. 30 Gmiiometrk Froblcms. 



that wa? bisected ; and even if it could, the walls of the 

 buildinc would prevent the angle from Ijeitig nicasured. The 

 three following problems show the method of obtaining this 

 angle when the instrument, for the reasons just given, has 

 necessarily been placed on one side of this centre. 



Problem I. 



In measuring the angles of the triangle ABC (fig. 4.), 

 suppose the angular point A to be in the centre of a steeple, 

 where the instrument cannot be placed to observe the quan- 

 tity of the angle BAG. 



Let D be a convenient place situated on the line AC for 

 fixinir the instrument, when on account of some impediment 

 it cannot be used at A. Then let the angles CBA, CBD be 

 observed at B, and let the angle CDB be observed at D : 

 these will be sufficient to determine the three angles of the 

 iriansile ABC. 



FoT CBA - CBD = DBA, and CAB = CDB - DBA; 

 whence, as A and CBA are known, ACB is known also. 



If D were the place of the steeple, and A the station fixed 

 upon, then CDB = CAB + ABD, which reduces the angle 

 A to that at D. But it seldom happens that the object C 

 can be seen from A through the steeple at D ; therefore in 

 aeneral it is necessary to choose the station D within the 

 triangle ABC, or on one side of A. 



If the angle DBA cannot be observed from B, some prac- 

 tical method must be had recourse to for determining it. 

 Thus, 



If Dp be let fall perpendicular to AB, then BD and Dp 

 being known, the correction DBA is readily obtained; for 



-— — ^ — - = — ^-t; — = the number of seconds in the angle 

 DBx ,1" DB '^ 



DBA. The side DB need not be known with extreme ac- 

 curacy for this purpose. 



In other cases, the following general equation may lead to 

 a solution : 



CD : '' n : : BD : 5, (D - D) : : BD + BD : ^, D. 

 — J, DJ 



in which it will be easy to substitute the value of any quan- 

 tity 



