Elements of Geometry and Trigonometry

represented by H, the surface of a zone or segment will

be represented by

2π × R× H, or π Χ Ρ Χ Η.

PROPOSITION IX. THEOREM.

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

+ πΧ

SXR = 4 × R2 × } R= π X R3 or π × D3. π

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

2 X RX H X † R = } π × R2 × H.

603. Scholium. The solidity of the spherical segment less than a hemisphere, and of 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 AO B, less the 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 surface 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 describe a sphere, and the semisquare a cylinder circumscribing the sphere.

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.

BOOK XI.

APPLICATIONS OF GEOMETRY TO THE MENSURATION OF PLANE FIGURES.

DEFINITIONS.

607. MENSURATION OF PLANE FIGURES is the process of determining the areas of plane surfaces.

608. The AREA of a figure, or its quantity of surface, is determined by the number of times the given surface contains some other area, assumed as the unit of measure.

609. The MEASURING UNIT assumed for a given surface is called the superficial unit, and is usually a square, taking its name from the linear unit forming its side; as a square whose side is 1 inch, 1 foot, 1 yard, &c.

Some superficial units, however, have no corresponding linear unit; as the rood, acre, &c.

610. TABLE OF LINEAR MEASURES.