any point D in the circumference of the circle; join CD, OD, PD. Then PD=√(PC2 + CD); and PC and CD are constant, therefore PD is constant. Suppose a great circle to pass through the points P and D; then the chord PD is constant, and therefore the arc of a great circle intercepted between P and D is constant for all positions of D on the circle AB.

Thus the distance of a pole of a circle from every point of the circumference of the circle is constant, whether that distance be measured by the straight line joining the points, or by the arc of a great circle intercepted between the points.

7. The angle subtended at the centre of a sphere by the arc of a great circle which joins the poles of two great circles is equal to the inclination of the planes of the great circles.

Let O be the centre of the sphere, CD, CE the great circles intersecting at C, A and B the poles of CD and CE respectively.

Draw a great circle through A and B, meeting CD and CE at M and N respectively. Then AO is perpendicular to OC, which is a line in the plane OCD; and BO is perpendicular to OC, which is a line in the plane OCE; therefore OC is perpendicular to the plane AOB (Euclid, xI. 4); and therefore OC is perpendicular to the lines OM and ON, which are in the plane AOB. Hence MON is the angle of inclination of the planes OCD and OCE. And the angle AOBAOM-BOM = BON – BOM=MON.

circles is meant the angle Thus, in the figure of the

8. By the angle between two great of inclination of the planes of the circles. preceding Article, the angle between the great circles CD and CE is the angle MON. In the figure to Art. 6, since PO is perpendicular to the plane ACB, every plane which contains PO is at right angles to the plane ACB. Hence the angle between any circle and a great circle which passes through its poles is a right angle.

9. Two great circles bisect each other.

For since the plane of each great circle passes through the centre of the sphere, the line of intersection of these planes is a diameter of the sphere, and therefore also a diameter of each great circle; therefore the great circles are bisected at the points where they meet.

10.

The arc of a great circle which is drawn from a pole of a great circle to any point in its circumference is a quadrant, and is at right angles to the circumference.

Let P be a pole of the great circle ABC; the arc PA is a quadrant and is at right angles to ABC.

For let O be the centre of the sphere, and draw PO. Then PO is at right angles to the plane ABC, because P is the pole of

ABC, therefore POA is a right angle, and the arc PA is a quadrant. And because PO is at right angles to the plane ABC, the angle between the planes POA and ABC is a right angle; therefore the arc PA is at right angles to AC.

11. If the arcs of great circles joining a point P on the surface of a sphere with two other points A and C on the surface of the sphere which are not at opposite extremities of a diameter be each of them equal to a quadrant, P is a pole of the great circle through A and C. (See the figure of the preceding Article.)

For suppose PA and PC to be quadrants, and O the centre of the sphere; then since PA and PC are quadrants, the angles POC and POA are right angles. Hence PO is at right angles to the plane AOC, and P is a pole of the great circle AC.

12. Great circles which pass through the poles of a great circle are called secondaries to that circle. Thus, in the figure of Art. 7 the point C is a pole of ABMN, and therefore CM and CN are parts of secondaries to ABMN. And the angle between CM and CN is measured by MN; that is, the angle between any two great circles is measured by the arc they intercept on the great circle to which they are secondaries.

13. If from a point on the surface of a sphere there can be drawn two arcs of great circles, not parts of the same great circle, which are at right angles to a given circle, that point is a pole of the circle.

For, since the two arcs are at right angles to the given circle, the planes of these arcs are at right angles to the plane of the given circle, and therefore the line in which they intersect is perpendicular to the plane of the given circle, and is therefore the axis of the given circle; hence the point from which the arcs are drawn is a pole of the circle.

14. To compare the arc of a small circle subtending any angle at the centre of the circle with the arc of a great circle subtending the same angle at its centre.

B

b

P

Let ab be the arc of a small circle, C the centre of the circle, P the pole of the circle, O the centre of the sphere. Through P draw the great circles PaA and PbB, meeting the great circle of which P is a pole, at A and B respectively; draw Ca, Cb, OA, OB. Then Ca, Cb, OA, OB are all perpendicular to OP, because the planes ab and AOB are perpendicular to OP; therefore Ca is parallel to OA, and Cb is parallel to OB. Therefore the angle acb= the angle AOB (Euclid, XI. 10). Hence,

arc ab

arc AB

radius Ca ̄ radius 04' (Plane Trigonometry, Art. 18);

15. Spherical Trigonometry investigates the relations which subsist between the angles of the plane faces which form a solid angle and the angles at which the plane faces are inclined to each other.

16. Suppose that the angular point of a solid angle is made the centre of a sphere; then the planes which form the solid angle

will cut the sphere in arcs of great circles. Thus a figure will be formed on the surface of the sphere which is called a spherical triangle if it is bounded by three arcs of great circles; this will be the case when the solid angle is formed by the meeting of three plane angles. If the solid angle be formed by the meeting of more than three plane angles, the corresponding figure on the surface of the sphere is bounded by more than three arcs of great circles, and is called a spherical polygon.

17. The three arcs of great circles which form a spherical triangle are called the sides of the spherical triangle; the angles formed by the arcs at the points where they meet are called the angles of the spherical triangle. (See Art. 8.)

18. Thus, let O be the centre of a sphere, and suppose a solid angle formed at O by the meeting of three plane angles. Let

A

B

AB, BC, CA be the arcs of great circles in which the planes cut the sphere; then ABC is a spherical triangle, and the arcs AB, BC, CA are its sides. Suppose Ab the tangent at A to the arc AB, and Ac the tangent at A to the arc AC, the tangents being drawn from A towards B and C respectively; then the angle bAc is one of the angles of the spherical triangle. Similarly angles formed in like manner at B and C are the other angles of the spherical triangle.

19. The principal part of a treatise on spherical trigonometry consists of theorems relating to spherical triangles; it is therefore