28 Goniometric Problems. 



horizonlal refraction have been made, and compared with 

 the terrestrial arc, a sufficiently accurate rule may be de- 

 duced for correcting the observations in all cases, without 

 finding how much it amounts to in every instance. Colonel 

 Mudge has taken it at -fVth of the contained arc (Trigono- 

 metrical Survey, parti, p. 3 73); and as the instruments 

 which this ingenious gentleman made use of were of the 

 best kind and used with the utmost care, there seems to hz 

 greater reason to depend upon his results than upon those 

 of any other person. 



In some geodesic operations it happens that we have to 

 resolve a irlangle ivk'ich has one angle very obtuse or very 

 acute. Let ABC (fig. 2.) be an oblique angled triangle, 

 whose vertical angle A is very obtuse ; and having given the 

 two sides together with this angle, let it be required to de- 

 termine the base BC. 



Continue BA to D, and draw CD perpendicular to BD ; 



then BC- = BD" -f CD% but BD* = BA* + AD* +2 AB. 



AD therefore BC*=:AB* + AD* + 2 AB. AD + CD*: and 



since AD* -f CD* = AC^ .-. BC^ = AB* + AC'- + 2 AB 



X AD. But rad. : AC : : r, DAC or BAG : AD = AC x 



c, BAC, which being substituted for its equal, we get BC* 



= AB^ + AC* +2AB. AC. c,BAC. But (Cagnoli, 134.) 



^^^ BA«.^ BAC^ „ , ." , ^ 



c, BAC = 1 f- Sec. : lakmg the first 



two terms, which are sufficient for our purpose, and putting 

 them for their equals, we have BC^ = AB* + AC' + 2 AB, 



AC (I - '■=-—-) = AB^- + AC* 4- 2 AB. AC - ABx 



AC.BAC«=(AB+AC)* -AB. AC.BAC*: extracting the 



square root on both sides of the equation, and neglecting the 



powers of BAC above the square, we get BC =AB + AC — 



AB. AC.BAC* , , , T,A^ • 1 • 



— T-iTr, — rr^ j hut as the atigle BAC is commonlv orivcn 

 3(AB + AL) ' ° " =" 



in minutes and seconds, we must divide it by R' or R" ; 



therefore, taking the latter, putting S = the sum of the 



sides, and P their product, 



P BAC* 



p,p _ c L V 



Which 



