Go?uo77ietnc Prollems, 33 



equals in the first equation, we have CAB = BDC + 

 AD/. (BDC + ADC) AD j, C D A 

 ABj, 1" ACi'V^' 



We might here extend our inquiry to the investigation of 

 expressions for those particular cases where it is necessary 

 to consider the forms of the bases of steeples or other build- 

 ings, the situation of their sides with respect to the sides of 

 the triangles that are measuring, and the distance of the in- 

 strument from the walls of the building; but the variety is- 

 so very numerous, that only a compapstively small number 

 could possibly be given here : it was therefore thought best 

 to leave this to the observer himself, who will adapt his so- 

 lution to the particular case that may occur. 



Problem TV.* 



Let ABD, ABS, and SBD (fig. 9.), be three planes per- 

 pendicular to each other, so that the three angles at B may 

 be right angles ; it is required to find an eqaation expressing 

 the relation subsisting between the angles ASB, BSD, and' 

 ASD. 



If we consider SB as radius, we shall have BD = SB/, 



BSD; BA = SB^ ASB; SD = SB/, BSD; SA = SB/, 



ASB ; and AD* = AB^ + BD- = SB^ /% ASB + SB"- 1\ 



AS*+SD'-AB^ 

 BSD; but by trigonometry c, ASD = 9 a o An ~ 



P, ASB + r\ BSD - f, ASB - f-- BSD , . 

 ^TT asB/BSD ^ ' ^"'^ smce/S -/• 



• This problem may be resolved by spherics in the following manner: 

 Let SBDA (fig. 10.) be the same figure as fig. J), liaving BD perpendicular 

 to the plane SBA ; and suppose Z to be the zenith of the station S, then the 

 arc ZC will be tlie complement of the angle of elevation DSB, and the arc CE 

 will measure the oblique angle DSA, also the arc ZE will be 90"; lience we 

 liave the three sides ZC, CE.EZ, to find the quantity of the angle CZE, which 

 it is evident will be equal to the angle BSA ; for it is proved by the writers on 

 spherics, that the spheric angle CZE is equal to the angle formed by the two 

 tangents intersecting at Z; but these two tangents would be parallel to tlie 

 lines A.S,BS, intersecting at S, therefore the angle CZE is equal to the angle 

 BSA. The triangle in this instance being retlilateral, we may consult Cag- 

 noli'i Trigonometry, p. 2j'1, art.')40 ; or MaiLcIyiie's lutrod. lo Taylor's I..og. 

 p. -12. 



Vol. 28. No. log. J«ne I8O7. C = 1, 



