2 S. L. Penfield — Stereographic Projection. 



obtained according as four- place or seven-place logarithm 

 tables are employed ; while for some very exact geodetic com- 

 putations, where small fractions of a second must be taken into 

 consideration, ten-place logarithm tables are at times made use 

 of. The advantages of graphical methods over numerical cal- 

 culations are numerous, and are fully appreciated by engineers 

 and others who deal extensively with measurements and prac- 

 tical results derived therefrom. 



The writer would be one of the last to claim that numerical 

 calculations can be dispensed with, yet he contends that, for a 

 large number of problems, especially those where the data are 

 not very exact, results obtained by graphical methods are in 

 every way as serviceable as those secured by calculation. 

 Then, too, it is possible to make computations by graphical 

 methods wholly without the use of formulas and tables, and 

 the processes can be carried out intelligently by persons who 

 have had no special mathematical training, provided only that 

 they have an appreciation of measurements expressed in terms 

 of degrees and fractions. Many advantages to be derived 

 from the use of the stereographic projection will naturally 

 suggest themselves during the course of this paper. In sub- 

 sequent paragraphs some of these advantages will be set forth, 

 and results obtained by plotting will be given, in order that 

 an idea of the accuracy of the method may be obtained. 



General Principles of the Stereographic Projection. — In 

 this method of projection all points and arcs on the surface of 

 a sphere are projected on a flat surface (the plane of the pro- 

 jection) passing through the center of the sphere, the pole or 

 point to which everything is projected being located on the 

 surface of the sphere and at right angles to the plane of the 

 projection. Often the equatorial plane is chosen as the plane 

 of the projection, and the pole to which everything is then 

 projected is the south pole. Under the foregoing conditions, 

 it is also customary to represent only the features of the upper 

 half of the sphere (the northern hemisphere) in the projection, 

 although, as will be shown, the projection may be carried out 

 beyond the equator so as to include the southern hemisphere 

 as well. Projections are likewise frequently made upon a 

 plane passing through some north and south meridian, in 

 which case the pole of the projection will be located upon the 

 equator, at right angles to the plane of the projection. As will 

 be shown, projections can be made without difficulty upon any 

 desired plane. A most important feature of the stereographic 

 projection is that all angular distances and directions, which 

 can be plotted and measured only with difficulty on a spherical 

 surface, appear on the flat surface of the stereographic projec- 

 tion in such relations that they may be easily plotted and 

 measured. This is true of no other method of projection. 



