64 



U. S. COAST AND GEODETIC SURVEY. 



In figure 15 let DBEA be the primitive circle and let AB 

 be the line of measures; g is the given point. Construct 

 Cg' equal to Cg and draw Eg' from the point of sight E 

 and prolong it to meet the primitive circle at G; then DG 

 is the arc distance, since all points of polar distance DG 

 are projected into the circle of which the arc gg' forms a 

 part. Therefore, the great circle distance of Cg and Cg' 

 are equal; DG is evidently the polar distance of g' , and 

 hence also of g. If the given point lies on the line of 

 measures the construction is the same as that given for 

 the determination of the great circle distance oi g\ 



Fig. 16.— Projection of a circle with given projection of pole and given polar distance oa 

 stereographic projection. 



Problem 2. — ^To construct the projection of a given circle, 

 its polar distance and the projection of its pole being 

 given: 



In figure 16 let T' be the projection of the pole. NESW 

 is the primitive circle with NS passing through P' and 

 with WE perpendicular to NS; NS is then the line of 

 measures, with W as the point of projection. Draw 

 WP'P and from P lay off the arcs Pp and Pq equal to the 

 given polar distance. Draw Wp and Wq, thus locating 



