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. III. 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 slant height. IV. Two cylinders or two cones, are similar, when their axes are to each other as the diameters of their bases. V. If, in the circle ACD which forms the base of a cylinder, a polygon ABCDEM be inscribed, a right prism, constructed on this base ABCDEM, 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, etc., of the prism, being perpendicular to the plane of the base, K N I H 2 D C B M F A are evidently included in the convex surface of the cylinder. Hence the prism and the cylinder touch one another along these edges. VI. 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. I Q R F S H Y G D K Р A N B L M Let M, N, etc., be the points of contact in the sides AB, BC, etc.; and through the points M, N, etc., let MX, NY, etc., 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. VII. A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within. called the centre. 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. VIII. A great circle of the sphere, is a section which passes through the centre; a small circle, one which does not pass through it. IX. A plane is a tangent to a sphere, when their surfaces have but one point in common. X. 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. XI. 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. XII. The altitude of a zone, or of a segment, is the distance between the two parallel planes, which form the bases of the zone or segment. XIII. While the semicircle DAE, 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. THEOREM I. The lateral or convex surface of a cylinder has for its measure the product of its circumference into its altitude. I D H C K N E A B In the cylinder, suppose a right prism to be inscribed, having a regular polygon for its base. The lateral surface of this prism has for its measure the perimeter of its base multiplied by its altitude (B. VI., T. I.). When the number of sides of the polygon forming the base of the inscribed prism is indefinitely increased, its limit will become the circle constituting the base of the cylinder (B. IV., T. VIII., S.). We may, therefore, regard a cylinder as a right prism having a regular polygon of an infinite number of infinitely small sides for its base; and since the lateral surface of a right prism will always have for its measure the perimeter of its base into its altitude, it follows that the lateral surface of a cylinder has for its measure the circumference of its base into its altitude. THEOREM II. The volume of a cylinder has for its measure the product of its base into its altitude. As in the last Theorem, if we regard a cylinder as a right prism, having a regular polygon of an infinite number of sides for its base, and recall to mind that the volume of a right prism is the product of its base into its altitude (B. VI., T. XII.), we shall at once see that the volume of a cylinder is equal to the product of its base into its altitude. Cor. I. Cylinders of the same altitude are to each other as their bases; and cylinders of the same base are to each other as their altitudes. Cor. II. Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases. For the bases are as the squares of their diameters; and the cylinders being similar, the diameters of their bases (D. IV.) are to each other as the altitudes: hence the bases are as the squares of the altitudes; consequently, the bases multiplied by the altitudes, or the cylinders themselves, are as the cubes of the altitudes. Scholium. Let R be the radius of a cylinder's base, and H the altitude. The area of the base (B. IV., T. XIV., S.) will be R2; and the volume of the cylinder will be R2 × H, or R2H, where 3.141592, etc. = T THEOREM III. B The convex, or lateral surface of a cone, is equal to the product of the circumference of its base into half its slant height. Let the circle whose radius is OA be the base of the cone, S its vertex, and SA its slant height. Then will the convex surface of the cone have for its measure circ. OA × 4SA. A E C D For, conceive a regular polygon inscribed in the circle OA; and on this polygon, as a base, a pyramid having S for its vertex, to be constructed. The lateral surface of this pyramid will have for its measure the perimeter of the polygon, constituting its base, into one half of SF its slant height (B. VI., T. XV.). When the number of sides of the inscribed polygon is indefinitely increased, its perimeter will be limited by the circumference of the circle, its slant height will be limited by the slant height of the cone, and the limit of the lateral surface will be the convex surface of the cone. A cone may thus be regarded as a right pyramid, having a regular polygon of an infinite number of infinitely short sides for its base. And since the lateral surface of a pyramid will always have for its measure the perimeter of its base into half its slant height, however great may be the number of sides in the polygon forming its base, it follows, that the convex surface of a cone has for its measure the circumference of its base into half its slant height. Scholium. Let L be the slant height of a cone, R the radius of its base. The circumference of this base will be 2πR; and the convex surface of the cone will be 2TR × 1L, or RL. THEOREM IV. The convex surface of a truncated cone is equal to its side multiplied by half the sum of the circumferences of its two bases. In the plane SAB which passes through the axis SO, draw the line AF perpendicular to SA, and equal to the circumference having AO for its radius; join SF, and draw DH parallel to AF. From the similar triangles SAO, SDC, we have AO:DC::SA: SD; and by the similar triangles SAF, SDH, hence, AF: DH::SA: SD; AF: DH:: AO: DC; But, by or (B. IV., T. XIV.), as circ. AO is to circ. DC. construction, AF circ. AO; hence DH = circ. DC. the triangle SAF, measured by AF × SA, is equal to the surface of the cone SAB, which is measured by circ. AO × SA. For a like reason, the triangle SDH is equal to the surface of the cone SDE. Therefore the surface of the truncated cone ADEB is equal to that of the trapezium ADHF; but the latter is measured by AD × (AF+DH) (B. III., T. XXIII., C. II.). Hence the surface of the truncated cone ADEB is equal to its side AD multiplied by half the sum of the circumferences of its two bases. Scholium. If a line AD, lying wholly on one side of the line OC, and in the same plane, make a revolution around OC, the surface described by AD will have for its measure the lines AO, DC being perpendiculars, drawn from the extremities of the axis OC. |