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. D K N M E c 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. K N E D B F 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. = THEOREM III. B The convex, or lateral surface of a cone, is equal to the prod uct 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 SA. A E 8 C 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 2R; and the convex surface of the cone will be 2′′R × 4L, 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 × (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. (AF÷DH) 2 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. For, if AD and OC are produced till they meet in S, the surface described by AD is evidently that of a truncated cone having AO and DC for the radii of its bases, the vertex of the whole cone being S. Hence this surface will be measured as above. This measure will always hold good, even when the point D falls on S, and thus forms a whole cone; and also when the line AD is parallel to the axis, and thus forms a cylinder. In the first case, DC would be nothing; in the second, DC would be equal to AO and to IK. Cor. I. Through I, the middle point of AD, draw IKL parallel to AB, and IM parallel to AF: it may be shown, as above, that IM = circ. IK; but the trapezium ADHF = AD × IM = AD x circ. IK. Hence it may also be asserted, that the surface of a truncated cone is equal to its side multiplied by the circumference of a section at equal distances from the two bases. C B I A Cor. II. The point I being the middle of AB, and IK a perpendicular drawn from the point I to the axis, the surface described by AB, by the last Corollary, will have for its measure AB circ. IK. Draw AX parallel to the axis; the triangles ABX, OIK will have their sides perpendicular each to each, namely, OI to AB, IK to AX, and OK to BX; hence these triangles are similar, and give the proportion F MK N ОР AB: AX or MN:: OI: IK, or as circ. OI to circ. IK; hence AB × circ. IK = MN x circ. OI. Whence it is plain that the surface described by the partial polygon ABCD is measured by (MN+NP+PQ) × circ. OI, or by MQ × circ. OI; hence it is equal to the altitude multiplied by the circumference of the inscribed circle. Cor. III. If the whole polygon has an even number of sides, and if the axis FG passes through two opposite vertices F and G, the whole surface described by the revolution of the half polygon FACG will be equal to its axis FG multiplied by the circumference of the inscribed circle. This axis FG will, at the same time, be the diameter of the circumscribed circle. |