Hence,

A = ↓ S (AB + EF) + } $ (BC+FG) + etc. ... (366); or, AS (AB + EF + BC + FG + CA + GE); or, A = †S [(AB + BC + CA) + (EF + FG + GE)]; or, A = 1S (p + P).

Q. E. D.

148. COROLLARY 1. The lateral surface of the frustum of a right cone is equal to one-half the product of the slant height by the sum of the circumferences of the bases.

For, the frustum of a cone is the limiting case of the inscribed frustum of a pyramid; and accordingly theorem (147) must be true alike for the frustum of the pyramid and for the frustum of the cone.

149. COROLLARY 2. The lateral surface of the frustum of a right cone may be found by multiplying its slant height by the circumference of the right section equidistant from the bases.

For, the circumference of the right section equidistant from the bases equals one-half the sum of the circumferences of the bases (135).

150. SCHOLIUM. Let the lines AB and CD be drawn in the same plane, and let M be the middle point of AB. Let AC, ME, and BD be perpendiculars drawn to CD. A Now, if ABDC be revolved about CD as an axis, the frustum of a right circular cone will be generated, AC, BD, and ME generating respectively an upper base, a lower base, and a mid-B dle right section. If we represent the lateral surface of the frustum by A, then will

M

A = ВA. 2 π. ME... (149).

D

·E

151.

EXERCISES.

THEOREM. The volume of two similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases.

152.

R

THEOREM. The surfaces, lateral or total, of two similar cones of revolution are to each other as the squares of their altitudes, or as the squares of the radii of their bases. Use diagram of (151).

1.53. DEFINITION. Similar pyramids are such as have the same number of faces, each face of the one being similar to a corresponding face of the other, and similarly placed with respect to adjoining faces.

154. THEOREM. With respect to their triedral angles, two similar pyramids are mutually equiangular. Use diagram of (155). Also see (39), (40), and (43).

155. THEOREM. Two similar pyramids are to each other as the cubes of their corresponding edges.

Place the equal triedral angles A and A' in coincidence, and draw from the common vertex A, a perpendicular to the base BCD; thus representing the altitude of the pyramids.

156.

THEOREM. In any truncated triangular prism ABC-F, the sum of the three pyramids ABC-F, ABC-E, and ABC-D is equivalent to the truncated prism.

157. COROLLARY 1 to (156). The volume of any truncated right triangular prism ABC-F is equal to the product of its base ABC by one-third the sum of its lateral edges AD, BE, and CF.

158. COROLLARY 2 to (156). The volume of any truncated triangular prism ABC-F is equal to the product of the right section GHK, by one-third the sum of the lateral edges AD, BE, and CF.

159.

THEOREM. Two triangular pyramids having a triedral angle of the one equal to a triedral angle of the other, are to each other as the products of the edges including the equal triedral angles.

Place the equal triedral angles in coincidence at 0. Draw

CP and C'P' perpendicular to the face OA'B'.

SECTION IV.

THE SPHERE.

DEFINITIONS.

160. A sphere is a solid bounded by a curved surface,

every point of which is equally distant from a fixed point within. The fixed

P

point is the centre. Any line from the centre to the surface is a radius. Any line through the centre limited by the surface is a diameter.

161. A plane or a line is tangent to a sphere when it touches the surface of the sphere in only one point.

C

B

P

From the above definitions the student may readily deduce the following:

162.

163.

164.

Radii of the same or equal spheres are equal.

Diameters of the same or equal spheres are equal.

Two spheres of equal radii may be made to coincide by placing their centres in coincidence.

165. A sphere may be generated by the revolution of a semicircle about the diameter as an axis.

166. Sections of a sphere through the centre are equal circles of the same radius as the sphere.

called great circles.

167. Any great circle bisects the sphere.

168.

Two great circles bisect each other.

Such circles are