GEOMETRY. BOOK VI. O F SOLID S. DEFINITIONS. 1 Every solid bounded by planes is called à polyedron. 2. The planes which bound a polyedron are called faces. The straight lines in which the faces intersect each other, are called the edges of the polyedron, and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron. 3. Two polyedrons are similar, when they are contained by the same number of similar planes, and have their polyedral angles equal, each to each. 4. A prism is a solid, whose ends are equal polygons, and whose side faces are parallelograms. Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side faces of the prism are the parallelograms DH, DK, EF, B K H AG, and BH. These are called the convex, or lateral surface of the prism Of the Prism. 5. The altitude of a prism is the distance between its upper and lower bases: that is, it is a line drawn from a point of the upper base, perpendicular, to the lower base. 7. A prism whose base is a triangle, is called a triangular pris; if the base is a quadrangle, it is called a quadrangular prisrr; if a pentagon, a pentagonal prism; if a hexagon a hexagonal prism; &c. 8. A prism whose base is a parallelogram, and all of whose faces are also paralelograms, is called a parallelopipedon. If all the faces are rectangles, it is called a rectangular parallelopipedon. 9. If the faces of the rectangular parallelopipedon are squares, the solid is called a cube: hence, the cube is a prism bounded by six equal squares Of the Pyramid. 10. A pyramid is a solid, formed by several triangles united at the same point S, and terminating in the different sides of a polygon ABCDE. The polygon ABCDE, is called the base of the pyramid; the point S, is called the vertex, and the triangles ASB, BSC, CSD, DSE, and ESA. form its lateral, or convex surface. E S B 11. A pyramid whose base is a triangle, is called a trangular pyramid; if the base is a quadrangle, it is called a quadrangular pyramid; if a pentagon, it is called a petagonal pyramid; if the base is a hexagon, it is called a hexagonal pyramid; &c. 12. The altitude of a pyramid, is the perpendicular let fall from the vertex, upon the plane of the base. Thus, SO is the altitude of the pyramid S-ABCDE. 13. When the base of a pyramid is a regular polygon, and he perpendicular SO passes through he middle point of the ase, the pyramid is called a right pyramid, and the line SO is called the axis. 16. A Cylinder is a sold, described by the revolution of a rectangle, AEFD, about a fixed side, EF. As the rectangle AEFD, turns around the side EF, like a door upon its hinges, the lines AE and FD describe circles, and the line AD describes the convex surface of the cylinder. E S The circle described by the line AE, is called the lower base of the cylinder, and the circle described by DF, is called the upper base. Of the Cylinder. The immovable line EF is called the axis of the cylir der A cylinder, therefore, is a round body with circular ends 17. If a plane be passed through the axis of a cylinder, it will intersect the cylinder in a rectangle, PG, which is double the revolving rectangle DE. M 18. If a cylinder be cut by a plane parallel to the base, the section will be a circle equal to the base. For, while the side FC, of the rectangle MC, describes 19. If a polygon be inscribed in the. lower base of a cylinder, and a corresponding polygon be inscribed in the upper base, and their vertices be joined by straight lines, the prism thus formed is said to be inscribed in the cylinder. |