Thus, if the triangle SAB, right-angled at A, be turned about the side SA, as an axis, it will generate the cone S-CD.BE.

The fixed line SA, is called the axis of the cone; the curved surface generated by the hypothenuse SB, is called the convex surface of the cone; the circle generated by the side AB, is called the base of the cone; and the point S, is called the vertex of the cone; the distance from the vertex to any point in the circumference of the base, is called

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H

the slant height of the cone; and the perpendicular distance from the vertex to the plane of the base, is called the altitude of the cone.

The line SB, which generates the convex surface, is, in any position, called an element of the surface; the elements are all equal, and any one is equal to the slant height; the axis is equal to the altitude.

Any line of the generating triangle SAB, as GH, which is perpendicular to the axis, generates a circle whose plane is perpendicular to the axis: hence, any section of a cone by a plane perpendicular to the axis, is a circle. Any section SBC, made by a plane through the axis, is an isosceles triangle, double the generating triangle.

6. A TRUNCATED CONE is that portion of a cone included between the base and any plane which cuts the

one.

When the cutting plane is parallel to the plane of the base, the truncated cone is called a FRUSTUM OF A CONE, and the intersection of the cutting plane with the cone is called the upper base of the frustum; the base of the cone is called the lower base of the frustum.

If the trapezoid HGAB, right-angled at A and G, be revolved about AG, as an axis, it will generate a frustum of a cone, whose bases are ECDB and FKH, whose altitude is AG, and whose slant height is BH.

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7. SIMILAR CONES are those which may be generated by similar right-angled triangles revolving about homologous sides.

The axes of similar cones are proportional to the radii of their bases (B. IV., D. 1); they are also proportional to any other homologous lines of the cones.

8. A pyramid is said to be inscribed in a cone, when its base is inscribed in the base of the cone, and when its vertex coincides with that of the cone.

The lateral edges of the inscribed pyramid are elements of the surface of the circumscribing cone.

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9. A pyramid is said to be circumscribed about a cone, when its base is circumscribed about the base of the cone, and when its vertex coincides with that of the cone.

In this case, the cone is inscribed in the pyramid.

The lateral faces of the circumscribing pyramid are tangent to the surface of the inscribed cone, along lines which are called elements of contact.

10. A frustum of a pyramid is inscribed in a frustum of a cone, when its bases are inscribed in the bases of the frustum of the cone.

The lateral edges of the inscribed frustum of a pyramid are elements of the surface of the circumscribing frustum of a cone.

11. A frustum of a pyramid is circumscribed about a frustum of a cone, when its bases are circumscribed about those of the frustum of the cone.

Its lateral faces are tangent to the surface of the frustum of the cone, along lines which are called elements of contact.

12. A SPHERE is a volume bounded by a surface, every point of which is equally distant from a point within called the centre. A sphere may be generated by a semicircle revolving about its diameter as an axis.

13. A RADIUS of a sphere is a straight line drawn from the centre to any point of the surface. A DIAMETER is a straight line through the centre, limited by the surface. All the radii of a sphere are equal: the diameters are also equal, and each is double the radius.

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14. A SPHERICAL SECTOR is a volume generated by a sector of the semicircle that generates the sphere. The surface generated by the arc of the circular sector is the base of the sector. The other bounding surfaces are either surfaces of cones or planes. The spherical sector generated by ACB is bounded by the surface generated by the arc AB and the conic surface generated by BC; the sector generated by BCD is bounded by the surface generated by BD and the conic surfaces generated by BC and DC, and so on.

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15. A plane is TANGENT TO A SPHERE when it touches it in a single point.

16. A ZONE is a portion of the surface of a sphere included between two parallel planes. The bounding lines

of the sections are called bases of the zone, and the distance between the planes is called the altitude of the zone. If one of the planes is tangent to the sphere, the zone has but one base.

17. A SPHERICAL SEGMENT is a portion of a sphere included between two parallel planes. The sections made by the planes are called bases of the segment, and the distance between them is called the altitude of the segment.

If one of the planes is tangent to the sphere, the segment has but one base.

The CYLINDER, the CONE, and the SPHERE, are sometimes called THE THREE ROUND BODIES.

PROPOSITION I. THEOREM.

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

Let ABD be the base of a cylinder whose altitude is H: then is its convex surface equal to the circumference of its base multiplied by the altitude.

For, inscribe in the cylinder a prism whose base is a regular polygon. The convex surface of this prism is equal to the perimeter of its base multiplied by its altitude (B. VII., P. I.), whatever may be the number of sides of its base. But, when the number of sides is infinite (B. V., P. X., Sch.), the convex surface of the prism coincides with that of the cylinder, the perimeter

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of the base of the prism coincides with the circumference of the base of the cylinder, and the altitude of the prism is the same as that of the cylinder: hence, the convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude; which was to be proved.

Cor. The convex surfaces of cylinders having equal altitudes are to each other as the circumferences of their bases.

PROPOSITION II. THEOREM.

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

Let ABD be the

base of a cylinder whose altitude is H; then is its volume equal to the product of its base and altitude.

For, inscribe in it a prism whose base is a regular polygon. The volume of this prism is equal to the product of its base and altitude (B. VII., P. XIV.), whatever may be the number of sides of its base. But, when the number of sides is infinite, the prism coincides with the cylinder, the base of the prism with the base of the cylinder, and the altitude of the prism is the same as that

H

of the cylinder: hence, the volume of the cylinder is equal to the product of its base and altitude; which was to be proved.

Cor. 1. Cylinders are to each other as the products of their bases and altitudes; cylinders having equal bases are to each other as their altitudes; cylinders having equal altitudes are to each other as their bases.