Now, ABCDEXSO measures the solidity of the pyramid SABCDE, and abcdexSo measures that of the pyramid Sabcde (P. 17); hence, two similar pyramids are to each other as the cubes of their homologous edges.

GENERAL SCHOLIUMS.

1. The chief propositions of this Book relating to the solidity of polyedrons, may be expressed in algebraical terms, and so recapitulated in the briefest manner possible. 2. Let B represent the base of a prism; H its altitude: then,

solidity of prism=BxH.

3. Let B represent the base of a pyramid; H its altitude: then,

solidity of pyramid=Bx}H.

4. Let H represent the altitude of the frustum of a pyramid, having the parallel bases A and B; VAXB is the mean proportional between those bases; then

solidity of frustum=}H(A+B+√A×B.)

5. In fine, let P and p represent the solidities of two similar prisms or pyramids; A and a, two homologous edges:

BOOK VIII.

THE THREE ROUND BODIES.

DEFINITIONS.

1. A CYLINDER is a solid which may be generated by the revolution of a rectangle ABCD, turning about the immovable side AB.

In this movement, the sides AD, BC, continuing always perpendicular to AB, describe the equal circles DHP, CGQ, which are called the bases of the cylinder; the side CD, describing, at the same time, the convex surface.

The immovable line AB is called

the axis of the cylinder.

Р

E

N

M

K

G

Every section MNKL, made in the cylinder, by a plane, at right angles to the axis, is a circle equal to either of the bases. For, whilst the rectangle ABCD turns about AB, the line KI, perpendicular to AB, describes a circle, equal to the base, and this circle is nothing else than the section made by a plane, perpendic ular to the axis at the point I.

Every section QPHG, made by a plane passing through the axis, is a rectangle double the generating rectangle ABCD.

2. SIMILAR CYLINDERS are those whose axes are proportional to the radii of their bases: hence, they are gen erated by similar rectangles (B. IV., D. 1).

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 F 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.

A

P

M

B

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

S

F

H

G

E

A

B

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

S

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

E

B

The cone is

D

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.