BI.VES OF A SPHERICAL ANGLE.] MATH EM ATICS. SPHERIC AL TRIGONOMETRY. 



657 



CHAPTER XV. 

 SPHERICAL TRIGONOMETRY. 



BEFORE reading the following Chapter, the student will 

 do well to reperuse that on Spherical Geometry already 

 given (p. 596, <tc.) He will there find the definitions 

 enunciated, and the chief properties of spherical triangles 

 proved, which are employed as the premises, and whence 

 the formulas of the following pages are deduced. It has 

 been stated that the chief applications of this science 

 are found in practical astronomy and geodesy ; also it is 

 observed, on p. 598, that the side of a spherical triangle 

 measures the angle it subtends at the centre of the 

 sphere, and hence is spoken of as an angle. Now it is 

 to be remarked, that in practical astronomy, the mea- 

 surements made by the various instruments are invari- 

 ably the angles subtended at the eye of the observer, by 

 arcs of the great sphere ; for instance, the altitude of a 

 star is measured directly as an angle, so that in these 

 cases the radius of the sphere never enters into con- 

 sideration. But in the case of measurements on the 

 earth's surface, if we have a distance measured along a 

 great circle in miles or yards, which is to enter into our 

 calculations, we must determine the arule these yards 

 or miles subtend at the earth's centre : thus if a be the 

 length in question, r the radius of the earth, the angle, 



g 

 then = where is in circular measure. If con- 



tain n, then n = 



Further it will be observed, 



that in case the sides of a spherical triangle are small 

 compared with the radius of the sphere, the triangle does 

 not differ sensibly from a plane triangle '.e.g., a triangle 

 oil the earth's surface, the sides of which are each about 

 a mile long, will not differ sensibly from a plane triangle^ 

 unless the measurements are made with very refined in- 

 struments. Hencftit is manifest that the plane triangle 

 is the limit of a spherical triangle, and accordingly we 

 shall find that the formulas for the solution of spherical 

 triangles are quite analogous to those that have been 

 already deduced for the solution of plane triangles (p. 

 616, tt stq.), and we shall see that the latter can be deduced 

 from the former by considering the plane triangle as the 

 limit of the spherical triangle. 



N.B. The following results already proved on p. 599 

 are very important. Let A, B, C, a, 6, <-, be the angles 

 and sides of any spherical triangle, and A', B', C", o', '/,', 

 the angles and sides of the corresponding polar triangle. 



Then A +a' = B + 6'-C-fc' = 180 

 And A' -fa =B'+6 -C*+c = 180 



We shall employ this notation for the angles and sides 

 of a spherical triangle, and of its polar triangle, through- 

 out the following pages. 



(1). To show that the Sinet of the A ngltt of a Spherical 

 Triangle are proportioned to the Sines of the opposite 

 lidet. 

 Let A B C be the triangle, O the centre of the sphere ; 



join O A, O B, O C ; Fig. i. 



through A draw a plane 



A N P perpendicular to 



O B, cutting the plane 



AOBin AN and BOG 



in N P, these lines are 



perpendicular to O B, 



and the angle ANP 



measures the inclina- 

 tion of the planes, and 



is .'. equal to the angle 



B of the triangle. 



Through A draw ano- 

 ther plane AMP perpendicular to C, cutting A O C 



in A M, C O B in M P, and A M P in A P, then A M 



VOL. L, 



and M P are perpendicular to O C, the angle A M P is 

 equal to the angle C of the triangle, and A P is perpen- 

 dicular to the plane BOA, and .'. APN and APM 

 are each right angles. Hence 



AP , . _ AP . sin. B AM 

 Sin. B = r-s? and sin. C= .^-f . . - p_= 7-==- 

 AN AM sin. C AN 



Also since c is the angle A B and 6 the angle A O C 



AN AM . sin. b AM 



sin. c = ^r.-. and sin. o= ?T-T. . . . = TTVT 



OA OA sin. c AN 



sin. B sin. 6 sin. B 

 or - 



sin. C 



sin. c sin. c 



sin. b sin. c 



The same proof holds good of the other sides and angles. 



sin. A sin. B sin. C 



Hence - = - r = . . . . (1). Q.E.D. 



sin. a sm. 6 sin. c 



COR. 1. Suppose a, b, c, to be the lengths of the 

 sides BC, CA, AB, then the angles denoted in formula 



(1) by o, 6, c, are in Circular measure, r being 

 the radius of sphere. Hence 



Bin. B 



b 

 Bin.. 



r 



sin. - 



c 

 n.jr 



c 



r 



b_ 

 c 



r. 



Now in the limiting case when r is infinite, - 



and b - - and .'. (Plane Trig., Art. 47) 



b e 



sm. sin. 



1, and 1. .".in the limit 



!2l___ the formula for plane triangles (p. 627). 

 sin C c 



COR. 2. If through O and A P a plane be drawn 

 cutting the surface of the sphere in Ap, then Ap is per- 



A P 

 pendicular to B C, and Q-Y equals the sine of A O P : 



i. e., is equal to the sine of Aj, which we will call p. 



AP . AN AP 

 ISow sin. B sm. c = + - ^ = 8m . Ap. 



(2). 



cos a cos. 6 cos. c 

 . , - 



As before, let O be the centre of the sphere, and ABC 

 Fig. . 



.'. sin. B sin c = sin. p 

 (2). To prove the Formula cos. A 



