TRIGONOMETRY. 



Now from N° ( I. ) we have 



fin. a fin. i fin. c 



finTA "" fin. B ~ fin. C" 



Clearing thcfe equations of their denominators, and re- 

 fpeftively adding and fubtrafting them, tiiere refults 



fin. c (fin. A + fin. B) = fin. C (fin. a + fin. i) 



fin. t (fin. A - fin. B) = fin. C (fin. a - fin. i) 



Dividing each of thefe two equations by the preceding, 

 there will be obtained, 



fin. A -t- fin. B fin. C fin. a + fin. i 



cof. A -j- cof. B 

 fin. A — fin. B 



fin.C 



fin.C 



fin. (a + i) 

 fin. a — fin. i 



cof. A + cof. B I — cof. C fin. (a -f 4) 



Confequently, from the relation eftabliftied under Arithmetic 

 of Sines, 



[a-i] 



tan. i (A + B) = cot. I C 



cof. I 



tan. 



(A 



B) = cot. 



cof. \{a + b) 

 -i) 



fin. i 



fin. -i (a ■)- A) 



And thefe equations, expreffed as analogies, are. 



cof. i 

 fin. 





;cof. i(3-5)::cot. iC :tan. i(A + B) 

 fin. i (a-i)::cot.iC : tan. i (A - B) 



Thefe analogies being applied to the fupplemental tri- 

 angle, by putting i8o° - A, 180° — B, &c. for a, i, &c. 

 we have 



cof. i (A + B) :cof.i(A - B)::tan.if:tan.i(a + i) 

 cof. -I (A + B) :cof.i(A- B)::tan.i<::tan.i(a-A) 



From a due confideration of thefe four analogies, it 

 refults, 



1. That i (A — B) < 90°, or that the difference of 

 two angles of a fpherical triangle is lefs than 1 80°. 



2. That i (a + i) and i (A + B) are always of the 

 fame afFeftion. 



3. That the difference of two fides is always lefs than 

 180°. 



4. That [a — i) and (A — B) have always the fame 

 fign ; whence it follows, that the greateft angle is oppofite 

 to the greateft fide, and reciprocally. 



To thefe it may be added, 



5. That the leaft angle is oppofite to the lead fide, and 

 the mean angle to the mean fide. 



One or other of thefe obfervations will ferve to remove 

 the ambiguity in the doubtful cafes where a, b, and B, or 

 A, B, and b, are given. 



We may now colleft the moil commodious of thefe 

 theorems, and prefent in one place all that will be 

 ufually required in the folution of oblique-angled fpherical 

 triangles. 



Sin. A 



fin^B _ fin. C 



fiiiT* 



fin. 



2. Tan. i A 



Tan. 



~ ^ Ifin. H* i- 



= •'{ 



b - c) fin. i (a -(- 



*) 



} 



[a + c 



Tan. 



Tan. 



Tan. 



i C 



hb 



Tan. 



S. Ta 

 9. Tan. 



10. Tan. 



11. Tan. 



12. Tan. 



13. Tan. 



14. Tan. 

 Vol. XXXVI. 



b - 



= V 



= v 



= ^/ 



= y 



^ tan. 



fm 



fiiiTi 



fin. k {" + c 



— a) fin. A (a + c + 4) 

 _ a) fm. i (« + ^ - 01 



- b) fir 



i {n 



+ * 



b) fin. I [b + c - a) 



\ 



tfin. i (a + * - fin. ^ [a + b + 

 r- cof. i (B -h C - A) cof. I (A + B + C)\ 

 I cof. A (A +~B - C) coi. i (A + C - B) j 

 f- cof. ^ (A 4 - C - B) cof. i (A - f B 4- C) i 

 1 cof. X (B + C - A)~Tof. i (A +'B - C) J 

 r- cof. i (A + B - C) co f. HA + B + C)7 

 1 cof. I (A 4- C - B) coirT-(B + C - A) 1 



— tan. ^ c 



:= tan. -k a 



= tan. i b 



Si tan. i b 



- cot. i C 



^ (A 

 fin. X (B - A) 

 fin. i (B + A) 

 cof. X (B - A) 



cof. i (B + 



fin. 



I(C 



A) 

 B) 



fin. He + B) 

 cof. X (C - B) 

 cof. I (C + B) 

 fin. \ (A - C) 

 fm. 4 (A -f C) 



cof. X (A - CO 



cof. X (A" H- C) 



fin. \ (* - •') 

 fin"; yr* + d) 



M ni 



15. Tan. 



