S. L. Penfield — StereograpMc Projection. 17 



tire 10, j?to«= 33° 0' and p' to b = 58° 15' ; hence atob = 

 88° 45', or the supplement of the sum of the two angles. 



Without the aid of the special protractor described in the 

 next paragraph, it is a laborious task to find the two projected 

 vertical circles passing through a and b, figure 10 ; though by 

 making use of the protractor No. I and scale No. 2, figure 3, 

 they can be plotted without great difficulty. 



Stereographic Protractor No. II. — To facilitate measure- 

 ments of the angular distances between points on a stereographic 

 projection, or measurements of the sides of spherical triangles, a 

 special protractor has been devised which is represented in 

 plate I, without reduction, and will be designated as the Stereo- 

 graphic Protractor, or Protractor No. II. The essential fea- 

 tures of this protractor are as follows : The circle of 14 ctn diame- 

 ter, divided into degrees, corresponds with the divided circle 

 shown, much reduced, in figure 3. On one-half of the circle, the 

 projected arcs of vertical small circles, p. 11, have been con- 

 structed for every degree. Since on a 14 cm -circle the stereo- 

 graphically projected circles are very near together, the even 

 degrees have been represented by full lines and the odd degrees 

 by dashed lines ; also the arcs of every fifth and tenth degree 

 are engraved somewhat heavier than the others. These details, 

 however, are not essential, but simply serve to make the use of 

 the protractor somewhat easier. On the other half of the cir- 

 cle only the arcs of every fifth and tenth degree are repre- 

 sented. This half of the protracter is really superfluous, but 

 for approximate measurements it will at times be found more 

 convenient than the other half where the arcs are crowded. 

 For convenience the protractor is numbered in two directions, 

 from 0° to 180°. Further, in order to have the protractor 

 really practical, it should be printed or engraved on some trans- 

 parent material ; transparent celluloid has been found to satisfy 

 every requirement. 



To use the protractor in finding the distance from a to b, 

 figure 10, for example, lay it (best with the printed side down) on 

 the drawing, and bring the 0° and 180° points to correspond 

 with the antipodal points p and p' on the divided circle, then 

 note the distances p to a and p to b in degrees and fractions, 

 and the difference between the two readings will equal the 

 angular distance from a to b. 



To measure from any given point p on the divided circle to 

 a point a within the circle, it is not necessary to go through 

 the operation of constructing the arc of a great circle through 

 p and a, as represented in figure 8, p. 14 ; merely place the 

 protractor with its 0° and 180° points on p and p', and then 

 note on the protractor the projected arc which corresponds 

 most nearly to a. 



Am. Jour. Sci.— Fourth Series Vol. XI, No. 1.— January, 1901. 

 2 



