Cor. 2. Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases.

For, the bases are as the squares of their radii (B. V., P. XIII.), and the cylinders being similar, these radii are to each other as their altitudes (D. 2): hence, the bases are s the squares of the altitudes; therefore, the bases multiplied by the altitudes, or the cylinders themselves, are cubes of the altitudes.

as the

PROPOSITION III. THEOREM.

The convex surface of a cone is equal to the circumference of its base multiplied by half the slant height.

Let SACD be a cone whose base is ACD, and whose slant height is SA: then will its convex surface be equal to the circumference of its base multiplied by half the slant height.

For, inscribe within it a right pyramid. The convex surface of this pyramid is equal to the perimeter of its base multiplied by half the slant height (B. VII., P. IV.), whatever may be the number of sides of its base. But when the number of sides of the base is infinite, the convex surface coincides with that of the

B

S

cone, the perimeter of the base of the pyramid coincides with the circumference of the base of the cone, and the slant height of the pyramid is equal to the slant height of the cone : hence, the convex surface of the cone is equal to the circumference of its base multiplied by half the slant height; which was to be proved.

PROPOSITION IV. THEOREM.

The convex surface of a frustum of a cone is equal to half the sum of the circumferences of its two bases multiplied by the slant height.

Let BIA-D be a frustum of a cone, BIA and EGD its two bases, and EB its slant height: then is its convex surface equal to half the sum of the circumferences of its two bases multiplied by its slant height.

B

E

di

For, inscribe within it the frustum of a right pyramid. The convex surface of this frustum is equal to half the sum of the perimeters of its bases, multiplied by the slant height (B. VII., P. IV., C.), whatever may be the number of its lateral faces. But when the number of these faces is infinite, the convex surface of the frustum of the pyramid coincides with that of the cone, the perimeters of its bases coincide with the circumferences of the bases of the frustum of the cone, and its slant héight is equal to that of the cone: hence, the convex surface of the frustum of a cone is equal to half the sum of the circumferences of its bases multiplied by the slant height; which was to be proved.

Scholium. From the extremities A and D, and from the middle point 7, of a line AD, let the lines AO, DC, and IK, be drawn perpendicular to the axis OC: then will IK be equal to half the sum of 40 and DC. For, draw Dd and li, perpendicular to AO: then, because Al is equal to ID, we shall have Ai equal to id (B. IV., P. XV.), and consequently to ls; that is, 40 exceeds IK

as much as IK exceeds DC: hence, IK is equal to the half sum of AO and DC.

OC, as an

of a cone

generate a

Now, if the line AD be revolved about axis, it will generate the surface of a frustum whose slant height is AD; the point will ircumference which is equal to half the sum of the circumerences generated by A and D: hence, if a straight line Je revolved about another straight line, it will generate a surface whose measure is equal to the product of the generating line and the circumference generated by its middle point.

This proposition holds true when the line AD meets OC, and also when AD is parallel to OC.

The volume of a cone is equal to its base multiplied by one-third of its altitude.

Let ABDE be the base of a cone whose vertex is S, and whose altitude is So: then will its volume be equal to the base multiplied by one-third of the altitude.

For, inscribe in the cone a right pyramid. The volume of this pyramid is equal to its base multiplied by onethird of its altitude (B. VII., P. XVII.), whatever may be the number of its lateral faces. But, when the number of lateral faces is infinite, the pyramid coincides with the cone, the base of the pyramid coincides with that of the

B

cone, and their altitudes are equal: hence, the volume of a cone is equal to the base multiplied by one-third of the altitude; which was to be proved.

Cor. 1. A cone is equal to one-third of a cylinder having an equal base and an equal altitude.

Cor. 2. Cones are to each other as the products of their bases and altitudes. Cones having equal bases are to each other as their altitudes. Cones having equal altitudes are to each other as their bases.

PROPOSITION VI. THEOREM.

The volume of a frustum of a cone is equal to the sum of the volumes of three cones, having for a common altitude the altitude of the frustum, and for bases the lower base of the frustum, the upper base of the frus tum, and a mean proportionul between the bases.

Let BIA be the lower base of a frustum of a cone, EGD its upper base, and OC its altitude: then will its volume be equal to the sum of three cones whose common altitude is OC, and whose bases are the lower base, the upper base, and a mean proportional between them.

For, inscribe a frustum of a right

of faces is infinite, the frustum of the pyramid coincides with the frustum of the cone, its bases with the bases of the cone, the three pyramids become cones, and their altitudes

are equal to that of the frustum; hence, the volume of the frustum of a cone is equal to the sum of the volumes of three cones whose common altitude is that of the frustum, and whose bases are the lower base of the frustum, the apper base of the frustum, and a mean proportional between them; which was to be proved.

PROPOSITION VII. THEOREM.

Any section of a sphere made by a plane, is a circle.

Let C be the centre of a sphere, CA one of its radii, and AMB any section made by a plane: then will this section be a circle.

For, draw a radius CO perpendicular to the cutting plane, and let it pierce the plane of the section at 0. Draw radii of the sphere to any two points M, M', of the curve which bounds the section, and join these points with 0: then, because the radii см, CM' are equal, the points

M

B

M, M', will be equally distant from 0 (B. VI., P. V., C.); hence, the section is a circle; which was to be proved.

Cor. 1. When the cutting plane passes through the centre of the sphere, the radius of the section is equal to that of the sphere; when the cutting plane does not pass throngh the centre of the sphere, the radius of the section will be less than that of the sphere.

A section whose plane passes through the centre of the sphere, is called a great circle of the sphere. A section whose plane does not pass through the centre of the sphere,