alsc equal. Now, because the plane angles are like arranged about the triedral angles, these perpendiculars lie in the same direction; hence the point B will fall on the point E, and the solid angles will exactly coincide. SCHOLIUM 3. When the planes of the equal angles are not like disposed about the triedral angles, it would not be possible to make these triedral angles coincide; and still it would be true that the planes of the equal angles are equally inclined to each other. Hence, these triedral angles have the plane and diedral angles of the one, equal to the plane and diedral angles of the other, each to each, without having of themselves that absolute equality which admits of superposition. Magnitudes which are thus equal in all their component parts, but will not coincide, when applied the one to the other, are said to be symmetrically equal. Thus, two triedral angles, bounded by plane angles equal each to each, but not like placed, are symmetrical triedral angles. BOOK VII. SOLID GEOMETRY. DEFINITIONS. 1. A Polyedron is a solid, or volume, bounded on all sides by planes. The bounding planes are called the faces of the polyedron, and their intersections are its edges. 2. A Prism is a polyedron, having two of its faces, called bases, equal polygons, whose planes and homologous sides are parallel. The other, or lateral faces, are parallelograms, and constitute the convex surface of the prism. The bases of a prism are distinguished by the terms, upper and lower; and the altitude of the prism is the per pendicular distance between its bases. Prisms are denominated triangular, quadrangular, peni angular, etc., according as their bases are triangles, quadrilaterals, pentagons, etc. 3. A Right Prism is one in which the planes of the lateral faces are perpendicular to the planes of the bases. 4. A Parallelopipedon is a prism whose bases are parallelograms. 6. A Cube or Hexaedron is a rectangular parallelopipedon, whose faces are all equal squares. 7. A Diagonal of a Polyedron is a straight line joining the vertices of two solid angles not adjacent. 8. Similar Polyedrons are those which are bounded by the same number of similar polygons like placed, and whose homologous solid angles are equal. Similar parts, whether faces, edges, diagonals, or angles, similarly placed in similar polyedrons, are termed homologous. 9. A Pyramid is a polyedron, having for one of its faces, called the base, any polygon whatever, and for its other faces triangles having a common vertex, the sides opposite which, in the several triangles, being the sides of the base of the pyramid. 10. The Vertex of a pyramid is the common vertex of the triangular faces. 11. The Altitude of a pyramid is the perpendicular distance from its vertex to the plane of its base. 12. A Right Pyramid is one whose base is a regular polygon, and whose vertex is in the perpendicular to the base at its center. This perpendicular is called the aris of the pyramid. 13. The Slant Height of a right pyramid is the perpen. dicular distance from the vertex to one of the sides of the base. 14. The Frustum of a Pyramid is a portion of the pyramid included between its base and a section made by a plane parallel to the base. Pyramids, like prisms, are named from the forms of their bases. 15. A Cylinder is a body, having for its ends, or bases, two equal circles, the planes of which are perpendicular to the line joining their centers; the remainder of its surface may be conceived as formed by the motion of a line, which constantly touches the circumferences of the bases, while it remains parallel to the line which joins their centers. We may otherwise define the cylinder as a body ger.. erated by the revolution of a rectangle about one of its sides as an immovable axis. The sides of the rectangle perpendicular to the axis generate the bases of the cylinder; and the side opposite the axis generates its convex surface. The line joining the centers of the bases of the cylinder is its axis, and is also its altitude. If, within the base of a cylinder, any polygon be inscribed, and on it, as a base, a right prism be constructed, having for its altitude that of the cylinder, such prism is said to be inscribed in the cylinder, and the cylinder is said to circumscribe the prism. Thus, in the last figure, ABCDEc is an inscribed prism, and it is plain that all its lateral edges are con tained in the convex surface of the cylinder. If, about the base of a cylinder, any polygon be circumscribed, and on it, as a base, a right prism be constructed, having for its altitude that of the cylinder, such prism is said to be circumscribed about the cylinder, and the cylinder is said to be inscribed in the prism. Thus, ABCDEFe is a circum- F scribed prism; and it is plain that n m E d B the line, mn, which joins the points of tangency of the sides, EF and ef, with the circumferences of the bases of the cylinder, is common to the convex surfaces of the cylinder and prism. 16. A Cone is a body bounded by a circle and the surface generated by the motion of a straight line, which constantly passes through a point in the perpendicular to the plane of the circle at its center, and the different points in its circumference. The cone may be otherwise defined as a body gene rated by the revolution of a right-angled triangle about one of its sides as an immovable axis. The other side of the triangle will generate the base of the cone, while the hypotenuse generates the convex surface. The side about which the generating triangle revolves is the axis of the cone, and is at the same time its altitude. If, within the base of the cone, any polygon be inscribed, and on it, as a base, a pyramid be constructed, having for its vertex that of the cone, such pyramid is said to be inscribed in the cone, and the cone is said to circumscribe the pyramid. Thus, in the accompanying figure, V-ABCDE, is an inscribed pyramid, and it is plain that all its lateral edges are contained in the convex surface of the cone. If, about the base of a cone, any polygon be circumscribed, and on it, as a base, a pyramid be constructed, having E B for its vertex that of the cone, such pyramid is said to be circumscribed about the cone, and the cone is said to be inscribed in the pyramid. |