III. Let B', B" be the respective bases of a circumscribed and corresponding inscribed prism, V, V" their respective volumes, and H their common altitude.

Now by increasing indefinitely the number of lateral faces of the prisms, and consequently the number of sides of their bases, the difference B'-B" can be made as small as we please.

$490

Hence (B'-B') x H can be made as small as we please.

$187

Hence its equal V-V" can be made as small as we please.

But the volume of the cylinder is always intermediate be tween V' and V". Ax. IC

Therefore the difference between the volume of the cylinder and either Vor V" can be made as small as we please. But V and V" can never equal the volume of the cylinder. Ax. IC

Therefore the volume of the cylinder is the common limit of V and V".

8 185

Q. E. D

934. The lateral area of a cylinder is equal to the product of the perimeter of a right section and an element.

GIVEN the cylinder AD', of which P is the perimeter of the right section FGHIJ, E an element, and S the lateral area.

Inscribe in the cylinder a prism. Let P' be the perimeter of its right section and S' its lateral area.

Its lateral edge is equal to E.

Hence

S' P'x E.

8 545

$649

Now let the number of lateral faces of the prism be in

935. Def.-The altitude of a cylinder is the perpendicular distance between its bases.

936. COR. I. The lateral area of a right cylinder is equal to the product of the perimeter of its base by its altitude.

937. COR. II. Let H denote the altitude, R the radius, S the lateral area, and T the total area of a cylinder of revolution.

and

S=2πRH,

T=2πRH+2π R = 2πR(H+R).

938. Def. Similar cylinders of revolution are cylinders formed by the revolution of similar rectangles about homologous sides.

939. COR. III. The lateral areas, or the total areas, of two similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of their radii.

940. The volume of a cylinder is equal to the product of its base and altitude.

GIVEN-a cylinder, of which B is the base, H the altitude, and V the volume.

Circumscribe about the cylinder a prism. Denote its base

by B' and its volume by V'.

Its altitude is H.

Hence

V'B' x H.

$545

8676

Now let the number of lateral faces of the prism be indefinitely increased.

941. COR. I. Let H be the altitude, R the radius, and V the volume of a circular cylinder.

942. COR. II. The volumes of two similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii.

943. Def.-A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is in the plane of the base of the cone.

944. Def.-A pyramid is circumscribed about a cone when its lateral faces are tangent to the cone and its base is in the plane of the base of the cone.