3. If, in the circle ABCDE, which forms the base of a cylinder, a polygon ABCDE be inscribed, and a right prism, constructed on this base, and equal in altitude to the cylinder; then, the prism is said to be inscribed in the cylinder, and the cylinder to be circumscribed about the prism.

The edges AF, BG, CH, &c., of the prism, being perpendicular to the plane of the base, are contained in the convex surface of the cylinder; hence, the

prism and the cylinder touch one another along these edges.

4. In like manner, if ABCD is a polygon, circumscribed about the base of a cylinder, a right prism constructed on this base, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, and the cylinder to be inscribed in the prism.

F

A

P

Μ

B

Z

Y

H

Let M, N, &c., be the points of contact in the sides AB, BC, &c.; and through the points M, N, &c., let MX, NY, &c., be drawn perpendicular to the plane of the base: these perpendiculars will then lie both in the surface of the cylinder, and in that of the circumscribed prism; hence, they will be their lines of contact.

5. A CONE is a solid which may be generated by the revolution of a right-angled triangle SAB, turning about the immovable side SA.

In this movement, the side AB describes a circle BDCE, called the base of the cone; the hypothenuse SB describes the convex surface of the cone.

The point S is called the vertex of the cone, SA the axis, or the altitude, and SB the slant height.

Every section HKFI, made by a

F

АН

G

K

E

B

A

The frustum may be generated by the revolution of the trapezoid ABHG, turning about the side AG. The immovable line AG is called the axis, or altitude of the frustum, the circles BDC, HFK, are its bases, and BH its slant height.

7. SIMILAR CONES are those whose axes are proportional to the radii of their bases: hence, they are generated by similar right-angled triangles (B. IV., D. 1).

8. If, in the circle ABCDE, which forms the base of a cone, any polygon ABCDE is inscribed, and from the vertices A, B, C, D, E, lines are drawn to S, the vertex of the cone, these lines may be regarded as the edges of a pyramid whose base is the polygon ABCDE and vertex S. The edges of this pyramid are in the convex surface of the cone, and the pyramid is said to be inscribed in the cone. also said to be circumscribed about the pyramid.

9. The SPHERE is a solid terminated by a curved surface, all the points of which are equally distant from a point within, called the centre.

The sphere may be generated by the revolution of a semicircle DAE, about its diameter DE: for, the surface described in this movement,

F

A

S

E

D

B

The cone is

G

B

E

by the semicircumference DAE, will have all its points equally distant from its centre C.

10. Whilst the semicircle DAE, revolving round its diameter DE, describes the sphere, any circular sector, as DCF, or FCA, describes a solid, called a spherical sector.

11. The radius of a sphere is a straight line drawn from the centre to any point of the surface; the diameter or axis is a line passing through the centre, and terminated, on both sides, by the surface.

All the radii of a sphere are equal; all the diameters. are equal, and each is double the radius.

12. It will be shown (P. 7,) that every section of a sphere, made by a plane, is a circle: this granted, a great circle is a section which passes through the centre; a small circle, is one which does not pass through the centre.

13. A plane is tangent to a sphere, when it has but one point in common with the surface.

14. A zone is the portion of the surface of the sphere included between two parallel circles, which form its bases. If the plane of one of these circles becomes tangent to the sphere, the zone will have only a single base.

15. A spherical segment is the portion of the solid sphere, included between two parallel circles which form its bases. If the plane of one of these circles becomes tangent to the sphere, the segment will have only a single base.

16. The altitude of a zone, or of a segment, is the distance. between the planes of the two parallel circles, which form the bases of the zone or segment.

17. The Cylinder, the Cone, and the Sphere, are the three round bodies treated of in the Elements of Geometry.

PROPOSITION I. THEOREM.

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

Let CA be the radius of the base of a cylinder, and H its altitude; denote the circumference whose radius is CA by circ. CA: then will the convex surface of the cylinder be equal to circ. CAX H.

the polygon, multiplied by the altitude H (B. VII., P. 1). Let now the arcs which are subtended by the sides of the polygon be continually bisected, and the number of sides of the polygon continually doubled: the limit of the perimeter of the polygon is circ. CA (B. 5, P. 12, s. 2), and the limit of the convex surface of the prism is the convex surface of the cylinder. But the convex surface of the prism is always equal to the perimeter of its base multiplied by H; hence, the convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude.

PROPOSITION II. THEOREM.

The solidity of a cylinder is equal to the product of its base by its altitude.

Let CA be the radius of the base of the cylinder, and H the altitude. Let the circle whose radius is CA be denoted by area CA: then will the solidity of the cylinder be equal to area CAX H.

For, inscribe in the base. of the cylinder any regular polygon BDEFGA, and construct on this polygon a right prism having its altitude equal to H, the altitude of the cylinder: this prism will be

inscribed in the cylinder. The

solidity of this prism will be

equal to the area of the poly

gon multiplied by the altitude H (B. VII., P. 14).

Let now the number of sides of the polygon be continually increased, as before described; the solidity of each new prism will still be equal to its base multiplied by its altitude: the limit of the polygon is the area CA, and the limit of the prisms, the circumscribed cylinder. But the solidity of each new prism is equal to the base multiplied by the altitude: therefore, the solidity of the cylinder is equal to the product of its base by its altitude.

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

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. 13); and the cylinders being similar, the radii of their bases are to each other as their altitudes (D. 2); hence, the bases are as the squares of the altitudes; therefore, the bases multiplied by the altitudes, or the cylinders themselves, are as the cubes of the altitudes.

Scholium. Let R denote the radius of a cylinder's base and H the altitude; then we shall have,

surface of base=">R2,

convex surface=2TM×R ×H,

solidity=XRH.