Gon'wriietrlc Prollems. 31 



thy that is not known, in terms of other quantities that are 

 known. 



The Theorcni, p. 29, will sometimes find its apphcatioa 

 in this reduction. < 



If two stations, D and d, be chosen any where in the 

 lines AC, AB, and the angles CDd and BdD be observed, 

 then ISO" - BdD = A^^D and 1S0° - CD J = ADa, but 

 180° - (AD J + AdD) = B//D - CDc^ - iso'^ = BAG. 



If the station d be so chosen that the angle Cd B may be 

 equal fo the angle CDB, then the three angles of the tri- 

 angle ABC may be obtained without actually measuring 

 either of them. For, when CDB = CdQ a circle will pass 

 throuo-h the four points CDf?B and BCD + BdD - ISO" = 

 CDd\ CBd; therefore BCD = 180° - B<iD, also CB^' = 

 180° - CDJand A = CDd + BdD - ISO"; whence the 

 three angles A, B, C, are obtained by only measurin" the 

 two CDd and BJD. 



Pro! I em 11. 



When the point D is not situated in either of the lines 

 AC or AB, but within the triangle. 



In this case (fig. .5.) we have BDE — DBA = BAE, and 

 CDE - DCA = EAC ; consequently, BDE + CDE — 

 (DBA + DCA) = BAE + EAC; whence we obtain the 

 angk; BAC = BDC - (DBA + DCA). 



If the steeple were at D and the station at A, then the 

 angle BDC = BAC + DBA + DCA. 



Pruldcm HI. 



When the sta'.ion D is without the triangle on one sije 

 of A (fig. 6). 



In this case, CFB = FAB + FBA = CDF + FCD; 

 therefore FAB = CDF + FCD — FBA; whence we get 

 the angle CAB = CDB -)- ACD - DBA. 



If A be situated on the other side of D (fig. 7)j then, by 

 the same mode of proceedir.g, the angle CAB = CDB -f 

 ABD - DCA. 



When the small angles ACD, ABD cannot be measured 

 from the stations B and C, some sunplc method must be re- 

 sorted to for finding them as before observed. 



The 



