CHAPTER XI.

SPHERICAL TRIGONOMETRY.

315. A sphere is a solid generated by the revolution of a semicircle about its diameter, and is such that all points on its surface are equally distant from a point within it, called the centre, which is the middle point of the fixed diameter round which the generating semicircle revolves.

316. A spherical triangle is a figure formed on the surface of a sphere by the mutual intersection of three great circles. The intercepted parts of each of the three great circles between the other two, form the three sides of the triangle.

317. The angles of a spherical triangle are the plane angles contained by tangents to each of the arcs at the point of their intersection; or they are the inclination of the planes of the great circles which contain the angles.

Let ABC be a spherical triangle formed by the portions AC,

E

A

B

a

CB, and BA of three great

circles contained between each of their intersections with the other two; and let the lengths of these in degrees be called a, b, and c; also let O be the centre of the sphere, so that OB, OC, and OA

are all equal; then the arc a is the measure of the angle BOC, b is the measure of the angle AOC, and c is the measure of BOA.

318. Let the angle BCA be a right angle, and consequently the plane OBC perpendicular to the plane OAC; and in OB take any point D; and in the plane OBC draw DE perpendicular to OC, which will therefore be perpendicular to every straight line passing through E in the plane AOC; draw through E, EF perpendicular to OA, and join DF: then the angles DEO, DEF, EFO are right angles. Again, since OFE is a right angle,

then but

OF2 + FE2 = OE2; to each add DE2,

OF2 + FE2 + DE2 = OE2 + DE2;

OE2 + DE2 = OD2, and FE2 + DE2 = FD2; hence, substituting, we have OF2 + FD2 = OD2; and therefore (Euc. I. 48) OFD is a right angle; also EF and DF being each perpendicular to OA, the common intersection of the planes BOA and COA, the angle DFE is the measure of the angle A of the triangle; and therefore, angle A = ▲ DFE.

319. Now, from the four right-angled triangles DEF, FEO, OED, and DOF, by applying Articles 122, 123, 124, and 125 of Plane Trigonometry, we obtain the following important relations between the sides and angles of a right-angled spherical triangle,

Fifth, Taking the product of the two expressions in (4), and observing that cot. a sin. a = cos. a, and that cot. b sin. b = cos. b, and that cos. a cos. b is, by (1), equal to cos. c; we have

cos. ccot. A cot. B.

(5)

Sixth, Dividing the first expression in (2) by the second in (3), and the second in (2) by the first in (3), and observing that sin. a÷tan. a = cos. a, that sin. b÷tan. b = cos. b, and that from (1) cos. cos. a = cos. b, and cos. cos. b = COS. a; we obtain, after a little reduction,

320. The above ten results can be reduced to the following very symmetrical forms, from which Napier derived his rules for the circular parts-namely:

cos. c = cos. a cos. b = cot. A cot. B.

and from (6) and (3) we have

cos. A sin. B cos. a

=

cos. B sin. A cos. b

tan. b cot. c.

(7)

NAPIER'S RULES FOR THE CIRCULAR PARTS.

321. If in a right-angled spherical triangle the right angle be left out, and the two sides containing it, with the complements of the two acute angles and of the side opposite the right angle, be called the five circular parts; and if these parts be taken three and three, in all the ten ways that this can be done, and one of these parts be called the middle part, the other two will either be adjacent to it, or they will be each separated from it by another part, in which case they are called opposite parts;

then the two following rules contain all the ten expressions in (7):

RULE I.—Sin. (middle part) = the product of the cosines of the opposite parts.

RULE II.-Sin. (middle part) of the adjacent parts.

=

the product of the tangents

322. In applying these rules, it is necessary to observe that the cos. tan. and cot. of an angle greater than 90° is negative; hence, from (1) and (5), it appears that c is greater than 90°, when a and b or A and B are one greater and the other less than 90°; and that c is less than 90°, when a and b or A and B are both less or both greater than 90°. Also from (4), it appears. that A and B are greater or less than 90°, according as a and b are greater or less than 90°; or the sides are of the same affection as the opposite angles.

323. Some may perhaps find it most convenient to associate the results obtained above with the corresponding definitions in Plane Trigonometry; the following can easily be derived from the above ten formulæ :

Solution.

In solving this or any question in right-angled Spherical Trigonometry, we must take each of the sought parts along with the two given parts; this will form three groups of three terms each -namely, A, B, a, Aca, and Aba. In each of these we take either a known or an unknown part for the middle part, and then apply Napier's Rules; thus, in the group A, B, a, the complement of A is the middle part, and a and the complement of B are opposite parts; for B is separated from A by the side c, and a is separated from A by the side b: and therefore,

cos. A sin. B cos. a; wherefore, sin B=

cos. A
COS.

In the second group, Aca, a is the middle part, and the complements of A and c are the opposite parts; and therefore,

In the third group, Aba, b is the middle part, and a and the complement of A are the adjacent parts; and therefore,

NOTE.

The check here introduced is to find one of the

three unknown parts in terms of the other two; and if the