1836.] On the Problem of the Three Points in Surveying, 321 



VI. — On the Practical application of the Problem of the Three Points 

 in Surveying.— By Lieutenant J. Campbell, Assistant Surveyor 

 General. 



Zf. 1. 



D 



Faje 32/ 



* The problem of the three points as it is called by the French, or 

 the Townly problem by the English, is that in which three points are 

 given in position, and the angles subtended by them form a third point, 

 the distances from which to the other three are required to be found. 



This problem appears by the English surveyors to have been 

 thought of but little use in practice, and has been paid but little atten- 

 tion to, the methods generally given for its solution being operose and 

 inconvenient in practice. 



There are several methods of 

 solving this problem, and, pass- 

 ing over the geometrical me- 

 thod which is given in almost 

 all elementary works on Tri- 

 gonometry, I shall proceed to 

 make a few remarks on the ge- 

 neral formula for its solution, 

 obtained by analytical investi- 

 gation, as given by Galbraith in 

 his Mathematical and Astrono- 

 mical tables at page 48, and in 

 most French works. Let a 

 and b (fig. I) be two of the 

 given sides of the triangle, and 

 7T and P be the angles sub- 

 tended from the station D by 

 the points A,B, C, D— being the 

 required station. The values 

 of the unknown angles x and y 

 being found, all the required 

 quantities may easily be com- 

 puted j the formula given is 



cot x — 



sin 7T . cot R 



+ cotR 



sin P . cos R 

 or thus. Cot x = cot R. ( 



a . sin 7r 



.6 . sin P . cos R 



but this will be found an inconvenient method, and very subject to 

 error, as it is necessary to be careful to apply the cosine R with its 

 proper sign. 



The value of R = x -J- y is known by attending to the si- 



