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S. L. Penfield — Stereographic Projection. 



27 



center of the United States, figure 25, not from east to west 

 along the 40th parallel, but on the arc of a great circle, the 

 edges of the map are about 45° (one-eighth of a circumference) 

 apart. An arc of 45°, a b, figure 26, appears in stereographic 

 projection as a line a' b'. A stereographic projection, however, 

 can be made on a tangent plane a" b", in which case the 

 linear distance from a" to b" will be twice that of a' to b', 

 while the area on a tangent plane will be four times that of a 

 plane passing through the center. As far as distortion is con- 

 cerned, however, it makes no difference whether a map is made 

 small on a central plane, or with four times the area on a 

 tangent plane, the proportions of the two maps remain the 

 same. Considering the radius of the circle, figure 26, as unity, 

 the distance from a to b, 45° along the circumference, is 0*785, 

 while from a" to b" it is 0*828, a difference of only 0*043. 

 These figures indicate that the distortion resulting from the 

 stereographic projection of a small portion of a sphere upon a 

 plane, for example figure 25, can not be very great. 



Figure 27 shows a stereographic protractor based upon a 

 14 cm circle, the same as that employed in making the map of 

 the United States, figure 25, and sufficiently large to cover all 

 portions of the map. The only great and small circles which 



come into consideration are the ones 

 near the center of protractors II and 

 III, plate I and figure 13. The small 

 circles on one half of the protractor 

 indicate degrees, and on the other 

 half statute miles. A semicircular 

 protractor with the small circles indi- 

 cating either degrees or miles would 

 answer every purpose. Such a pro- 

 tractor would have to be centered on 

 a map of the United States, corre- 

 sponding to figure 25, at 95° W., 

 40° N. ; that is, at the center of the 

 projection. By turning the protrac- 

 tor, some great circle can be found 

 running through any two points under consideration ; and by 

 noting the positions of the points with reference to the small 

 circles, their distance apart may be determined, either in degrees 

 or miles. 



By increasing the width of the map of the United States, 

 figure 25, sixteen times, an ordinary sized atlas sheet, eighteen 

 inches in length, would result. In that case the projection would 

 be based upon a fundamental circle of 224 cm (about 7-J feet) 

 diameter, and to plot the stereographically projected meridians 

 and parallels on such a scale would present no difficulty. The 



