R

SIMILAR SOLIDS.

Find the

Ex. 723. The dimensions of a trunk are 4 feet, 3 feet, 2 feet. dimensions of a trunk similar in shape that will hold four times as much.

Ex. 724. By what number must the dimensions of a cylinder be multiplied to obtain a similar cylinder (i) whose surface shall be n times that of the first; (ii) whose volume shall be n times that of the first?

Compare the

Ex. 725. A pyramid is cut by a plane parallel to the base which passes midway between the vertex and the plane of the base. volumes of the entire pyramid and the pyramid cut off.

Ex. 726. The height of a regular hexagonal pyramid is 36 feet, and one side of the base is 6 feet. What are the dimensions of a similar pyramid whose volume is that of the first?

Ex. 727. The length of one of the lateral edges of a pyramid is 4 meters. How far from the vertex will this edge be cut by a plane parallel to the base, which divides the pyramid into two equivalent parts?

Ex. 728. A lateral edge of a pyramid is a. At what distances from the vertex will this edge be cut by two planes parallel to the base that divide the pyramid into three equivalent parts?

Ex. 729. A lateral edge of a pyramid is a. At what distance from the vertex will this edge be cut by a plane parallel to the base that divides the pyramid into two parts which are to each other as 3: 4?

Ex. 730. The volumes of two similar cones are 54 cubic feet and 432 cubic feet. The height of the first is 6 feet; what is the height of the other?

Ex. 731. Two right circular cylinders their heights. Their volumes are as 3: 4.

have their diameters equal to Find the ratio of their heights.

Ex. 732. Find the dimensions of a right circular cylinder 15 as large as a similar cylinder whose height is 20 feet, and diameter 10 feet.

Ex. 733. The height of a cone of revolution is H, and the radius of its base is R. Find the dimensions of a similar cone three times as large. Ex. 734. The height of the frustum of a right cone is the height of the entire cone. Compare the volumes of the frustum and the cone.

Ex. 735. The frustum of a pyramid is 8 feet high, and two homologous edges of its bases are 4 feet and 3 feet, respectively. Compare the volume of the frustum and that of the entire pyramid.

BOOK VIII.

THE SPHERE.

PLANE SECTIONS AND TANGENT PLANES.

737. DEF. A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre.

738. A sphere may be generated by the revolution of a semicircle ACB about its diameter AB as an axis.

739. DEF. A radius of a sphere is a straight line drawn from the centre to the surface.

A diameter of a sphere is a straight line passing through the centre and limited by the surface.

740. All the radii of a sphere are equal, and all the diameters of a sphere are equal.

741. DEF. A line or plane is tangent to a sphere when it has one, and only one, point in common with the surface of the sphere.

742. DEF. Two spheres are tangent to each other when their surfaces have one, and only one, point in common.

PROPOSITION I. THEOREM.

743. Every section of a sphere made by a plane is a circle.

Let O be the centre of the sphere, and ABD any section made by a plane.

To prove that the section ABD is a circle.

Proof. Draw the radii OA, OB, to any two points A, B, in the boundary of the section, and draw OC 1 to the section.

The rt. AOAC and OBC are equal.

For OC is common, and OA: = ОВ.

§ 151

§ 740

§ 128

But A and B are any two points in the boundary of the section; hence, all points in the boundary are equally distant from C, and the section ABD is a circle.

§ 216 Q. E. D.

744. COR. 1. The line joining the centre of a sphere to the centre of a circle of the sphere is perpendicular to the plane of the circle.

745. COR. 2. Circles of a sphere made by planes equally distant from the centre are equal.

For ACAO- OC2; and 40 and OC are the same for all equally distant circles; therefore, AC is the same.

746. COR. 3. Of two circles made by planes unequally distant from the centre, the nearer is the greater.

747. DEF. A great circle of a sphere is a section made by a plane which passes through the centre of the sphere.

748. DEF. A small circle of a sphere is a section made by a plane which does not pass through the centre of the sphere.

749. DEF. The axis of a circle of a sphere is the diameter of the sphere which is perpendicular to the plane of the circle. The ends of the axis are called the poles of the circle.

750. COR. 1. Parallel circles have the same axis and the same poles.

751. COR. 2. All great circles of a sphere are equal.

752. COR. 3. Every great circle bisects the sphere.

For the two parts into which the sphere is divided can be so placed that they will coincide; otherwise there would be points on the surface unequally distant from the centre.

753. COR. 4. Two great circles bisect each other.

For the intersection of their planes passes through the centre, and is, therefore, a diameter of each circle.

754. COR. 5. If the planes of two great circles are perpendicular, each circle passes through the poles of the other.

755. COR. 6. Through two given points on the surface of a sphere an arc of a great circle may always be drawn.

For the two given points together with the centre of the sphere determine the plane of a great circle which passes through the two given points.

§ 496 If, however, the two given points are the ends of a diameter, the position of the circle is not determined.

§ 494 756. COR. 7. Through three given points on the surface of a sphere one circle may be drawn, and only one. § 496

757. DEF. The distance between two points on the surface of a sphere is the arc of the great circle that joins them.

PROPOSITION II. THEOREM.

758. The distances of all points in the circumference of a circle of a sphere from its poles are equal.

Let P, P' be the poles of the circle ABC, and A, B, C, any points

in its circumference.

To prove that the great circle arcs PA, PB, PC are equal. Proof. The straight lines PA, PB, PC are equal.

Therefore, the arcs PA, PB, PC are equal.

§ 514

§ 241

In like manner, the great circle arcs P'A, P'B, P'C' may be proved equal.

Q. E. D.

759. DEF. The distance on the surface of the sphere from the nearer pole of a small circle to any point in the circumference of the circle is called the polar distance of the circle.

760. DEF. The distance on the surface of the sphere of a great circle from either of its poles is called the polar distance of the circle.

761. COR. The polar distance of a great circle is a quadrant; that is, one fourth the circumference of a great circle.