44.72 1 44.721 59.628 139.628 20 vers. Gine. 2792.560 •4 686.9792 area of Lune. Ex. 2. Thę chord is 20, and versed sines 10 and 2. Requited the area of the lune. Anf. 128.522. Ex. 3. The length of the chord is 48, and the heights of the segments 18 and 7. What is the area? Anf. 405.8676 Note. If semicircles be described on the three sides of a rightangled triangle, as diameters, then will the triangle be equal to the two lunes on the legs, taken together. DEFINITIONS. are figures that have length, breadth, and thicknefs. 2. The boundaries of solids are superficies. 3. A solid angle is that which is made by the meeting of more than two plane angles in the same point, and which are not in the same plane. 4. Similar solids are such as have their angles similar, and which are contained by the fame number of similar planes. 5. A cube is a solid contained by six equal squares. Fig. 8;. 6. A parallelopipedon is a solid having fix rectangular fides, every opposite pair of which are equal and parallel each to each. Fig. 86. 7. A prism is a solid whose sides are parallelograms, and is either triangular, square, pentagonal, &c. according to the figure of its end. Fig. 87. 8. A cylinder is a round folid, whose bafes are equal circles. Fig. 83. 9. A pyramid is a solid, whose base is a plane figure, and its sides triangles, whose vertices meet in a point, called the vertex of the pyramid, and is either triangular, square, pentagonal, hexagonal, &c. according to the figure of its base. Fig. 89. 10. A cone is a pyramid, having a circular base, and is described by the revolution of a right-angled triangle about one of its legs. It is either right-angled, acute-angled, or obtuseangled, according as the revolving leg is equal to, greater, or less than the other. Fis: 90. 11. The 11. The fixed leg is called the axis of the conè. 12. A sphere, or globe, is described by the revolution of de semicircle about its diameter; the centre and diameter of the sphere are the same as those of the revolving semicircle. Fig. 91. 13. A segment of any folid is a part cut off the top by a plane parallel to the base. The frustum of a solid is that part which remains after the fegment is cut off. Fig. 92. 14. The prismoid is a solid resembling the fruftum of a pyramid, having parallel bases, and these bases both rectangles, but disproportional. Fig. 93. 15. A zone is that part of a sphere between two parallel planes. Fig. 94. PROBLEM I. Fig. 85. To.find the fuperficies of a cube RULE. Multiply the area of one of its fides by 6, and the product will be the area of the cube. EXAMPLE 1. Required the superficies of a cube, whose fide is 14 inches, 14 14 56 6 1176 Anf. Ex. 2. How many square yards are in the fuperficies of a cube, whose fide is 5: feet ? Ans. 20 sq. yds. I feeo. Ex. 3. How many square feet are in the superficies of a cube, whose fide is 18 inches ? Anf. 137 sq. feet. PROBLEM II. To find the solidity of a cube. RULE. Multiply the length, breadth, and thickness continually, and the product is the solidity. EXAMPLE I. What is the solidity of a cube, whose fide is 8 feet? Ex. 2. Required the folidity of a cube, the side being 15 feet Anf. 3375 feet. Ex. 3. Required the folidity of a cube, whose fide is 35 yards. Anf. 34-328125 cub. yds. Ex. 4. How many yards digging are in a cubical celler 12 feet deep? A:1f. 64 cub. yds. Ex. 5. How many solid yards are in a cubical cellar, whose fide is 10 feet? Arif. 37., cib. udson PROBLEM III. To find the superficies of a parallelopipedon, or prisin, and of the cylinder. RULE. Multiply the perimeter of the end by the length ; to the product add twice the area of the end, and the sum will be the sun perficies. EXAMPLE I. Required the superficies of a parallelopipedon, whose length is 72 feet, breadth 3 feet, and thickness 2 feet. 2+2=4 3 3+3=6 10 perimeter. 6 area of one end. 72 2 2 720 12 area of both ends. 732 feet. Ex. 2. Required the surface of a parallelopipedon, whose length is 72 feet, breadth 5, and depth 4 feet. Anf. 1336 sq. feet. Ex: 3. What is the superficies of a parallelopipedon, whose length is 15, breadth 6, and thickness 4 inches ? Anf. 2 feet 5 inches. Ex. 4. Required the surface of a triangular prisın, whose length is 10 feet, and fides 3, 4, 5 feet. Ang. 132 feet. Ex. 5. Required the superficies of a prism, when the length is 32 feet, and the end a pentagon, whose side is 6 feet. Anf. 1150.037 Ex. |