generated by the arc radius (P. XIII., C.). a circular sector, the volume which it generates is a spherical sector, and the surface generated by the arc is a zone : hence, the volume of a spherical sector is equal to the zone. which forms its base multiplied by one-third of the radius

BC multiplied by one-third of the But this portion of the semicircle is

Cor. 2. If we denote the volume of a sphere by V, and its radius by R, the area of the surface will be equal to 4 R2 (P. X., C. 1), and the volume of the sphere will be equal to 4 R2 R; consequently, we have,

hence, the volumes of spheres are to each other as the cubes of their radii, or as the cubes of their diameters.

Scholium. If the figure If the figure EBDF, formed by drawing lines from the extremities of the arc BD perpendicular to CA, be revolved about CA, as an axis, it will generate a segment of a sphere whose volume may be found by adding to the spherical sector generated

D

A

B

M

by CDB, the cone generated by CBE, and subtracting from their sum the cone generated by CDF. If the arc BD is so taken that the points E and F fall on opposite sides of the centre C, the latter cone must be added, instead of subtracted: hence,

segment EBDF=zone BD×3CD+BE2 ×}CE—«DF2 ׇCF.

The surface of a sphere is to the entire surface of the circumscribed cylinder, including its bases, as 2 is to 3: and the volumes are to each other in the same ratio.

Let PMQ be a semicircle, and PADQ a rectangle, whose sides PA and QD are tangent to the semicircle at P and Q, and whose side AD, is tangent to the semicircle at M. If the semicircle and the rectangle be revolved about PQ, as an axis, the former will generate a sphere, and the latter a circumscribed cylinder.

1o. The surface of the sphere is to the entire surface of the cylinder, as 2 is to 3.

For, the surface of the sphere is equal to four great circles (P. X., C. 1), the convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude (P. I.); that is, it is equal to the circumference of a great circle multiplied by its diameter, or to four great circles (B. V., P. XV.); adding to this the

M

A

two bases, each of which is equal to a great circle, we have the entire surface of the cylinder equal to six great circles: hence, the surface of the sphere is to the entire surface of he circumscribed cylinder, as 4 is to 6, or as 2 is to 3; which was to be proved.

2o. The volume of the sphere is to the volume of the cylinder as 2 is to 3.

For, the volume of the sphere is equal to πR3 (P. XIV., C. 2); the volume of the cylinder is equal to its base multiplied by its altitude (P. II.); that is, it is equal to

TR2 × 2R, or to

R3: hence, the volume of the sphere is to that of the cylinder as 4 is to 6, or as 2 is to 3; which was to be proved.

Cor. The surface of a sphere is to the entire surface of a circumscribed cylinder, as the volume of the sphere is to volume of the cylinder.

Scholium. Any polyedron which is circumscribed about a sphere, that is, whose faces are all tangent to the sphere, may be regarded as made up of pyramids, whose bases are the faces of the polyedron, whose common vertex is at the centre of the sphere, and each of whose altitudes is equal to the radius of the sphere. But, the volume of any one of these pyramids is equal to its base multiplied by onethird of its altitude: hence, the volume of a circumscribed polyedron is equal to its surface multiplied by one-third of the radius of the inscribed sphere.

Now, because the volume of the sphere is also equal to its surface multiplied by one-third of its radius, it follows that the volume of a sphere is to the volume of any circumscribed polyedron, as the surface of the sphere is to the surface of the polyedron.

Polyedrons circumscribed about the same, or about equal spheres, are proportional to their surfaces.

If we denote the convex surface of a cylinder by S, its volume by V, the radius of its base by R, and its altitude by H, we have (P. I., II.),

If we denote the convex surface of a

cone by, S, its

volume by V, the radius of its base by R, its altitude by H,

and its slant height by H', we have (P. III., V.),

If we denote the convex surface of a frustum of a cone by S, its volume by V, the radius of its lower base by R, the radius of its upper base by R', its altitude by H, and its slant height by H', we have (P. IV., VI.),

If we denote the surface of a sphere by S, its volume and its diameter by D, we have (P. X., C. 1, XIV., C. 2, XIV., C. 1),

by V, its radius by R,

If we denote the radius of a sphere by R, the area of any zone of the sphere by S, its altitude by H, and the volume of the corresponding spherical sector by V, we shall have (P. X., C. 2),

If we denote the volume of the corresponding spherical segment by V, the radius of its lower base by R', the radius of its upper base by R", the distance of its lower base from the centre by H', and the distance of its upper base from the centre by H", we have (P. XIV., S.),

1/2

V = }≈(2R2 × H+ R''2 × H'' = R22 × H')

1 A SPHERICAL ANGLE is an angle included between the arcs of two great circles of a sphere meeting at a point. The arcs are called sides of the angle, and the point at which they meet is called the vertex of the angle.

The measure of a spherical angle is the same as that of the diedral angle included between the planes of its sides. Spherical angles may be acute, right, or obtuse.

2. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by arcs of three or more great circles. The bounding arcs are called sides of the polygon, and the points in which the sides meet are called vertices of the polygon. Each side is supposed to be less than a semi-circumference.

Spherical polygons are classified in the same manner as plane polygons.

3. A SPHERICAL TRIANGLE is a spherical polygon of three sides.

Spherical triangles are classified in the same manner as plane triangles.

4. A LUNE is a portion of the surface of a sphere bounded by semi-circumferences of two great circles.

5. A SPHERICAL WEDGE is a portion of a sphere bounded by a lune and two semicircles meeting in a diameter of the sphere.