jS. L. Penfield — Stereographic Projection. 21 



I and II, figure 4 and plate I, practically all kinds of problems 

 in spherical trigonometry can be plotted and solved by graphical 

 methods. Two additional protractors, however, are so conve- 

 nient that some space may be well devoted to them. 



Protractor No. Ill, figure 13, is a combination of great and 

 small circles on the same sheet. Every fifth degree only is 

 given, the even degrees being indicated by full, and the odd 

 degrees by dashed, lines. It is printed on transparent celluloid 

 and is intended to be used for making an approximate meas- 

 urement of the distance between any two points on a stereo- 

 graphic projection. At the center, there is a small hole, and 

 if a sheet upon which a stereographic projection has been made 

 is pierced at its center, the protractor can be quickly centered 

 upon the sheet by means of a needle passed through the two 

 holes. Thus centered, the protractor is turned until the two 

 points, the distance between which is to be measured, fall either 

 on an arc of a great circle or at proportionately equal distances 

 between the arcs of two great circles. The distances of the 

 two points from either zero point of the protractor is then 

 noted, as indicated in degrees by the stereographically projected 

 small circles, and the difference gives the distance between the 

 points. Generally speaking, it is not essential to know upon 

 what great circle the two points are located. They must be on 

 some great circle, and the 0° and 180° points of the protractor 

 locate approximately where this great circle intersects the 

 divided circle. It is from the antipodal points thus ascertained 

 that the readings of the protractor are made. 



Given a chart or map in stereographic projection and a pro- 

 tractor, as just described, less than two minutes are required to 

 shift the protractor to the proper position and measure the 

 distance between any two points. Measurements thus made 

 with a protractor graduated only to every fifth degree can be 

 regarded as but approximately correct. They will seldom be 

 more than a degree from the truth, and will average less than 

 half a degree. For quick and approximate work, therefore, 

 this protractor will be very serviceable. 



The great circles for every fifth degree might well appear 

 on that half of protractor No. II, plate I, on which the small 

 circles for every fifth degree are engraved. The advantage, 

 however, of having the two kinds of circles on the same pro- 

 tractor was not appreciated until after the plate for protractor 

 No. II had been engraved. 



Protractor No. IV. — In this protractor, figure 14, the great 

 circles of every alternate degree (second, fourth, etc.) are repre- 

 sented. On the small scale adopted, it would bring the arcs 

 too close together to represent every degree. The arcs are not 

 numbered, but those corresponding to every tenth degree are 

 represented by lines heavier than the rest, so that they may be 



