TRIGONOMETRY. 



Subftitute thi3 value of cot. c for it in Equation 3. 



N" ( IV. ) and it will become 



fin. a cot. <p' cof. B = cof. a cof. B + fm. B cot. C. 



_ (cot. e' fin. a — cof. a) cof. B 

 Whence cot. C = ^ jj^^-g = 



(cot. <p' fin. a — cof. a) cot. B ; or 

 cot. B fin. (a — ?>') 



Subftitute the value of cot. C for it in equation N°{IV.) 

 and it will reduce to 



cot. c 



cot. C = 



fin. (f' 



(XVI.) 



_ cot. a cof. (B — f") 

 — cof. ip" 



4. Given a, c, C, to find B. 

 Determine if" from N° (XVIII.) ; then 

 cot. c cof. ^ 



It may be obferved, that equation N° ( XV. ) is analogous 

 to N° (VIII.) ; which (hews, that the operation here per- 

 formed, is equivalent to letting fall a perpendicular arc 

 from the angle A to the bafe a ; the fubfichary arc if' be- 

 ing the fegment adjacent to the given angle B. 



2. Given B, C, Cf to find a. 



Here ?i' muft be found by N" (XV.) and then from 

 N° (XVI.) we have 



cot. C fin. a' 

 fin. (a - ip') = 



cof. (B - 9") = 



cot. a 



(XIX.) 



(XX.) 



cot. B 



(XVII.) 



Whence a becomes known. 

 3. Given B, C, a, to find c. 



Find a dependent angle (p", by making 



cot. C 

 cot. C = cof. a tan. ffl", or — ;^- = tan. ip", 



cof. a 



cot. 



cot. 



cot. 



V 



(XVIII.) 



i (A + B - C) 



from which B becomes known. 



Cafe 4. — Of thefe four parts ; "u/z. the three angles and 

 a fide, (fuppofe i;,) any three being given, to find the 

 fourth. 



This cafe comprifes three problems. 



I. Given three angles to find a fide. 



Suppofe Equation i. N° (XI.) to be applied to the fo- 

 lution of the fupplemental triangle, by changing a, b, c, and 

 C, into a', b', c\ and C. Then to bring it back to the tri- 

 angle propofed, let there be fubftituted for a', b', c', and 

 C, the correfponding values 180° - A, 180" - B, 180° 

 — C, and 180° — c; thofe equations will then be tranf- 

 formed into the following, which are applicable to the pre- 

 fent problem. 



* = V 



f cof. 



I- cof. A (B -f C -f A 

 f cof. X (B + C 

 1- cof. A (B + 



cof. 



(A - B + C) 



cof. 



- A) cof. i 



i (B + C 

 (A + B 



A) 

 C) 



fcof. i 



(B -h C -f A 

 A -I- C - BV cof. 



cof. i 



I (B + 



A -h C - B) 



C - A) 



^ I- cof. I (B -)- C 4- A) cof. A (A + B - C 



(XXI.) 



fin. A a 



The following are the expreflions for the fines of the half angles ; viz. 



| cof. A (A + B 4- C) cof. A (B + C 

 ^ I _ fin. B fin. C '^ 



f cof. A (A -f B -I- C) cof. A (A + C 



A)1 



fin. 



hi 



- B) 



fin. A f — ^ 



f cof. 



— fin. A fin. C 

 (A + B -h C) cof. A (A 



4- B 



C) 



fin. A. fin. B 



(XXII.) 



It may here be obferved, that notwithftanding the deno- 

 minators are negative in the above expreflions, the whole 

 fraftion under the radical are always pofitive. 



2. Given A, B, c, to find C. 



Here, by applying in like manner the equations N° ( XII. ) 

 and N°(XIII.) to the fupplemental triangle, we ftiall 

 have 



cot. (? = cof. c tan. B'; (XXIII.) 



from which the fubfidiary angle ip may be determined, and 

 thence 



cof.C = ^°^-^f-'^-"\ (XXIV.) 



3. Given B, C, and c, to find A. 

 Find (p from N° (XXIII.) then from N° (XXIV.) 



there refults 



(fin. A - (?) = 



cof. C fin. 9 

 EoTB 



(XXV.) 



from which A becomes known. 



0/ the Analogies of Napier. — In the introduftion to the 



prefent article we had occafion to mention the analogies of 

 Napier, which it may not be amifs to illuftrate before we 

 proceed any farther in our inveftigations. Thefe analogies 

 are four fimple and elegant formulae, which we owe to the 

 celebrated inventor of logarithms, of which two ferve to 

 determine any two angles of a fpherical triangle by means 

 of two oppofite fides and their included angle ; while the 

 other two ferve to find any two fides by means of their 

 oppofite angle and the contained fide. They, therefore, 

 together with equation N° ( I. ), will ferve for the fohition of 

 all the cafes of obhque-angled fpherical triangles. The in- 

 veftigation of thefc analogies may be given as follow. 



If from Equation i. N° (II.) cof. c be exterminated, 

 there will refult, after a little reduftion, 



cof. A fin. c = cof. a fin. 6 — cof. C fin. a cof. b ; 



and, by a fimple permutation of letters, 



cof. B fin. c = cof. b fin. a — cof. C fin. 6 cpf. a. 



Adding thefe equations together, and reducing them, we 



have 



fin. c (cof. A -f cof. B) = ( I 



cof. C) fin. (a + b). 

 Now 



