SPHERICAL TRIGONOMETRY.

CHAPTER I.

ON CERTAIN PROPERTIES OF SPHERICAL TRIANGLES.

1. DEF. DEF. A Sphere is a solid bounded by a surface of which every point is equally distant from a fixed point, called the centre of the sphere.

DEF. The line joining the centre with any point in the surface is called the radius of the sphere.

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

Let ABCD be a sphere of which the center is 0; AFCG the curve in which a plane cutting the sphere inter- A sects its surface; OE a perpendicular from O upon the cutting plane.

Join E with F any point in AFCG, and join FO. Then since OE is perpendicular to the cutting plane, it is perpendicular to EF, a line in that plane;

B

N

E

F

EF = √(OF2 – OE), a constant quantity.

Now E is a fixed point in the cutting plane, and F is any point in the curve AFC. Therefore AFC is a circle whose center is E and radius is EF. Euclid, 1. Def. 15.

COR. If the cutting plane pass through the centre of the sphere, EO vanishes, and EF becomes equal to OF, the radius of the sphere.

3. DEF. The circle in which a sphere is cut by a plane is called a great, or a small, circle according as the cutting plane passes, or does not pass, through the centre of the sphere.

COR. Since the radius of every great circle is equal to the radius of the sphere, Art. 2, all great circles of the sphere must be equal.

NOTE. Unless the contrary be expressly mentioned, when hereafter we speak of an arc of a sphere, an arc of a great circle is meant.

4. DEF. SPHERICAL TRIGONOMETRY investigates the relations subsisting between the angles of the plane faces which form a solid angle, and the angles at which the plane faces themselves are inclined to one another.

For the sake of convenience the angular point is made the centre of a sphere, which being cut by the plane faces containing the solid angle, presents on its surface a figure whose sides are arcs of great circles.

Let ABC be a triangle of this kind, whose sides AB, BC, CA, are formed by the intersection of the planes AOB, BOC, COA, with the surface of a sphere of which O is the centre. The angle of any face, as AOB, arc AB

is

Pl. Trig. Chap. vi.; and the angle

E

b

C

a

B

contained between any two faces, as BAO and CAO, is the `angle contained between AD and AE, which are lines drawn in the planes BAO and CAO at right angles to their intersection 40. Eucl. XI. Def. 6.

The lines AD, AE, being at right angles to the radius OA, and lying in the planes of the arcs AB and AC, are tangents to those arcs.

5. After certain properties of spherical triangles have been proved, it will not, in pursuing further investigations, be requisite to represent in our figures the centre of the sphere on which the triangles are described.

NOTE. It must, however, never be forgotten, that when the words "the angle BAC" or "the angle A" occur, the angle meant is that of the inclination of the planes passing through O, the centre of the sphere, and the arcs of AB and AC, and that this is the angle contained between two lines drawn from any point in AO at right angles to it, and respectively lying in the planes AOB and AOC. Also, whenever the expression "the sine of AB" occurs, the sine is meant of the angle which the arc AB subtends at the centre of the circle of which it is a portion.If this circle be a great circle, its centre is also the centre of the sphere.

6. In the following pages the angle BAC will commonly be indicated by the letter A, and the angle subtended by BC, the side of the triangle opposite to BAC, will be indicated by a. The other parts of the triangle BAC will be represented in like manner.

7. DEF. If OE, which is perpendicular to the plane AFC, (Fig. Art. 2.) be produced both ways to meet the surface of the sphere in B and D, these points are respectively called the nearer and the more remote poles of the circle AFC, and the straight line BOD is called the axis of the circle AFC.

8. The pole of a circle is equally distant from every point in its circumference. (Fig. Art. 2.)

Join B with F any point in AFCG. Then, BEF being a right angle,

= BE2 + OF2 - OE2; a constant quantity.

And F being any point in AFCG, B is equally distant from every point in the circumference of that circle.

Similarly, DF2 = DE2 + EF2

· DE2 + OF2 – OE; a constant quantity.

Hence D is equally distant from every point in the circumference of AFCG.

Again; let BF be an arc of a great circle passing through B and F. Then since BF is constant, and in equal circles, equal circumferences are subtended by equal straight lines, Euclid III. xxix, the arc BNF is constant,—and also, because the radii of all great circles are equal, the angle BOF subtended at the centre of the sphere, is constant.

Hence it appears that every point in the circumference of a circle of a sphere is equally distant from the pole of the circle; whether the distance be estimated by the length of a straight line joining the point with the pole, or by the arc of a great circle connecting the same points, or by the angle which this arc subtends at the centre of the sphere.

9. Since BO is at right angles to the plane AFC, every plane through BO is at right angles to that plane. Hence, the angle between any circle whatever and a great circle passing through its pole, is a right angle.

10. If AFCG become a great circle, OE vanishes, E coincides with O, and BOF becomes a right angle.

Now since BO is perpendicular to the plane AOF, the angle BOF is a right angle, and BF is a quadrant.

Also, since BOF is a plane passing through BC, which is a line perpendicular to the plane AOF, BOF is at right angles to AOF, and therefore the angle BFA— which, Art. 4, is the inclination of the two planes to one another, is a right angle.

G

B

N

Hence it appears, that if a quadrant (FB) be drawn at right angles to a great circle AFC from any point F in it; the extremity of the quadrant is the pole of the circle.

COR. If B be the pole of AFC, AO and FO are at right angles to BO, and AOF is the inclination of AOB and FOB; .. Art. 5, LAF LABF.

DEF. Great circles which pass through the pole of a great circle are called secondaries to that circle.

11. If from a point on the surface of a sphere there can be drawn two arcs (which are not parts of the same circle) at right angles to a given circle, that point is the pole of the circle. (Fig. Art. 10.)

Let B be the point, BA and BF arcs through A and F, which are points in the circle AFC; and let BA and BF be at right angles to AF.

Then since BA and BF are at right angles to AF, and are not parts of the same circle, the planes BAO and BFO must intersect, and their intersection BO is at right angles to AOF, Eucl. xI. 19;—therefore B is the pole of AFC.

COR. Hence also it appears, that the intersection of two arcs drawn at right angles to a given circle through any two points in it, is the pole of that circle.

12. The angle subtended at the centre of the sphere by the arc which joins the poles of two great circles, is the inclination of the planes of the circles.

Let the given circles be FD and FE intersecting in F, A and B their respective poles, and ABD the circle through A and B.

Now 40 is perpendicular to OF,a line in the plane DOF,

And BO is perpendicular to OF,a line in the plane EOF;

.. OF is perpendicular to the plane AOB, and therefore to OD and OE, which are lines in that plane;

.. DOE is the angle of inclination of the planes FOD, FOE.

And AOB = AOD – BOD = 90° – BOD

Сок. Arts. 5, 4.

BOE - BOD

= DOE.

Hence also it appears that AB = ▲ DE = ▲ DFE.