GEOMETRY. 363 



From the points A and B as centres, and with 

 any opening of the compasses greater than half 

 A B, describe arches cutting each other in c and d. 

 Draw the hne c d ; and the point E, where it cuts 

 A B, will be the middle required. 



Prob. 2. To raise a perpendicular to a given line 

 A B, from a point given at C. 



Case 1. When the given point is near the middle 

 of the line. On each side of the point C, take any 

 two equal distances C ^ and Ce; from d and e, 

 with any radius or opening of the compasses greater 

 than C df or C e, describe two arcs cutting each 

 other in Jl Lastly, through the points^ C, draw 

 the line J^C, and it will be the perpendicular re- 

 quired. 



Case 2. When the point is at or near the end of 

 the line. Take any point d, above the line, and 

 with the radius or distance dCy describe the arc 

 e CJf cutting A B in e and C. Through the centre 

 d, and the point e, draw the line e df, cutting the arc 

 e Cfinf. Through the points^ C, draw the line 

 fC, and it will be the perpendicular required. 



Prob. 3. From a given point f, to let fall a per- 

 pendicular upon a given line A B. 



From the point J^ with any radius, describe 

 the arc de^ cutting A B in e and d. From the 

 points e dy with the same or any other radius, 

 describe two arcs cutting each other in g. Through 

 the points yand g\ draw the liney^'; andjTC will 

 be the perpendicular required. 



Prob. 4. To make an angle equal to another 

 angle which is given, as « B ^. 



From the point B, with any radius, describe the 

 arc a b, cutting the legs B «, B Z), in the points a 

 and b. Draw the line D e, and from the point D, 

 with the same radius as before, describe the arc ej^ 



