and. Longitude of the Observatory on the Calton Hill. 259 



West Lomond, in latitude 56° 14.' 57" N., longitude 3° 17' 4" 

 W,, was 68° 40' 46": what is the latitude and longitude of the 

 Observatory ? 



Let P be the north pole, C the Calton Hill, L the "West 

 Lomond, M the Isle of May Light, and 

 B North Berwick Law; also ECFG a 

 parallel of latitude passing through C, and 

 PE, PC, PF, and PG, meridians passing 

 through L, C, B, and M. 



Then by the problem originally pro- 

 posed by Townley, let the marginal figure 

 be constructed, and apply the principles of 

 spherical trigonometry; since the lines em- 

 ployed are arcs on the surface of the earth, 

 which in this case may be considered as a 

 sphere. 



Now by the Trigonometrical Survey we 

 have, — 



West Lomond, latitude... 56° 14' 57" N. long. 3° 



Isle of May Light 56 11 22 2 



North Berwick Law 56 3 8 2 



Whence are obtained PL = 33° 45' 3" LPB = 34 53 

 PB = 33 5Q 52 LPM = 44 17 

 PM = 33 48 38 BPM = 9 24 



1. In the triangle LPB, there are given the sides PL, PB, 

 and the angle LPB, to find LB = 22' 4.4"-48 = 26-17 English 

 miles. 



2. In the triangle LPM are given PL, PM, and the angle 

 LPM, to find LM = 24' 52"-88 = 28-632 English miles. 



3. In the triangle BPM are given PB, PM, and the angle 

 BPM, to find BM = 9' 45"-56 = 11-23 English miles. 



4. In the triangle PLB are given all the sides to find the 

 angles PLB and PBL = 121° 3' 52", and 58° 27' 10" respec- 

 tively. 



5. In the triangle LBM are given all the sides now found 

 by computation to determine the angles LBM = 90° 51' 40", 

 and BML = 66° 2' 54". Also LBM-LBD = 90° 51' 40" 

 -68° 40' 46" = DBM = 22° 10' 54"; and 180°-(BLD + 

 LBD) = 180° -82° 29' 34" = 97° 30' 26" = LBD. BLM 

 will also be found to be 23° 5' 28"; whence BLM — DLB = 

 23° 5' 28"— 13° 48' 48" = 9° 16' 40". 



LikewiseLDwillbefound=21'22"-08,LDM = 128°28'28", 



LMI) = LMC = 42° 14' 52". But LMC4- LCM = 1 10° 55' 38", 



and consequently CLM=69° 4' 22"; CLM-BLM = 69° 4' 22" 



2 L 2 -23" 



