272 Penfield — Use of the Stereographic Projection for 



north and S the south pole. By means of lines running from 

 S to the 10° graduation marks of the circle the several points, 

 40° either side of the center, are projected to the line a ft, 

 where they fix the points of a stereographic scale. As far as 

 may be told by the eye, the scale a ft appears equally spaced. 

 The lines of projection are also continued beyond the circle 

 to the intersection with the tangent a' ft', where a stereographic 

 scale is likewise determined, similar to a ft, but relatively twice 

 as long; or again the projection may be carried to a" ft", 

 parallel to a' ft' , where a similar, but still more lengthened, 

 scale is formed. At c d a scale has been interposed which 

 represents spaces equal in length to -g^-ths of the circumference 



19 



A<si 

 10 



minj a diameter as 100 units, 

 ° on arc of a great circle = 8- 



"2G66 



Increase expressed in per cent. 



0° to 10° on stereo 



graphic scale = 



8-74887 



0-25£ 



10 



- 20 " 





S-88383 



1-80 



20 



- 30 " 





9-16222 



4-99 



30 



- 40 " 





- 

 9-00210 



10-03 



~40 



- 50 " 



-u- 



10-233 75 



I ,,, 



50 



- 60 'i 





11-10428 



j 27-25 



60 



- 70 " 





12-28570 



J 4078 



70 



- 80 " 





1388921 



59 16 



80 



- 90 ." 





10-09a04 



J 



of the circle, or, in other words, the distances from space to 

 space on the line c d are the same as along the circumference 

 of the circle for distances of 10°, following the curvature. 

 Close inspection of the scales a' ft' and c d fails to show any 

 appreciable difference between the first lines (10°) either side 

 of the center. If therefore a country to be mapped is not over 

 20° wide, the point of tangency may be taken at its center and 

 the distortion resulting from the stereographic projection will 

 be scarcely appreciable. If a country is 40° wide, at 20° either 

 side of the center there will be perceptible, but still not very 

 great distortion, as indicated by difference in the scales a' ft' and 

 c d. Even if the distance across a map is 80°, the distortion 

 at 40° from the center is marked only near the periphery, and 

 there it is not sufficiently great to seriously impair the useful- 

 ness of the map. 



Still another way of representing the distortion is shown in 

 figure 19. If a sphere is 100 cm in diameter, its circumference 

 measures 314'15927 cm , and 10° of arc would equal 8'72666 cm . 

 Considering a map projected upon a tangent plane of such a 



