to the product of the diameter by the circumference of a great circle. 589. Cor. 1. The surface of a sphere is equal to the area of four of its great circles. For the area of a circle is equal to the product of the circumference by half the radius, or one fourth of the diameter (Prop. XV. Bk. VI.). 590. Cor. 2. The surface of a zone or segment is equal to the product of its altitude by the circumference of a great circle. For the surface described by the sides BC, CD of the inscribed polygon is equal to the product of the altitude GM by the circumference of the inscribed circle OI. If, now, the number of the sides of an inscribed polygon be indefinitely increased, its perimeter will equal the circle, and BC, CD will coincide with the arc BCD; consequently, the surface of the zone described by the revolution of BCD is equal to the product of its altitude by the circumference of a great circle. In like manner, the same may be proved true of a segment, or a zone having but one base. 591. Cor. 3. The surfaces of two zones, or segments upon the same sphere, are to each other as their altitudes; and any zone or segment is to the surface of the sphere as the altitude of that zone or segment is to the diameter. 592. Cor. 4. If the radius of a sphere is represented by R, and its diameter by D, its surface will be represented by 4π X R2, or π X D2. 593. Cor. 5. Hence, the surfaces of spheres are to eachother as the squares of their radii or diameters. 594. Cor. 6. If the altitude of a zone or segment is represented by H, the surface of a zone or segment will be represented by 595. The solidity of a sphere is equal to the product of its surface by one third of its radius. For a sphere may be regarded as composed of an indefinite number of pyramids, each having for its base a part of the surface of the sphere, and for its vertex the centre. of the sphere; consequently, all these pyramids have the radius of the sphere as their common altitude. Now, the solidity of every pyramid is equal to the product of its base by one third of its altitude (Prop. XX. Bk. VIII.); hence, the sum of the solidities of these pyramids is equal to the product of the sum of their bases by one third of their common altitude. But the sum of their bases is the surface of the sphere, and their common altitude its radius; consequently, the solidity of the sphere is equal to the product of its surface by one third of its radius. 596. Cor. 1. The solidity of a spherical pyramid or sector is equal to the product of the polygon or zone which forms its base, by one third of the radius. For the polygon or zone forming the base of the spherical pyramid or sector may be regarded as composed of an indefinite number of planes, each serving as a base to a pyramid, having for its vertex the centre of the sphere. 597. Cor. 2. Spherical pyramids, or sectors of the same sphere or of equal spheres, are to each other as their bases. 598. Cor. 3. A spherical pyramid or sector is to the sphere of which it is a part, as its base is to the surface of the sphere. 599. Cor. 4. Hence, spherical sectors upon the same sphere are to each other as the altitudes of the zones forming their bases (Prop. VIII. Cor. 3); and any spherical sector is to the sphere as the altitude of the zone forming its base is to the diameter of the sphere. 600. Cor. 5. If the radius of a sphere is represented by R, its diameter by D, and its surface by S, its solidity will be represented by SXR 4 × R2 × = R= X R3 or л × D3. 601. Cor 6. Hence, the solidities of spheres are to each other as the cubes of their radii. 602. Cor. 7. If the altitude of the zone which forms the base of a sector be represented by H, the solidity of the sector will be represented by π 2 X RX H× R= }π × R2 × H. 603. Scholium. The solidity of the spherical segment less than a hemisphere, and of B one base, formed by the revolution of a portion, A B C, of a semicircle about the radius OA, is equivalent to the solidity of the spherical sector formed by AOB, less the D solidity of the cone formed by OBC. The solidity of the spherical segment greater than a hemisphere, and of one base, formed by the revolution of ADE, is equivalent to the solidity of the spherical sector formed by AOD, plus the solidity of the cone formed by ODE. The solidity of the spherical segment of two bases formed by the revolution of CBDE about the axis AF, is equivalent to the solidity of the segment formed by ADE, less the solidity of the segment formed by ABC. PROPOSITION X.-THEOREM. 604. The surface of a sphere is equivalent to the convex urface of the circumscribed cylinder, and is two thirds of the whole surface of the cylinder; also, the solidity of the sphere is two thirds of that of the circumscribed cylinder. Let ABFI be a great circle of the sphere; DEGH the circumscribed square; then, if the semicircle ABF and the semi-square ADEF be revolved about the diameter AF, the semicircle will D B describe a sphere, and the semi- E A H The convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude (Prop. I.). But the base of the cylinder is equal to the great circle of the sphere, its diameter E G being equal to the diameter BI, and the altitude DE is equal to the diameter AF; hence, the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. This measure is the same as that of the surface of the sphere (Prop. VIII.); hence, the surface of the sphere is equal to the convex surface of the circumscribed cylinder. But the surface of the sphere is equal to four great circles of the sphere (Prop. VIII. Cor. 1); hence, the convex surface of the cylinder is also equal to four great circles; and adding the two bases, each equal to a great circle, the whole surface of the circumscribed cylinder is equal to six great circles of the sphere; hence, the surface of the sphere is or of the whole surface of the circumscribed sphere. In the next place, since the base of the circumscribed cylinder is equal to a great circle of the sphere, and its altitude to the diameter, the solidity of the cylinder is equal to a great circle multiplied by its diameter (Prop. II.). But the solidity of the sphere is equal to its sur face, or four great circles, multiplied by one third of its radius (Prop. IX.), which is the same as one great circle multiplied by of the radius, or by of the diameter; hence, the solidity of the sphere is equal to of that of the circumscribed cylinder. 605. Cor. 1. Hence the sphere is to the circumscribed cylinder as 2 to 3; and their solidities are to each other as their surfaces. 606. Cor. 2. Since a cone is one third of a cylinder of the same base and altitude (Prop. V. Cor. 1), if a cone has the diameter of its base and its altitude each equal to the diameter of a given sphere, the solidities of the cone and sphere are to each other as 1 to 2; and the solidities of the cone, sphere, and circumscribing cylinder are to each other, respectively, as 1, 2, and 3. |