PART II.

SPHERICAL TRIGONOMETRY.

SECTION I.

ON THE SPHERE.

DEF. I. A sphere is a solid bounded by a surface or superficies, of which every point is equally distant from a point within it, called the centre.

DEF. II. The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

1. PROP. Every section of a sphere by a plane is a circle.

Let AB be any section of a sphere made by a plane AB; O the centre of the sphere; from O let fall OC perpendicular to this plane; in the section AB, take any points D, E, F ;

A A

join OD, DC; OE, EC; OF, FC; then, since OC is perpendicular to the plane AB, it is perpendicular to every straight line, which meets it in that plane, (Euc. XI. Def. III.);

B

therefore, the angles OCD, OCE, OCF, are right angles; also, (Def. I.) OD = OE = OF; hence,

or the points D, E, F, lie in the circumference of a circle, of which the centre is C, and radius

= CD.

DEF. III. A great circle is one whose plane passes through the centre of the sphere.

2. Hence, a radius of a great circle is a radius to the sphere.

DEF. IV. A small circle is one whose plane does not pass through the centre.

3. In the above proposition, if OC = p, and radius of the spherer, the radius of the small circle = (x2 — p2)3. Two circles are said to be parallel, when their planes are parallel; or when the line drawn from the centre of the sphere perpendicular to the plane of one of them, is also perpendicular to the plane of the other. (Euc. XI. 14.)

4. PROP. A great circle may be drawn through any two points on the surface of a sphere, but not generally through more than two.

For the plane of a great circle must also pass through the centre of the sphere; and a plane may be made to pass through any three points, but not generally more than three. (Euc. XI. 2.) Since a small circle does not pass through the centre of the sphere, it may be drawn through any three given points (not in the plane of a great circle) on the spherical superficies.

5. PROP. Two great circles bisect each other.

For the intersection of their planes, being a straight line passing through the centre, is a diameter of the sphere, and is therefore a diameter of both circles; which are therefore bisected. (Euc, I. Def. XVIII.)

6. PROP. The inclination of two great circles of a sphere, is the angle made by their tangents at the point of intersection.

For each of these tangents being perpendicular to the radius in which the planes of the circles intersect, the angle contained by the tangents measures the inclination of the planes of the circles. (Euc. XI. Def. VI.)

DEF. V. A spherical angle is the angle at which two arcs of great circles intersect each other on the surface of a sphere, and is the same as the angle between the tangents to the two

arcs.

7. Hence, spherical angles, that are vertically opposite, are

equal; as are also the angles at the opposite points of intersection of great circles. (Euc. XI. 10.)

DEF. VI. If through the centre of a circle, whether great or small, a straight line be drawn perpendicular to its plane, the point in which, if produced, it meets the surface of the sphere, is called the pole of that circle.

8. Thus (art. 1), if OC, which is perpendicular to the plane of ABD, be produced both ways, meeting the surface of the sphere in the points P and p, these points are called the poles of ADB. It is evident that the line PCp passes through the centre of the sphere. In a small circle, the term pole is more usually applied to that point only, as P, which is nearest to the circle. If P be called the nearer pole of AB, then p may be termed the farther pole.

9. PROP. The pole of a circle is equally distant from every point in that circle, the distances being measured by arcs of great circles.

The same construction remaining as in art. 1, through D and E, draw two great circles PDp, PEp, meeting the

B

great circle A'D'B', whose plane is parallel to ADB, in D' and E'; draw OD', OE', PD', PE'; then, because OD' is