameter by .5236 1. What are the solid contents of 3 globes, their diameters being 13, 15, and 30 inches, respectively? Ans. 1150.3492; 1767.15; and 14137.2 cu. in. 1 ART. 326. TO FIND THE AREA OF THE SURFACE OF A BODY BOUNDED BY PLANE SURFACES. Rule. Find the area of the surfaces separately; then add. TO FIND THE AREA OF THE CURVED SURFACE OF A RIGHT CONE. Rule.-Multiply the circumference of the base by the slant height, and take half the product. TO FIND THE SURFACE OF A GLOBE. Rule.-Multiply the square of the diameter by 3.1416. 1. Each side of the base of a triangular pyramid is 5 ft. 4 in.; its slant height, from the vertex to center of each side of the base 7 ft. 6 in.: find the area of its surface. Ans. 60 sq.ft. 2. What is the convex surface of a cone, whose side is 25 ft., and diameter of the base 8 ft.? Ans. 333.795 sq. ft. 3. Find the area of the curved surface and base of a right cone, the slant height 4 ft. 7 in.; the diameter of its base 2 ft. 11 in. Ans. 27.679895+sq. ft. 4. If the earth be a perfect sphere 7912 mi. in diameter, what its superficial contents? Ans. 196663355.7504 sq.mi. ART. 327. GAUGING Is the method of finding the contents of any regular vessel, in gallons, bushels, barrels, &c. When the vessel is in the form of a cube or parallelopiped, apply this Rule. Take the dimensions in inches and multiply the length, breadth, and depth together; This product divided by 231, will give the contents in wine gal.; or, divided by 2150.4, will give the contents in bu. 1. How many and 4 ft. deep? wine gallons in a trough, 2. How many bushels in a box, 12 ft. 10 ft. deep? 10 ft. long, 5 ft. wide, Ans. 1496+gal. long, 6 ft. wide, and Ans. 578.57+bu. ART. 328. TO FIND THE CONTENTS OF A CISTERN, BOTH ENDS CIRCULAR, THE UPPER AND LOWER DIAMETERS EQUAL. Rule. Take all the dimensions in inches; then square the diameter, multiply this by the hight, and this product by .7854; this will give the contents in cubic inches, and this divided |