1 .-. 



NAVIGATION. 



[SOLUTION op rni ANGLES. 



of the triangle, thus making cos. C subtractive, we ahall 

 equally for both triangles, tho condition 



o 6 cos. C -f- c eo*. B 

 And similarly, 6 c eo*. A + o co*. 



c a co* B -f- 6 cos. A. 

 Let these be regarded a* three algebraical equations, in 

 which the unknown quantities (usually r, y, and f) are 

 eo*. A, co*. B, ami cos. C ; then, multiplying them by a, 

 b, e respectively, we have 



a* 06 oos. C -f- ac K * B 



6 - be co*. A + afccos. O 



e 1 ac co*. B -j- be cos. A ; 

 and subtracting each of these from the sum of the other 

 two, there results 



1 r . 6* + c* a* 

 b* + c* a 1 26e cos. A 8 - A "" 



o'+c 1 6'- 2acco.B 



cos. B = 



e 



cos. C = 



2ac 



2afr 



The expressions on the right furnish a complete solu 

 tion to the present problem ; but when the given sides 

 a, b, e, consist each of several places of figures, the cal- 

 culation of the fractions involves a good deal of arith. 

 radical work. On this account, means have been con- 

 trived to change the preceding forms into, others, that 

 shall consist exclusively of factors and divisors, without 

 any addition or subtraction operations ; so that the 

 expressions may be fitted for computation by logarithms: 

 how this is brought about we shall explain presently. 

 But, as logarithms may be dispensed with whenever 

 a, b, c are conveniently small numbers, we shall first 

 show the best form of using the above expression for 

 cos. A, in such a case. 



Subtract 1 from the fraction for cos. A; it then 

 become*. 



(b _ c )i _ > 



26e 



1 - 



(6 c + a) (b c o) 



26c 



and now adding the 1 subtracted, we have 

 ft- /J-aXfc c a) 



I':'-. A 



26c 



2bc w 



which is tolerably convenient for calculation without 

 logarithms : but a form somewhat preferable is given 

 subsequently. 



If the 1 were transposed and signs changed, the right- 

 hand member of the last equation would be equally 

 convenient for calculation with logarithms ; but the left- 

 hand member, which would then be 1 cos. A, has no 

 corresponding logarithm in the ordinary tables ;* the 

 object is then to change 1 cos. A into some equivalent 

 trigonometrical quantity, the log. of which is to be found in 

 the tables. A diagram will assist in showing how this is 

 to be effected 



Suppose A B, Fig. 10, to be the trigonometrical radius 

 Fi. 10. namely, A B = 1, and let 



B C D be the arc of a semicircle 

 to that radius, and B A C the 

 angle A ; then An bisecting the 

 chord B C, also bisects the arc 

 BCand the angle A,/. BC- 



^ jL il B 2 sin. i A. 



NowDCB, in a semicircle, 



being a right angle, we have (Euc. 8 of VI.) BD. Bm 

 BC'; that i*. since B D - 2, and Am - cos. A, 

 3(1 cos. A) - (2 sin. $ A)' .M cosT A = 2sin.JA, 



.'. .in. J A - 



be 



aa Tk, 



- A U IW th* ttrt'd-lint of the anirl.. A. A 



"r. M.ck.r'. 



' 



to K 1 "" ln 

 nt Prvrtt, ,/ 



a formula well adapted to computation by logarithms. 

 If f be put to represent J(o + 6 + c), then the two factors 

 in the numerator will be respectively band* e, 

 and we shall have the expression for sin. J A m t 

 compact form, 



OC 



Again : because as proved above J(l cos. A) sin. 1 J A, 

 if each be subtracted from 1, we shall have 



i (1 + cos. A) - 1 -ain.i A - co*.i A, 



..2cos.4A-l+oos. A. 

 (6-f-c)' o 

 2bc ~ 



26e 



Zbc 



and dividing (I) by (II), we have finally 



Either by the formula) marked (I), (II), (III), will 

 supply a rule for finding an angle from the sides being 

 k'ivm ; but it is better to work by the formula itself 

 to be guided by written directions. That marked (II) is 

 a little the shortest ; but when J A, that is, half the 

 sought angle, is foreseen to be very small, it will be 

 better to use tho formula (?) : the reason is that the 

 sines of angles very near 90, or, which is the same thing, 

 the cosines of angles very near 0, vary from each other 

 by such slight differences, that several of such angles 

 have equal claim to the same sine or Fr 



instance, suppose we were led to 9-9999998 for the 

 logarithmic sine or cosine of an angle to be found. Hy 

 tables calculated to seven places of decimals, this numW 

 is the log. sine of every angle, indifferently, between 89 

 56' 19* and 80 57' 8*; and, consequently, the log. cosine 

 of every angle between 2* 62* and 3' 41". In such 

 extreme cases the formula (III) is to be preferred, 

 although for a small angle (f) may be safely employed, 

 and (II) for a large one. In practice, however, these 

 ill-conditioned triangles, as they are called, are al. 

 avoided, if possible ; and in Navigation they are but 

 little likely to be met with. An error of a few seconds 

 in this subject is, however, a matter of no moment ; and 

 we advert to the peculiarity here, merely to explain to 

 the learner how it happens that, although the three 

 formulae above are all equally correct, yet in inquiries 

 where the minutest accuracy is necessary, one of them 

 may, in certain rare cases, be preferable to another. 

 The defect is not in the formula, but in the tables, 

 which being calculated to only six or seven places of 

 decimals, cannot, of course, mark distinctions of value 

 that affect only the decimals more remote. 



EXAMPLES. 



1. Given the sides a - 95-12, 6 - 162-34, and e = 98, 

 to find the angle A . 



By formula (I). 



(>-b)(,-c) 



ic ~ 



o- 95-12 



6 - 162 34 Arith. Comp. 7-7896 



c = 98 Arith. Comp. 8 0088 



sin. 4A 



2)355-40 



- 177-7: 

 6- 15-39 

 * - c - 79 73 



sin. J A, 16" 7* 



.'. A - 32 15' 



11872 

 1-9016 



2)18 8872 

 94436 



+ The n'udnit my compare the ibort formula with nimltar I-TPITM'' "n 

 it pagr 2I. in connection with the calculation of line*, i.-., from derived 

 brinuUr. Ki>. 



