4 S. L. Pen field — Stereoyraphic Projection. 



€, and d, respectively, while points more than 90° from 1ST 

 (110°, 135°, and 160°, for example), when projected to the 

 south pole and continued along the lines of projection until 

 they meet the plane of the equator, appear beyond the circle 

 at <?, and points still farther out as indicated by the direction 

 of the arrows. In stereographic projection, figure 2, the 

 equator appears as a circle, the north pole ^occupies a posi- 

 tion in the center, and a north and south meridian is projected 

 upon the plane of the equator as a straight line corresponding 

 to some diameter of the circle, it may be CD, or it may have 

 some other direction, C D' for example, depending, so to 

 speak, upon the longitude of the meridian. Points 20°, 45°, 

 70°, 90°, and 110° from iT, as measured on a north and south 

 meridian, appear in stereographic projection, figure 2, at &, b, c, 

 d, and e, or a' ', b\ c', d\ and e', respectively, the distances of 

 these points from N being equal to those of corresponding 

 points from the center, figure 1, provided that the diameters 

 of the two circles are the same. The diameter C D, figure 2, 

 represents a stereographically projected north and south 

 meridian, and the distances iTto a, b, c, d, and e, indicate 20°, 

 45°, 70°, 90°, and 110°, respectively, as measured on the 

 meridian. The true linear distances N to a, h, etc., are equal 

 to the tangents of half the angles under consideration, the 

 radius of the circle being regarded as unity. The foregoing 

 tangent relation is well illustrated in figure 1, and depends 

 upon an important principle of geometry ; namely, that two 

 lines within a circle meeting at the circumference (as any two 

 lines meeting at S, figure 1) make an angle with one another 

 which is measured by half the arc included between the lines 

 at the circumference. The distances W to a, c to d, and d to e, 

 each represent 20° in stereographic projection, and the rela- 

 tions of these distances should be carefully considered. Pro- 

 ceeding from the center, each stereographically projected degree 

 is somewhat greater than the one just before it, hence the dis- 

 tance c to d just within the equator is nearly twice as great as 

 iTto a near the pole, and d to e just beyond the equator is 

 more than twice as great as N to a. 



From a consideration of figure 2, it is evident that, although 

 any point on the stereographic projection (a, for example) 

 which is 20° from the pole has a fixed relative distance from 

 the center irrespective of the size of the circle, the absolute 

 distance of 20° from the pole will vary with the size of the 

 circle. Hence, the construction of stereographic projections 

 can be greatly facilitated by adopting some definite size for 

 the fundamental circle, and devising certain protractors and 

 scales, by means of which points occupying known positions 

 on the sphere may be quickly and accurately plotted. In 



