Both legs. The hy- By 23 Sph. Tr. AC. ing the comp. of AC middle part, R : Cos. AB :: Cos. BC : Cos. AC, which is less or greater than a quadrant, according las AB and BC are of the same or different affections, by 17 Sph. Tr. 2. The same, Either By 24 Sph. Tr. obl. angle and 16. 6 Eu., makA. ing AB middle part, T. BC:R::S. AB: Cot. A, of the same affection, as BC, by 16 Sp. Tr. Given, besides Case. Part. Sought. Solutions. the right angle. 6. 1. Both obl. an- The hy- By 24 Sph. Tr. gles. pothenuse and 16. 6 Eu, makA and C. AC. jing the comp. of AC middle part, R : Cot, A : : : Cot. 'C : Cos. AC, which is less or greater than a quadrant, according as A and Care of the same or different affections, by 17 Sph. Tr. The same. Either By 23 Sph. Tr. leg. and 16. 6 Eu.. mak-i AB. ing the comp. of C middle part, s. A: Cos. C: :R: Cos. AB, of the same affection as the angle C, by 16 Sph. Tr. 2. PROBLEM II. of the three sides and three angles of an oblique angled spherical triangle, any three being given, to find the rest. The cases of oblique spherical trigonometry, with their solutions, are as follow. Fig. 1. Fig 2. C B Case. Given. Sought. Solutions. 1.1. Two' The re The perpendicular CD being, in both sides maining figures, let fall from C on AB ; in the and an side. triangle ADC, right angled at D, AC angle AB. and Ă being given, to find AD, by 24 10pp0 Sph. Tr. taking the complement of A, as site middle part, Cot. AC: R :: Cos. A :T. AD; and by Cor. 2. 26 Sph. Tr. Cos. them. AC : Cos. ČB: : Cos. AD: Cos. DB ; and AB is the sum or difference of AD CB & and DB. A. one of AC, 2 By 27 Sph. Tr. S. BC: S. AC:: S. same. gle oppo-A: S. ABC. site the o- len side. ABC. 3. Thel The an- In the triangle ADC, right angled at same. gle inclu-D, AC and X being given, to find the |ded by the angle ACD. By 24. Sph. Tr. taking given the complement of AC as middle part, Isides. Cot. A:R:: Cos. AC : Cot. ACD, and ACB. by Cor. 4. 26 Sph. Tr. T. BC : T. AC :: Cos. ACD: Cos. BCD, and the angle ACB is equal to the sum or difference of the angles ACD and BCD. The o 2. 1. Two In the triangle ADC, right angled at sides & ther side. D, AC and A being given, find AD, as the in- CB. in case 1 part 1 above, and by Cor. 2. cluded 26 Sph. Tr. Cos. AD: Cos. DB : : Cos. angle. AC: Cos. BC. According as BD and DC are of the same or different affec-i AB & tions, BC is less or greater than a quadA. rant, by 17. Sph. Tr. AC, Case. Part. Given. Sought. Solutions. 2.2. Thel An an- In the triangle ADC, right angled at same. gle oppo-D, AC and A being given, find AD, as site ono offin case 1 part 1 above; and by Cor. 3. the given 26 Sph. Tr. S. DB : S. AD :: T. A : sides. T. B. According as AB is greater or ABC. less than AD, the angles A and B are of the same or different affections (19. Sph. Tr.). The o 3. 1. Two In the triangle ADC, right angled at jangles ther an-D, AC and A being given, find the angle and a gle. ACD as in case 1 part 3 above; and by side op- ACB. Cor. 1. 26 Sph. Tr. Cos. A : Cos. B : : posite S. ACD: S. BCD; and ACB is the sum one of or difference of ACD and BCD, accordthem. ing as the perpendicular CD falls within A, or without the triangle ACB, or (19. ABC Sph. Tr.), according as the angle A and & AC. ABC are of the same or different affections. 2. The The side In the triangle ADC, right angled at same. Ibetween D, AC and A being given, find AD, as in the given case 1 part 1 above ; and by Cor. 3. 26 angles. Sph. Tr, T, B: TA :: S. AD: S. DB ; AB. and AB is the sum or difference of AD and DB. 3. The The sido By 27. Spl. Tr. S. B: S. A:: S. AC same. opposite S. BC. the other given anIgle. BC. Case. Given. Sought. Solutions. 4.1. Twol A side In the triangle ADC, right angled at angles opp. one D, AC and A being given, find the anand the of the gle ACD, as in case 1 part 3 above; and side be- given an-by Cor. 4. 26 Sph. Tr. Cos. BCD : Cos. tween gles. ACD T. AC: T. BC. If BCD and them. BC. A, (and therefore, by 16. Sph. Tr. DB A, and DC), be of the same affection, BC ACB is less than a quadrant; if BCD and A, & AC. and therefore DB and DC, be of different affections, BC is greater than a quadrant (17 sph. Tr). The o 2. The In the triangle ADC, right angled at same. ther an-D, AC and A being given, find the angle igle. ACD, as in case 1 part 3 above ; and by ABC. Cor. 1. 26 Sph. Tr. S. ACD : S. BCD :: Cos. A': Cos. B. According as the angle ACB is greater or less than ACD, the perpendicular CD falls within or without the triangle ACB, and therefore the angles A and ABC are of the same or different affections (19 Sph. Tr). |