In the base of the cylinder inscribe any regular polygon, A B C D E F, and on this polygon construct a right prism of the same altitude with the cylinder. The prism will be inscribed in the convex surface of the cylinder. The solid contents of this prism are equal to the product of its base by its altitude (Prop. XIII. Bk. VIII.). B C Conceive now the number of the sides of the polygon to be indefinitely increased, until its perimeter coincides with the circumference of the circle ABCDEF (Prop. XII. Cor., Bk. VI.), and the solid contents of the prism will equal those of the cylinder. But the solid contents of the prism will still be equal to the product of its base by its altitude; hence the solid contents of the cylinder are equal to the product of its base by its altitude. D 576. Cor. 1. Cylinders of the same altitude are to each other as their bases; and cylinders of equal bases are to each other as their altitudes. π 578. Cor. 3. If the altitude of a cylinder be represented by H, and the area of its base by R2 × (Prop. XV. Cor. 2, Bk. VI.), the solid contents of the cylinder will be represented by R2 × × H. 577. Cor. 2. 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 radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each other as their altitudes (Art. 570); therefore the bases are as the squares of the altitudes; hence, the products of the bases by the altitudes, or the cylinders themselves, are as the cubes of the altitudes. PROPOSITION III.-THEOREM. 579. The convex surface of a cone is equal to the circumference of the base multiplied by half the slant height. S Let A B C D EF-S be a cone whose base is the circle A B C DEF, and whose slant height is the line SA; then its convex surface is equal to ABCDEF multiplied by SA. In the base of the cone inscribe any regular polygon, ABCDEF, and on this polygon construct a regular pyramid having the same vertex, S, with the cone. Then a right pyramid will be inscribed in the cone. B C A F E H D From S draw SH perpendicular to B C, a side of the polygon. The convex surface of the pyramid is equal to the perimeter of its base, multiplied by half its slant height, SH (Prop. XV. Bk. VIII.). Conceive now the arcs subtending the sides of the polygon to be continually bisected, until a polygon is formed having an indefinite number of sides; its perimeter will equal the circumference of the circle ABCDEF; its slant height, SH, will equal that of the cone, and its convex surface coincide. with the convex surface of the cone. But the convex surface of every right pyramid is equal to the perimeter of its base, multiplied by half the slant height; hence the convex surface of the cone is equal to the circumference of its base multiplied by half its slant height. 580. Cor. If SA represent the slant height of a cone, and R the radius of the base, then, since the circumference of the base is represented by 2 RX Bk. VI.), the convex surface of the sented by 2 R XXSA, equal to π (Prop. XV. Cor. 3, cone will be repreX RX SA. PROPOSITION IV.-THEOREM. 581. The convex surface of a frustum of a cone is equal to half the sum of the circumference of the two bases multiplied by its slant height. M L Let ABCDEF-M be the frustum of a cone, and AG its slant height; then the convex surface is equal to half the sum of the circumferences of the two bases ABCDEF, GHIKLM, multiplied by A G. B C For, inscribe in the bases of the frustum two regular polygons of the same number of sides, having their sides parallel, each to each. Draw the straight lines AG, BH, CI, &c., joining the vertices of the corresponding angles, and these lines will be the edges of the frustum of a pyramid inscribed in the frustum of the cone. The convex surface of the frustum of the pyramid is equal to half the sum of the perimeters of the two bases multiplied by its slant height, ON (Prop. XVII. Bk. VIII.). HOP D Conceive now the number of sides of the inscribed polygons to be indefinitely increased; the perimeters of the polygons will then coincide with the circumferences of the circles ABCDEF, GHIKLM; and the slant height, ON, of the frustum of the pyramid, will equal the slant height, A G, of the frustum of the cone; and the surfaces of the two frustums will coincide. But the convex surface of every frustum of a right pyramid is equal to half the sum of the perimeters of its two bases, multiplied by its slant height; hence, the convex surface of the frustum of the cone is equal to half the sum of the circumference of its two bases multiplied by half its slant height. 582. Cor. Through R, the middle point of the side KD, VOD draw the diameter RST, parallel to the diameter AQD, and the straight lines RU, KV, parallel to the axis PQ. Then, since DR is equal to R K, DU is equal to UV (Prop. XVII. Cor. 2, Bk. IV.); hence, the radius S R is equal to half the sum of the radii Q D, PK. But the circumferences of circles being to each other as their radii (Prop. XIII. Bk. VI.), the circumference of the section of which SR is the radius is equal to half the sum of the circumferences of which QD, PK are the radii; hence, the convex surface of a frustum of a cone is equal to the slant height multiplied by the circumference of a section at equal distances between the two bases. G Let ABCDEF-S be a cone, whose base is ABCDEF, and altitude SH; then its solidity is equal to ABCDEFX SH. T Q PROPOSITION V.-THEOREM. 583. The solidity of a cone is equal to the product of its base by one third of its altitude. S S E F In the base of the cone inscribe any regular polygon, ABCDEF, and on this polygon construct a regular pyramid, having the same vertex, A S, with the cone. Then a right pyraImid will be inscribed in the cone; and its solidity will be equal to the product of its base by one third of its altitude (Prop. XX. Bk. VIII.). B C Conceive, now, the number of sides of the polygon to be indefinitely increased, and its perimeter will become equal to the circumference of the cone, and the pyramid will exactly coincide with the cone. But the solidity of every right pyramid is equal to the product of the base by one H D third of its altitude; hence, the solidity of a cone is equal to the product of its base by one third of its altitude. 584. Cor. 1. A cone is the third of a cylinder having the same base and the same altitude; hence it follows,1. That cones of equal altitudes are to each other as their bases; 2. That cones of equal bases are to each other as their altitudes; 3. That similar cones are as the cubes of the diameters of their bases, or as the cubes of their altitudes. X R2 X H. 585. Cor. 2. If the altitude of a cone be represented by HI, and the radius of its base by R, the solidity of the cone will be represented by R2 XXH, or PROPOSITION VI. - THEOREM. 586. The solidity of the frustum of a cone is equivalent to the sum of three cones, having for their common altitude the altitude of the frustum, and whose bases are the two bases of the frustum, and a mean proportional between them. G M Let ABCDEF-M be the frustum of a cone; then will its solidity be equivalent to the sum of three cones having the same altitude as the frustum, and whose bases are the two bases of the frustum, and a mean proportional between them. B C For, inscribe in the two bases of the frustum two regular polygons having the same number of sides, and having their sides parallel, each to each. Let the vertices of the corresponding angles be joined by the straight lines BH, CI, &c., and there is inscribed in the Η L E L K D |