S. L. Penjield — Stereographic Projection. 11 



pole p, 36° from the equator, is readily located on a diameter 

 by means of protractor No. I, p. 7. On just what diameter 

 it should be placed would depend upon some determining 

 factor — for example, the longitude of the meridian. The 

 small circle which is to be projected is x° = 28° from p ; hence 

 it would intersect the north and south meridian at points 8° 

 and 64° from the equator, which points can be quickly located 

 on the same diameter as p by means of protractor No. I. All 

 that now remains to be done is to find the center point and 

 construct the circle. 



Another small circle in stereographic projection is illus- 

 trated by case II, figure 5, B. Here p' is some pole which 

 may be located anywhere within the circle. Draw a diameter 

 passing through p\ bring the base line of protractor No. I to 

 correspond with the diameter, and note the position of p'. In 

 the case under consideration, p' was found to.be 60° 20' from 

 N. The small circle described about p' is distant 45° from p' • 

 hence it will touch the diameter at 15° 20' = 60° 20'-45° 0', 

 and at 105° 20 / =60° 20'+45° 0'. The two points 15° 20' and 

 105° 20' can be located on the diameter by means of scale No. 

 3, figure 3, and it then becomes an easy matter to find the 

 center point and construct the circle. Numerous applications 

 of the principles of the stereographic projection of small 

 circles will appear during the course of this article. 



The Stereographic Projection of Vertical Small Circles. — 

 This is a problem deserving special consideration because of 

 its very general application. The point p, figure 6, A, is 

 located on the equator at the crossing of some meridian ; 

 hence the small circle described about it touches the north and 

 south meridians, passing through p at points a and b, equally 

 distant from p. The same demonstration that was employed 

 on p. 9, figure 5, A, for illustrating that a small circle on the 

 sphere is projected as a circle on the plane of the equator, 

 holds good in the present case. Figures 5, A, and 6, A, are 

 lettered alike, and the demonstration need not be repeated. 

 In order to construct a vertical small circle in stereographic 

 projection, figure 6, B, four points can be readily fixed upon. 

 Two of these are the projection upon some diameter, gey, of 

 the points a and b of the upper figure, the other two being 

 points on the equator at the desired distance, x°, from p. To 

 facilitate the projection of any desired vertical small circle, 

 scale No. 2 of figure 3 has been constructed, from which the 

 radius of the desired circle can be obtained. To construct, 

 therefore, a small circle 36° from p, as represented by figure 6, 

 B, draw a diameter x y through p, and upon it, by means of 

 the scale on the base line of protractor No. I, locate a point 

 36° from p. Then set a pair of dividers so that their points 



