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BOOK VIII.

THE CYLINDER, THE CONE, AND THE SPHERE.

DEFINITIONS.

1. A cylinder is the solid produced by the revolution of a rectangle ABCD, conceived to turn about the immovable side AB.

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

The immovable line AB is called the axis of the cylinder.

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Every section KLM, made in the cylinder, 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 circular plane, equal to the base, and this plane is nothing else than the section made perpendicular to the axis at the point I.

Every section PQGH, made through the axis, is a rectangle double of the generating rectangle ABCD.

2. A cone is the solid produced by the revolution of a right-angled triangle SAB, conceived to turn about the immovable side SA.

In this movement, the side AB describes a circular plane BDCE, named the base of the cone; the hypothenuse C SB describes its convex surface.

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3. If from the cone SCDB, the cone SFKH be cut off by a section parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone.

We may conceive it to be described by the revolution of a trapezium ABHG, whose angles A and G are right-angles, 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 is its side.

4. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases.

5. If in the circle ACD, which forms the base of a cylinder, a polygon ABCDE is inscribed, a right prism constructed on this base ABCDE and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or 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 evidently included in the convex surface of the

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cylinder; hence the prism and the cylinder touch one anoth

er along these edges.

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

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Let M, N, &c., be the points of A contact in the sides AB, BC, &c.;

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and through the points M, N, &c., let MX, NY, &c., be drawn perpendicular

to the plane of the base: those perpendiculars will evidently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of con

tact.

7. If the polygon ABCDEF, be inscribed in the circle ACE which forms the base of a cone, and upon this polygon a pyramid be constructed equal in altitude to the cone, and if S the vertex of the cone and A,B,C, &c., the vertices of the polygon, be joined, then will SA, SB, SC, &c., the edges of the pyramid, be in the convex surface of the cone; and the pyramid is said to be inscribed in the cone, or the cone to be circumscribed about the pyramid.

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8. The sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within called the centre.

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9. The radius of a sphere is a straight line drawn from the centre to any point in the surface; as CA, CF.

The diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface; as AB, DE. All the radii of a sphere are equal; all the diameters are equal, and each is double of the radius.

Note. It will be shown hereafter, that every section of the sphere is a circle. (See Prop. 7. 8.)

10. A great circle is a section which passes through the centre of the sphere; a small circle is one which does not pass through the centre.

11. The circumference of a sphere is the same as a great circle.

12. A plane is a tangent to a sphere, when their surfaces have but one point in common.

13. A zone is the portion of the surface of a sphere, included between two parallel planes which form its bases. One of those planes may be a tangent to the sphere; in which case, the zone has only a single base.

14. A spherical segment is that portion of the solid sphere included between two parallel planes which form its bases. One of these planes may be a tangent to the sphere; in which case, the segment has only a single base.

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

16. Whilst the semicircle DAE, (Def. 8,) revolving round its diameter DE, describes the sphere; any circular sector, as DCF, or FCH, describes a solid, which is named a spherical sector.

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

Let RC be the radius of the given cylinder's base, and H its altitude; also let the circumference whose radius is RC, be represented by circ. RC. We are to show that the convex surface of the cylinder is equal to

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with it. (9. 5. Cor. 1.) And since they are of the same altitu the convex surface of the one must coincide with the con"

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