BOOK EIGHTH. DEFINITIONS. 1. A cylinder is a solid, which may be produced by the revolution of a rectangle ABCD, conceived to turn about the immovable side AB. In this rotation, the sides AD, BC, continuing always perpendicular to AB, describe equal circular planes DHP, CGQ, which are called the bases of the cylinder; the side CD at the same time. describing the convex surface. The immovable line AB is called the axis of the cylinder. Every section KLM made in the cylinder, at rightangles to the axis, is a circle equal to either of the bases; for, while the rectangle ABCD revolves about AB, the line KI, perpendicular to AB, describes a circular plane, equal to the base, which is a section made perpendicular to the axis at the point I. Every section PQGH passing through the axis, is a rectangle, and is double of the generating rectangle ABCD. 2. A cone is a solid, which may be produced by the revolution of a B H K S E C right-angled triangle SAB, conceived to turn about the immovable side SA. In this rotation, the side AB describes a circular plane BDCE, named the base of the cone; and the hypothenuse SB, its convex surface. The point S is named the vertex of the cone; SA, axis or altitude. its Every section HKFI formed at right-angles to the axis, is a circle. Every section SDE passing through the axis, is an isosceles triangle double of the generating triangle SAB. 3. If, from the cone SCDB, the cone SFKH be cut off by a section parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone. We may conceive it to be described by the revolution of a trapezium ABHG, whose angles A and C are right, about the side AG. The immovable line AG is called the axis or altitude of the frustum; the circles BDC, HFK are its bases, and BH is its side. 4. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases. 5. If, in the circle ACD which forms the base of a cylinder, a polygon ABCDE is inscribed, a right prism, constructed on this base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism. The edges AF, BG, CH, &c., of the B prism, being perpendicular to the plane of the base, are evidently included in the convex surface of the cylinder: hence the prism and the cylinder touch one another along these edges. 6. In like manner, if ABCD is a polygon circumscribed about the base of a cylinder, a right prism, constructed on this base, ABCD, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, or the cylinder to be inscribed in the prism. Let M, N, &c., be the points of contact in the sides AB, BC, &c.; H N P G S and through the points M, N, &c., let MX, NY, &c., be drawn perpendicular to the plane of the base: those perpendiculars will evidently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of contact. 7. A sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre. 8. The radius of a sphere, is a straight line drawn from the centre to any point in the surface; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface. All the radii of a sphere are equal: all the diameters are equal, and double of the radius. 9. A great circle of the sphere, is a section which passes through the centre; a small circle, one which does not pass through it. 10. A plane is a tangent to a sphere, when their surfaces have but one point in common. 11. The pole of a circle of a sphere, is a point in the surface equally distant from all the points in the circumference of this circle. 12. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles. Those arcs, named the sides of the triangle, are always supposed to be each less than a semicircumference; the angles, which their planes form with each other, are the angles of the triangle. 13. A spherical triangle takes the name of right angled, isosceles, equilateral, in the same cases as a rectilineal triangle. 14. A spherical polygon is a portion of the surface of a sphere, terminated by several arcs of great circles. 15. A lune is that portion of the surface of a sphere which is included between two great semicircles meeting in a common diameter. 16. A spherical wedge, or ungula, is that portion of the solid sphere, which is included between the same great semicircles, and has the lune for its base. 17. A spherical pyramid is a portion of the solid sphere, included between the planes of a solid angle whose vertex is the centre; the base of the pyramid, is the spherical polygon intercepted by the same planes. 18. A zone is the portion of the surface of the sphere, included between two parallel planes, which form its bases. One of these planes may be a tangent to the sphere; in which case, the zone has only a single base. 19. A spherical segment is the portion of the solid sphere, included between two parallel planes which form its bases. One of those planes may be a tangent to the sphere; in which case, the segment has only a single base. 20. The altitude of a zone, or of a segment, is the distance of the two parallel planes, which form the bases of the zone or segment. 21. While the semicircle DAE, (Def. 7,) revolving round its diameter DE, describes the sphere; any circular sector, as DCF or FCH, describes a solid, which is named a spherical sector. Note. The cylinder, the cone, and the sphere, are the three round bodies treated of in the elements of geometry. PROPOSITION I. THEOREM. The solidity of a cylinder is equal to the product of its base by its altitude. Let CA be a radius of the given cylinder's base; H the altitude. Let surf. CA represent the area of the circle whose radius is CA: we are to show that the solidity of the cylinder is surf. CAX H. For, if surf. CAx H is not the meas H A G D H ure of the given cylinder, it must be the measure of a |