Gonmnetric Prollems. 29 



Which formula will be found extremely useful on a variety 

 of occasions. 



Let there now he given (see fig. 3.) two sides and the in- 

 cluded angle to deterinine the third side, tvhen the gii'C7i 

 angle is very acuta. 



By the same mode of investigation wc get here BC* = 



AB«' + AC« - 2 AB. AC. c, BAG ; and Cagn. 1 54, c, BAG 



BAC* 

 = 1 nearly; whence BC* = AB« + AC* — 



2AB.AC. (1 - -^^j -AB^ + AC^-2AB. AC. + ABx 



AC. BAC*= (AB-AC)*+ AB. AC. BAC% and extracting 



the root of both sides BC = AB - AC + ^-^^'^^^; 



2 (AB — AC) 



dividing BAG by R'' for the reason before given, putting D 



= dlff. ot sides, and P = product, we have 



BC = D + ^ X -^^,: 



From these tvv-o foraiuloe we easily derive others for find- 

 ing the angle in either of the two cases when the three sides 

 are given. Thus the angle being very oltuse, 



BAC" = R" ^il(S-BC); 



and when the angle is very acute, 



BAC" = R" i/^(BC-D). 



In carrying on a series of triangles it is usual to select 

 ?uch objects for the angular points as are most distinctly to 

 be seen from each other on account of their elevation ; and 

 among them it frequently happens that the pointed spire of 

 a steeple, or the flagstaff on the lop of a tower, is cliosen. 

 This object, perhaps, can be observed extremely well from 

 the other two angular points of the triangle, and the spire 

 or flagstaff bisected by the vertical wire of the telescope with 

 the greatest accuracy. But as it is desirable to measure all 

 three of the angles of each triangle, we often find, to our 

 great mortification, that when the instrument is removed to 

 this ihiid object it cannot be so placed as to have its centre 

 immediately under or over the centre of the flagstaff or spire 



that 



