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.

1°. 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 R3 (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

R2 × 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 polycdron.

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 alti. tude 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 II, 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 II, and its slant height by H', we have (P. IV., VI.),

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

S = 4πR2

V = { «R3 = }«D?

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

the area of

II,

and the

V, we

(9.)

• (10.)

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

V = } π (2 R2 × H + R' H' = R"2 × H")

(11.)

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 their point of intersection 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 three or more arcs of 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 two semi-circumferences of great circles.

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

6. A SPHERICAL PYRAMID is a portion of a sphere bounded by a spherical polygon and sectors of circles whose common centre is the centre of the sphere.

The spherical polygon is called the base of the pyramid, and the centre of the sphere is called the vertex of the pyramid.

7. A POLE OF A CIRCLE is a point on the surface of the sphere, equally distant from all the points of the cir cumference of the circle.

8. A DIAGONAL of a spherical polygon is an arc of a great circle joining the vertices of any two angles which are not consecutive.

PROPOSITION I. THEOREM.

Any side of a spherical triangle is less than the sum of

Let ABC be a

whose centre is 0 than the sum of the

the other two.

spherical triangle situated on a sphere then will any side, as AB, be less sides AC and BC.

For, draw the radii OA, OB, and OC: these radii form the edges of a triedral angle whose vertex is 0, and the plane angles included between them are measured by the arcs AB, AC, and BC (B. III., P. XVII., Sch.). But any plane angle, as AOB, is less than the sum of the plane angles AOC

B

and BOC (B. VI., P. XIX.): hence,

the arc AB is less than the sum of the arcs AC anl BC; which was to be proved.