5. A spherical triangle is the portion of the surface of a sphere included by the arcs of three great circles.

6. These arcs are called the sides of the triangle, and each is supposed to be less than half of the circumference.

7. The angles of a spherical triangle are the angles contained between the planes in which the sides lie. Or the angle formed by any two arcs of great circles is the angle formed by the planes of the great circles of which the arcs are a part,

8. A spherical polygon is the portion of the surface of a sphere bounded by several arcs of great circles. 9. A plane is said to be a tangent to a sphere when it contains only one point in common with the surface of the sphere.

10. A zone is the portion of the surface and a spherical segment, the portion of the volume of a sphere between two parallel planes, or cut off by one plane.

The circles in which the planes intersect the sphere are called bases of the zone or segment.

11. A lune is the portion of the surface of a sphere comprehended between two great semicircles.

12. A spherical wedge or ungula is the solid bounded by a lune and the planes of its two circles.

cle.

PROP. I.

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

1

Then, since OC is perpendicular to the plane XZ, it will be perpendicular to all straight lines passing through its foot in that plane. (Def. 3, Geometry of Planes.)

Hence the angles OCP, OCP, OCP3.

are right angles

..CP=CP,=CP,

are all points upon by def. 1, OP=OP,

Hence XPZ is a circle whose center is C, and every other section of a sphere made by a plane may, in like manner, be proved to be a circle.

Cor. 1. If the plane pass through the center of the sphere, then OC=0, and the radius of the circle will be equal to the radius of the sphere.

Cor. 2. Hence all great circles are equal to one another, since the radius of each is equal to the radius of the sphere.

Cor. 3. Hence, also, two great circles and their circumferences always bisect each other; for, since both pass through the center, their common intersection passes through the center, and is a diameter of the sphere and of each of the two circles.

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

Cor. 5. We can always draw one, and only one, great circle through any two points on the surface of a sphere; for the two given points and the center of the sphere give three points, which determine the position of a plane.

If, however, the two given points are the extremities of a diameter, then these two points and the center of the sphere are in the same straight line, and an infinite number of great circles may be drawn through the two points. (Prop. 3, Geom. of Planes.)

Distances on the surface of a sphere are measured by the arcs of great circles. The reason for this.is, that the shortest line which can be drawn upon the surface of a sphere, between any two points, is the arc of a great circle joining them, which will be proved hereafter.

PROP. II.

If a diameter be drawn perpendicular to the plane of a great circle, the extremities of the diameter will be the poles of that circle, and of all the small circles whose planes are parallel to it.

Let APB be a great circle of the sphere whose center is O.

Draw ZN, a diameter perpendicular to the plane of the circle APB.

A

P

P

Then Z and N, the extremities of this diameter, are the poles of the great circle APB, and of all the small circles, such as apb, whose planes are parallel to that of APB. Take any points P1, P2,

rence of APB.

N

P

B

in the circumfe

Through each of these points respectively, and the points Z and N, describe great circles, ZP,N, ZP,N. Join OP, OP, .

Then, since ZO is perpendicular to the plane of APB, it is perpendicular to all the straight lines OP1, drawn through its foot in that plane.

OP 2,

...

Hence all the angles ZOP, ZOP2, right angles, and .. the arcs ZP1, ZP2, quadrants.

Thus it appears that the points Z and N are at a quadrant's distance, and.. equally distant from all the points in the circumference of APB, and are .. the poles of that great circle.

Again; since ZO is perpendicular to the plane APB, it is also perpendicular to the parallel plane apb (Geometry of Planes, Prop. 14).

Hence the oblique lines Zp., Zp29

drawn

to p, p, in the circumference of apb, will be equal to each other. (Prop. 7, Geometry of Planes.) .. The chords Zp1, Zp29 arcs Zp1, Zp2,.

be equal.

being equal, the

which they subtend, will also

.. The point Z is the pole of the circle apb; and the point N is also a pole, the arcs Np,, &c., being supplements of the arcs Zp1, &c.

Def. The diameter of the sphere perpendicular to the plane of a circle is called the axis of that circle.

Cor. 1. Every arc P.Z, drawn from a point in the circumference of a great circle to its pole, is a quadrant, and this arc PZ makes a right angle with the arc APB. For, the straight line ZO being perpendicular to the plane APB, every plane which passes through this straight line will be perpendicular to the plane APB (Prop. 18, Geometry of Planes); hence the angle between these planes is a right angle, or, by def. 7, the angle of the arcs AP, and ZP, is a right angle.

Cor. 2. In order to find the pole of a given arc AP of a great circle, take P.Z, perpendicular to AP,,* and equal to a quadrant, the point Z will be a pole of the arc AP1; or, from the points A and P, draw two arcs AZ and P.Z perpendicular to AP1, the point Z in which they meet is a pole of AP1.

Cor. 3. Reciprocally, if the distance of the point Z from each of the points A and P, is equal to a quadrant, then the point Z is the pole of AP, and each of the angles ZAP, ZP,A is a right angle.

For, let O be the center of the sphere; draw the radii OA, OP1, OZ;

Then, since the angles AOZ, P,OZ are right angles, the straight line OZ is perpendicular to the straight lines OA, OP, and is .. perpendicular to their plane; hence, by the above prop., the point Z is the pole of

A perpendicular are to AP1 at P1 is described by means of its pole, which will be in APB, at a quadrant's distance from P1.

AP1, and.. (corol. 1), the angles ZAP, ZP,A are right angles.

Cor. 4. Great circles, such as ZA, ZP, whose planes are at right angles to the plane of another great circle, as APB, are called its secondaries; and it appears from the foregoing corollaries, that,

1. The planes of all secondaries pass through the axis, and their circumferences through the poles of their primary; and that the poles of any great circle may always be determined by the intersection of any two of its secondaries.

2. The arcs of all secondaries intercepted between the primary and its poles are = 90°.

3. A secondary bisects all circles parallel to its primary, the axis of the latter passing through all their

centers.

Cor. 5. Let the radius of the sphere = R, radius of small circle parallel to it = r. Distance of two cir

cles, or Oo= d.

Join Op, and let the arc Pip, in degrees and fractions of a degree, be sin. 4 and r= cos.

p.

expressed by . Then will d to the radius R, and we have

in which cos. o and sin. o express these trigonometrical lines to radius 1, the usual radius of the tables.

Cor. 6. Two secondaries intercept similar arcs of circles parallel to their primary, and these arcs are to each other as the cosines of the arcs of the secondaries between the parallels and the primary.

For the arcs of the parallels subtend at their respective centers, angles equal to the inclinations of the planes of the secondaries, and these arcs will, therefore (def. 55), be similar. Again: let pp. in the diagram be one of these arcs, and imagine another,

*The two following corollaries require a knowledge of the first principles of Trigonometry.