Geographical Maps and Sailing Charts. 273 



sphere, figure 18, a distance 10° out from the point of tangency 

 would be 8-T488T cm , or but 0*02221 cm longer than the distance 

 of 10°, following the curvature. In like manner in figure 19 

 other 10° spaces of a stereographic scale are expressed in lengths, 

 carefully plotted to scale, and the accompanying percentages 

 give the increase over the length of 10° of arc. Thus it may 

 be seen that in any of the stereographic hemispheres shown in 

 earlier pages of this article there is greater distortion in the 

 space between 70° and 90°, measured out from the center along 

 a radius, than from the center out to 70°. 



The foregoing examples serve to indicate that near the center 

 of a stereographic map of any hemisphere there must be very 

 little distortion, and in order to study the possibilities of a 

 stereographic map of limited area, it was decided to make a 

 projection of the United States on a large scale. The scale 

 adopted was based upon a sphere of 1*8 meters (5*9 feet) radius, 

 supposing the projection as made on a central plane, figure 16, 

 page 269, and this particular scale was chosen in order to give 

 a map corresponding in size with one issued by the U. S. Geo- 

 logical Survey, accompanying the Twenty-first Annual Report 

 of the Director, 1899-1900. The dimensions of the govern- 

 ment map are 27 j- by 16f inches, and parallels and meridians 

 are drawn two degrees apart. Although not stated, it bears 

 evidence of being based upon the Polyconic Projection which 

 is used by the U. S. Coast and Geodetic Survey. The central 

 meridian is 97° W. and its intersection with the parallel 39° !N". 

 is approximately the center of the map. 



Projection of a Map of the United States upon the Plane 

 of the Horizon at 39° JV., 97° W.— The essential details of 

 making this kind of a map are illustrated by figure 20. The 

 upper circle represents a vertical section (an elevation) through 

 the N. and S. poles of a sphere, along what is to be the 

 central meridian of the map. As 39° IS", is to be the center of 

 the map the JV.S. pole is inclined 51° (90°-39°) from the ver- 

 tical, and intersections of the meridian with the equator, EE\ 

 the poles N.S., and the trace of the central plane upon which 

 the map is projected, N'S', are indicated by the graduation of 

 the circle, i 3 , at 39° S., is the point of vision. Numerous 

 points on the central meridian are projected to the line JV'S', 

 thus determining points which are needed in the construction 

 of the map. The lower part of the figure represents the plane 

 upon which the map is made, and, taken in connection with 

 the elevation above, may be considered as a plan. The line 

 N" S" , parallel to N'S ', is the projected central meridian, JS" 

 and S" being respectively the stereographically projected north 

 and south poles. The circle 7, of the same size as the meridian 

 above, would bound the hemisphere, if the latter were fully 



