BOOK IX.

THE SPHERE, AND ITS PROPERTIES.

DEFINITIONS.

497. A SPHERE is a solid. or volume, bounded by a curved surface, all points of which are equally distant from a point within, called the centre.

The sphere may be con

ceived to be formed by the revolution of a semicircle, DAE, about its diameter, DE, which remains fixed.

498. The RADIUS of a sphere is a straight line drawn from the centre to any point in surface, as the line CB.

D

C

B

E

The DIAMETER, or AXIS, of a sphere is a line passing through the centre, and terminated both ways by the surface, as the line D E.

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

499. A CIRCLE, it will be shown, is a section of a sphere. A GREAT CIRCLE of the sphere is a section made by a plane passing through the centre, and having the centre of the sphere for its centre; as the section AB, whose centre is C.

500. A SMALL CIRCLE of the sphere is any section made by a plane not passing through the centre.

501. The POLE of a circle of the sphere is a point in the

surface equally distant from every point in the circumference of the circle.

502. It will be shown (Prop. V.) that every circle, great or small, has two poles.

503. A PLANE is TANGENT to a sphere, when it meets the sphere in but one point, however far it may be produced.

A

504. A SPHERICAL ANGLE is the difference in the direction of two arcs of great circles of the sphere; as AED, formed by the arcs EA, DE.

It is the same as the angle. resulting from passing two planes through those arcs; as the angle formed on the edge. EF, by the planes EAF, EDF.

505. A SPHERICAL TRIANGLE is a portion of the surface of a sphere bounded by three arcs of great circles, each arc being less than a semi-circumference; as A ED.

508. A LUNE is a portion of the surface of a sphere comprehended between semi-circumferences of two great circles; as AIGBDF.

These arcs are named the sides of the triangle; and the angles which their planes form with each other are the angles of the triangle.

509. A SPHERICAL WEDGE, or UNGULA, is that portion of a sphere comprehended between

506. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a plane triangle.

E

507. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by several arcs of great circles.

A

D

511. A SPHERICAL SEGMENT is a portion of the sphere cut off by a plane, or comprehended between two parallel planes.

512. The ALTITUDE of a ZONE or of a SPHERICAL SEGMENT is the perpendicular distance between the two parallel planes which comprehend the zone or segment.

In case the zone or segment is a portion of the sphere cut off, one of the planes is a tangent to the sphere.

PROPOSITION I. - THEOREM.

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513. A SPHERICAL SECTOR is a solid described by the revolution of a circular sector, in the same manner as the semicircle of which it is a part, by revolving round its diameter, describes a sphere.

514. A SPHERICAL PYRAMID is a portion of the sphere comprehended between the planes of a polyedral angle whose vertex is the centre.

The base of the pyramid is the spherical polygon intercepted by the same planes.

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

Let ABE be a section made by a plane in the sphere whose centre is C. From the centre, C, draw CD perpendicular to the plane ABE; and draw the lines CA, CB, CE, to different points of the curve ABE, which bounds the section.

The oblique lines CA, CB, CE are equal, being radii of the sphere; therefore they are equally distant from the perpendicular, CD (Prop. V. Cor., Bk. VII.). Hence, the lines DA, DB, DE, and, in like manner, all the lines drawn from

D to the boundary of the section, are equal; and therefore the section ABE is a circle whose centre is D.

A

D

B

516. Cor. 1. If the section passes through the centre of the sphere, its radius will be the radius of the sphere; hence all great circles are equal.

517. Cor. 2. Two great circles always bisect each other. For, since the two circles have the same centre, their common intersection, passing through the centre, must be a common diameter bisecting both circles.

518. Cor. 3. Every great circle divides the sphere and its surface into two equal parts. For if the two hemispheres were separated, and afterwards placed on the common base, with their convexities turned the same way, the two surfaces would exactly coincide.

519. Cor. 4. The centre of a small circle, and that of the sphere, are in a straight line perpendicular to the plane of the small circle.

520. Cor. 5. Small circles are less according to their distance from the centre; for, the greater the distance CD, the smaller the chord A B, the diameter of the small circle A B E.

521. Cor. 6. The arc of a great circle may be made to pass through any two points on the surface of a sphere; for the two given points and the centre of the sphere determine the position of a plane. If, however, the two given points be the extremities of a diameter, these two points.

and the centre would be in a straight line, and any number of great circles may be made to pass through the two given points.

PROPOSITION II. — THEOREM.

522. Any one side of a spherical triangle is less than the sum of the other two.

Let ABC be any spherical triangle; then any side, as A B, is less than the sum of the other two sides, AC, BC.

B

ό

For, draw the radii OA, OB, OC, and the plane angles AO B, AOC, COB will form a triedral angle, O. The angles AOB, AOC, COB will be measured by AB, AC, BC, the side of the spherical triangle. But each of the three plane angles forming a triedral angle is less than the sum of the other two (Prop. XIX. Bk. VII.). Hence, any side of a spherical triangle is less than the sum of the other two.

PROPOSITION III.-THEOREM.

523. The shortest path from one point to another, on the surface of a sphere, is the arc of the great circle which joins the two giren points.

Let ABD be the arc of the great circle which joins the points A and D; then the line ABD is the shortest path from A to D on the surface of the sphere.

For, if possible, let the shortest

path on the surface from A to D pass through the point C, out of the are of the great eircle A B D. Draw A C, DC, ares of great circles, and take DB equal to DC. Then in the spherical triangle ABDC the side ABD is less than the sum of the sides AC, DC (Prop. II.); and

B

D