246 Penfield — Use of the Stereographic Projection for 



areas on maps should appear as nearly as possible the same as 

 on a globe. This requires that the spaces included between 

 the projected parallels and meridians should be nearly the 

 same, proportionately, as on a sphere, and that angles are pre- 

 served, or but slightly distorted. Second, if a map is pub- 

 lished with a scale of miles, it should be possible by means of 

 it to measure distances between places with reasonable accu- 

 racy. This is a most important consideration, for most persons 

 must rely wholly upon maps for the estimation of distances. 

 The determination of distances by calculation, requiring a 

 knowledge of spherical trigonometry and the use of formulas 

 and tables of logarithms, is far too laborious to be generally 

 applicable. Lastly, the tracks of great circles, the directions 

 along which the shortest distances between places on a sphere 

 are measured, should be found easily on maps, and angles 

 should be preserved so that bearings may be taken. This last 

 consideration is a most important factor in navigation. In the 

 pages which follow it will be shown to what extent maps based 

 upon the stereographic projection fulfil the aforementioned 

 requirements. 



In a previous communication entitled " The Stereographic 

 Projection and its Possibilities, from a Graphical Stand- 

 point"* the principles of the projection have been set forth 

 by the present writer, and special scales and protractors were 

 described by means of which it was possible not only to plot 

 all kinds of problems in spherical trigonometry with facility 

 and accuracy, but, also, to measure the parts of triangles thus 

 plotted. A considerable portion of the paper was devoted to 

 a discussion of the possibilities of applying the principles of 

 the stereographic projection to map making, and the present 

 article may be considered as an elaboration of the ideas there 

 presented. It is not considered necessary to discuss again the 

 principles of the stereographic projection, but for the benefit 

 of those who may not be familiar with the subject a brief 

 statement of some of its essential features may not be out of 

 place. 



Essential Features of the Stereographic Projection. — Stereo- 

 graphic maps are based upon a very simple principle of 

 projection. The point from which the lines of projection 

 emanate, which may be regarded as the "point of vision, is at 

 the surface of the sphere, and it may be imagined that one is 

 looking as it were through a hollow earth at the surface oppo- 

 site. The plane upon which the projection or map is made 

 must be at right angles to the diameter of the sphere which 

 passes through the point of vision. It is generally convenient 



*This Journal (4), xi, pp. 1-24 and 115-144, 1901. 



