16 



S. L. Penfield — Stereographic Projection. 



a' just as many projected degrees beyond the divided circle as 

 a is within it. Thus, as measured on a stereographically pro- 

 jected north and south meridian, a' is antipodal to a, and the 

 problem of finding the center c and drawing a circular arc 

 through a, b, and a\ which is fully illustrated by the figure, is 

 too simple to need more detailed explanation. In some cases 

 it may prove easier to plot a line x y, as illustrated by figure 9, 

 and find upon it the point c by trial, for only one circular arc 

 can be found, which, passing through a and b, intersects the 

 divided circle at antipodal points p and p' . If the two points 

 within the circle are so located that the projected great circle 

 passing through them has a very long radius, the curved ruler 

 described later on can be quickly adjusted so that a circu- 

 lar arc may be drawn passing through them and intersecting the 

 divided circle at antipodal points. 



To measure the Angular Distance between any two points 

 on a Stereographic Projection. — To measure the Side of a 

 Spherical Triangle. — Let the two points a and b, be anywhere 

 within the divided circle, figure 10. Since the angular dis- 

 tance between any two points on a sphere is measured in 

 degrees along the arc of a great circle, it is first necessary to 

 construct a great circle passing through a and b, and thus locate 

 the antipodal points p and p' on the divided circle. It is now 

 possible to find some projected vertical small circle described 

 about p, which passes through a and serves as a measure of the 

 angular distance p to a; likewise a small circle described 

 about p' and passing through b, which serves to measure the 

 distance from p' to b. Knowing the angular distances p to a 

 and p' to b, the distance a to b is readily determined. In fig- 



