PARTICULARLY APPLICABLE TO SOJIE GEODETICAL PROBLEMS. Ill 



PROBLEM. (Figs. 1, 2, 3.) 



Three stations A, B, C are given in position ; also, there are given 

 the angles ADS, ADC which the lines joining A, one of them, and 

 B, C, the other two, subtend at a fourth station D in their plane ; 

 to determine the position of the fourth station, by a geometrical con- 

 struction. 



Solution. At the point B, in the line BA, make the angle ABd 

 equal to ADC the angle which AC subtends at D; observing, that 

 the line Bd must have such a position, that if the angle ABd were 

 placed on ADC, so that BA lay along DA ; then Bd would lie on 

 DC. Also, at the point C, in the line CA, make the angle ACd 

 equal to ADB, the angle which AB subtends at D; observing that 

 the position of Cd must be such as to admit of the angle ACd being 

 applied on ADB; join A, and d the intersection of the lines Bd, 

 Cd. By the theorem, the triangles ADC, ADB will be similar to 

 ABd, ACd respectively ; and as all the angles of these last are ma- 

 nifestly given, because their sides are given in position, therefore all 

 the angles of the triangles ADC, ADB will be known; the angle 

 DCA being equal to BdA; DBA to CdA; DA C to BAd; and 

 DAB to CAd. 



Scholium. Since the angle BdA is equal to DCA (fig. 2.) and 

 CdA to DBA, the angle BdC is equal to the sum of the angles 

 DCA, DBA. Now it may happen that their sum is equal to two 

 right angles ; then, Bd and Cd will be in one straight line ; and, 

 there being no intersection, nothing can be determined in respect to 

 the angles BAD, CAD. But in this case, since DCA, DBA make 

 two right angles, the points A, B, C, D are in the circumference of a 

 circle. Thus it appears, that when the point D is in the circumference 

 of a circle which passes through A, B, C, the problem is indeterminate. 



8. It is deserving of remark that this simple construction, by which 

 the point d is found, has served to change the proposed geodetical 

 problem into another which at first view appears easier of solution ; for 



