Of the Cone.

20. A cone is a solid, described by the revolution of a right angled triangle, ABC, about one of its sides, CB.

The circle described by the revolving side, AB, is called the base of the cone.

The hypothenuse, AC, is called the slant height of the cone, and the surface described by it, is called the convex surface of the cone.

A

B

The side of the triangle, CB, which remains fixed, is called the axis, or altitude of the cone, and the point C, the vertex of the cone.

21. If a cone be cut by a plane parallel to the base, the section will be a circle. For, while in the revolution of the right angled triangle SAC, the line

CA describes the base of the cone, its parallel FG will describe a circle. FKHI, parallel to the base. If from the cone S-CDB, the cone S-FKH be taken away, the remaining part is called the frustum of the cone.

22. If a polygon be inscribed in the base of a cone, and straigh lines be drawn from its vertices to the vertex of the cone, the pyramid thus formed is said to be inscribed in the cone. Thus, the pyramid S-ABCD is inscribed in the cone.

F

H

Ꮐ

E

B

Of the Sphere.

23. Two cylinders are similar, when the diameters of their bases are proportional to their altitudes.

24. Two cones are also similar, when the diameters of heir bases are proportional to their altitudes.

25. A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a certain point within called the centre.

Of the Sphere.

29. All diameters of a sphere are equal to each other; and each is double a radius.

30. The axis of a sphere is any line about which it re volves; and the points at which the axis meets the surfa re called the poles.

Of the Pism.

34. A spherical segment is a portion of the solid sphere in

cluded between two parallel planes. are its bases. If one of the planes is

the segment will have but one base.

These parallel planes tangent to the sphere,

35. The altitude of a zone or segment, is the distance be tween the parallel planes which form its bases.

THEOREM I.

The convex surface of a right prism is equal to the perimeter of its base multiplied by its altitude.

Let ABCDE-K be a right prism: then will its convex surface be equal to

(AB+BC+CD+DE+EA)×AF.

For, the convex surface is equal to the sum of the rectangles AG, BH, CI, DK, and EF, which compose it; and the area of each rectangle is equal to the product of its base

K

F

H

B

E

by its altitude. But the altitude of each rectangle is equal to the altitude of the prism: hence, their areas, that is, the con vex surface of the prism, is equal to

(AB+BC+CD+DE+EA)× AF;

that is, equal to the perimeter of the base of the prism multiplied by its altitude.

THEOREM II.

The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

Of the Prism.

Let DB be a cylinder, and AB the diameter of its base: the convex surface will then be equal to the altitude AD multiplied by the circumference of the base.

D

For, suppose a regular prism to be inscribed within the cylinder. Then, the convex surface of the prism will be equal to the perimeter of the base multiplied by the altitude (Th. i). But the altitude of the prism is the same as that of the cylinder; and if we suppose the sides of the polygon, which forms the base of the prism, to be indefinitely increased, the polygon will become the circle (Bk. IV. Th. xxiii. Sch.), in which case, its perimeter will become the circumference, and the prism will coincide with the cylinder. But its convex surface is still equal to the perimeter of its base multiplied by its altitude: hence, the convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

THEOREM III.

In every prism the sections formed by planes parallel to the base are equal polygons.

Let AG be any prism, and IL a section made by a plane parallel to the base AC: then will the polygon IL be equal to AC.

For, the two planes AC, IL, being parallel, the lines AB, IK, in which they intersect the plane AF, will also be parallel (Bk. V. Th. ix). like reason, BC and KL will be par

For a

F

G

K

D