Ml 



MATHEMATICS. SPHERICAL TRIGONOMETRY. 



and A, B, we can determine C, by formula (13) ; and 

 finally we can determine e by formula (1). 



It will be observed that in this case, since the result* 

 depend on our determining B from a given value of sin. 

 K, they will bo ambiguous, a* in the analogous case ol 

 plane triangle* (p. 619) ; for if B' be the value of B < 

 00", which we derive from formula (1) : then 180 B' 

 also satisfies formula (1). 



This amount of ambiguity depends on the data. For 

 lot ABC be a triangle, having the angle BAC v A, 

 A C - 6 and B C - 

 e. Produce A B and 

 AC to meet in A', 

 draw CB' - C I:. 

 Then the given data A. 

 belongs a* much to 

 the triangle ACB. 

 as to A C B'. Moreover it is plain that C B B' = C B' B, 

 and hence if C B' A = B', we shall have C B A = 180 

 15', the same conclusion that we derived from the 

 formula. 



(13). To solve the Fourth Case of Oblique- Angled Spherical 

 Triangles. 



In this we suppose that we have given A. B. and C. Then 

 x- is given from formula (15), and s from for- 

 mula (1C) ; hence we obtain a and b, and then we obtain 

 C. by formula (1). If we wish to obtain C without the 

 previous calculation of a and b, we must introduce a sub- 

 sidiary angle and proceed as in article (11). From 

 formula (4) we have 



Cos. C = cos. A cos. B -f- sin. A sin. B cos. c. 

 Assume sin.* 6 =* sin. A sin. B sin. 2 vr 

 And we shall obtain 



(14). To tolve the Fifth Case of Oblique-Angled Triangles. 

 In this case we will suppose that A. C. and c. are 

 given. We shall obtain sin. a from formula (1). Now, 

 formula (13) gives us 



A+C 



' B 



cotan. -x- 



Whence we obtain B, and a similar modification of 

 formula (15) will give us b ; or, having B, we may obtain 

 6 from formula (1). 



In this case a is determined from its sine, and there- 

 fore has two values, vvz. a' and 180" a'; and if both 

 these values are admissible, the case is ambiguous. 



In the triangle ABC, let A B = c, B A C = A and 

 B C A = C, and suppose B C = a, draw B C' -= a' ; then 

 if a' be greater than c, Fig. 7. 



! it is plain that A falls 

 between C and C' ; in 

 this case produce C'B 

 and C* C to meet in C*. 

 Thru the angle BCCf 

 BC / C-BC"C. - 

 Hence the data belong c 

 equally to the triangle B A C. and B A C* ; and the case 

 is ambiguous, provided c < a ; also it will be observed 

 that H C* - 180 B C' - 180 B C - 180 a' as 

 previously appeared from the calculation. If C be less 

 than B A, then C' would fall between B and A, and the 

 bore construction would be no longer possible. Hence 

 if c > a, the case is not ambiguous. 



(15). To to'.te Oi Sixth Gate of Oblique-Angled 

 Tn ingles. 



In this cane we have given A, B, O. We shall find a 



from formula (11), and b and c from similar formulas. 

 This case, however, never occurs in any of the pra 

 applications of spherical trigonometry. 



On the Solittion of Quadrantal Triangles. 



A quadrantal triangle (Spherical Geometry, Def. 14, 

 p. 699) has one side of 90, and .'. the correspomlin-^ 

 polar triangle is a right-angled triangle. From this con- 

 sideration it would be easy to modify Xaj/icr's Rule to 

 suit the cae of the quadrantal triangle. In practice, 

 however, it is better to treat them as oblique triai 

 on doing so it will be found practically that the cir- 

 cumstance of one side being equal to 90 will introduce 

 important simplifications. 



OH THE FORMULAS PECUXIAB TO GEODETICAL 

 OPERATIONS. 



We have already stated, in general terms, that the 

 science of spherical trigonometry finds one of its appli- 

 cations in Geodesy. It is to be observed that this appli- 

 cation possesses some peculiarities in consequence of the 

 sides of the triangles employed in a survey, on even the 

 largest scale, being small compared with the radius of the 

 earth, and consequently small when estimated in degrees 

 or minutes ; whereas, in astronomy, there is no limi- 

 tation imposed on the magnitudes of the sides of the 

 triangles employed. Our object in the present article is 

 to explain concisely the results of this limitation, and to 

 deduce certain formulas depending on it. 



(1C). To state the Object of a Trigonometrical Survey of a 

 Country. 



The object of the survey is, (1), to fix accurately the 

 relative positions of certain chief points in the country, 

 so as to lay them down on a map ; and, (2), having fixed 

 these chief points, then Fig. 8. 



by means of subsi- 

 diary operations, to lay 

 down in detail all the 

 minor features of the 

 country,its roads, rivers, 

 towns, hamlets, <fcc. The 

 accompanying figure 

 will be sufficient to il- 

 lustrate this matter for 

 our present purpose, 

 which is from an actual 

 survey. A is a place 

 called Ruckingo, B, High Nook, C, AUington, D, Lydd, 

 E, Fairlight Down, and F, Tenterden. The line A B 

 is measured very accurately, and is called the bnse line ; 

 and then the angles CAB, ABC, are measured ; from 

 these data, A C, and C B, can be calculated ; then C B 

 being known, the angles D C B, and C B D, can be 

 measured, and thus C 1), and D B, be determined. This 

 operation is to be continued for any number of triangles 

 whatever. It is usual, in the larger triangles, to mea- 

 sure all the three angles of any triangle, and not merely 

 the two at the base ; this is done with a view of keeping 

 a check upon the various errors to which all observations 

 are liable. 



When the triangulating has been continued for some 

 distance, it is necessary to compare the calculated I< 

 of a line that has been fixed upon, and then nn-asm 

 the coincidence of the two results is a verification of all 

 previous measurements and calculations ; hence such a 

 line is called a base of verification. It is usual to choose 

 stations that are from ten to twenty miles a] 'art ; also it 

 is usual to choose for a base line a line of about four or 

 five miles long. In late French surveys only two 

 of verification have been used. The accuracy attar 

 in practice will be appreciated when the fact is state,! 

 that, in some English bases of verification, of four or 

 livi! miles long, the computed and measured lengths have 

 differed only by one or two inches. The operations of a 

 trigonometrical survey are then two, (1), the 

 mi-lit of base lines ; (2), the measurements of angles. 

 We will proceed to consider each of these. 



