height 11 feet, length at top 1020 yards, at base 960 yards: find its volume.

69. The altitude of a prism is 9 feet and the perimeter of the base 6 feet: find the altitude and perimeter of the base of a similar prism one-third as great.

70. A pyramid is cut by a plane parallel to the base and midway between the vertex and base: find the ratio of the volume cut off to the whole volume.

71. The length of one of the lateral edges of a pyramid is 16 inches. How far from the vertex will this edge be cut by a plane parallel to the base, which divides the pyramid into two equivalant parts?

72. The height of the frustum of a pyramid is 16 inches, and two homologous edges of its bases are 8 and 6 inches: find the ratio of the volume of the frustum to that of the entire pyramid.

73. Show that the volume of a regular tetraedron is equal to the cube of its edge multiplied by.

74. Show that the volume of a regular octaedron is equal to the cube of its edge multiplied by † √2.

75. Find the volume of a regular icosaedron whose edges are each 20 feet.

76. Verify Euler's Theorem in the case of all the regular polyedrons.

77. Find the number of edges in a pyramid, and in a prism, on a polygon of n sides as base.

PROBLEMS.

78. Given three indefinite straight lines in space which do not intersect, to construct a parallelopiped which shall have three of its edges on these lines.

79. To cut a solid tetraedral angle by a plane, so that the section shall be a parallelogram.

BOOK VIII.*

THE SPHERE.+

CIRCLES OF THE SPHERE AND TANGENT PLANES.

DEFINITIONS.

657. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within it, called the centre.

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

658. A radius of a sphere is a A

straight line drawn from the centre to

any point of the surface.

A diameter of a sphere is a straight

B

line passing through its centre, and terminating at both ends in the surface.

All the radii of a sphere are equal; and all the diameters are equal, since each is the sum of two radii.

659. A line or plane is tangent to a sphere when it has but one point in common with the surface of the sphere. 660. Two spheres are tangent to each other when their surfaces have but one point in common.

661. The common point is called the point of contact, or point of tangency.

662. Two spheres are concentric when they have the same centre.

663. Two spheres are equal when they have equal radii.

* This book treats of the properties and relations of those parts of the surface of a sphere which are bounded by arcs of great circles.

+ In teaching this subject, as well as in teaching Spherical Trigonometry, the class-room should be furnished with a spherical black-board, on which the student should draw the diagrams of spherical surfaces.

cle.

Proposition 1. Theorem.

664. Every section of a sphere made by a plane is a cir

Hyp. Let ACB be a plane section

of the sphere whose centre is 0.
To prove that ACB is a O.
Proof. Draw 00' to the plane
ACB meeting it in O', and draw OA,
OC to any two pts. in the section.

Since OA and OC are radii of the

P

B

sphere,

P'

And since OA and OC are equal oblique lines from 0 to the plane ACB,

.. O'A O'C.

But A and C are any two pts. in the section ACB.

(498)

Q.E.D.

.. the curve ACB is a O whose centre is O’. 665. DEF. The circular section of a sphere by a plane is called a circle of the sphere.

If the plane passes through the centre of the sphere, the section is a circle whose radius is equal to the radius of the sphere. This circle is called a great circle of the sphere.

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

666. DEF. The two points in which a diameter of the sphere, perpendicular to the plane of a circle, meets the surface of the sphere, are called the poles of the circle, and this diameter is called the axis of the circle; thus, P, P′ are the poles of the circle ACB, and PP' is its axis.*

*P' is said to be antipodal to P.

667. COR. 1. The line through the centre of a circle of the sphere, perpendicular to its plane, passes through the centre of the sphere. (664) Therefore, the axis of a circle passes through its centre (666), and all parallel circles have the same axis and the same poles.

668. Cor. 2. All great circles of a sphere are equal. (665) 669. Cor. 3. All small circles at equal distances from the centre of the sphere are equal; and of two circles unequally distant from the centre, the nearer is the larger, and conversely.

670. COR. 4. Every great circle bisects the sphere and its surface.

For, if the two parts are separated, and then placed on the common base, with their convexities turned the same way, their surfaces will coincide, since all the points of either are equally distant from the centre.

(657)

671. COR. 5. Any two great circles bisect each other. For, the common intersection of their planes passes through the centre of the sphere, and is a diameter of each circle.

672. Cor. 6. An arc of a great circle may be drawn through any two given points on the surface of a sphere. For, the two given pts. with the centre of the sphere determine the plane of a great O passing through the two given pts.

(484)

If the two pts. are the extremities of a diameter, the position of the circle is not determined: for the two given pts. and the centre being in the same st. line, an indefinite number of planes can be drawn through them.

(482)

673. COR. 7. An arc of a circle may be drawn through any three given points on the surface of a sphere.

For, the three pts. determine a plane.

(484)

NOTE. By the distance between two points on a sphere is meant the arc of a great circle joining them.

Proposition 2. Theorem.

674. All points in the circumference of a circle of a sphere are equally distant from each of its poles. Hyp. Let P, P' be the poles of the O ACB, where A, C, B are any pts. on its Oce.

To prove arcs PA, PC, PB equal. Proof. Since PP' is a line through the centre of the ACB to its plane,

E

P

B

H

P/

.. the st. lines PA, PC, PB are equal.

(496)

(198)

Q.E.D.

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

Similarly, the arcs P'A, P'C, P'B are equal.

675. DEF. The distance from any point in the circumference of a circle to its nearest pole is called the polar distance of the circle.

676. COR. 1. The polar distance of a great circle is a quadrant. Thus, PE, PG are quadrants, for they are the measures of the rt. s POE, POG, whose vertices are at the centre of the sphere.

(237)

677. COR. 2. If a point P on the surface of a sphere is at a quadrant's distance from the two points E, G, of an are of a great circle, it is the pole of that arc.

For, the s POE, POG are rt. /s; .. the radius OP is I to the plane of the arc EG (500); and .. P is the pole of the arc EG.

(666)

678. SCH. In Spherical Geometry, the term quadrant usually means a quadrant of a great circle.