S. L. Penfield — Stereographic Projection. 23 



of a stereographic projection ; more especially, however, to 

 locate the antipodal points where the great circle crosses the 

 divided circle. To perform this operation, the protractor is 

 centered by means of a needle over a sheet upon which a 

 stereographic projection has been made. It is then revolved 

 nntil any two points of the projection are on, or proportionately 

 distant from, the same great circle, when, by means of a pencil, 

 marks are made on the drawing where the base line of the 

 protractor crosses the divided circle. From the antipodal 

 points thus located, measurements to the two points within 

 the circle can be made by means of protractor No. II. In the 

 majority of cases it is not necessary to draw the arc of a great 

 circle passing through two points within the circle. To 

 measure the distance between them, however, it is necessary 

 to locate the antipodal points on the engraved circle. On 

 page 15, figure 9, it was shown how a great circle passing 

 through any two points within the circle of a stereographic 

 projection may be plotted. Protractor No. IV furnishes a 

 quick, though perhaps not quite so accurate a method of accom- 

 plishing the same result. 



The Engraving of the Scales and Protractors. — It seems 

 best to give a brief account of the methods employed in making 

 the scales and protractors. The graduation of scale No. 3, 

 figure 3, repeated in part on the base line of protractor No. 

 I, figure 4, was calculated to a fraction of a millimeter by 

 means of a simple tangent relation ; for example, the distance 

 ■of the 20° line from the center equals the radius of the circle 

 multiplied by the natural tangent of 10°, page 4. The 

 method of preparing scale No. 1, figure 3, may be illustrated 

 by referring to figure 7, A* page 13. The distances (from the 

 center) of the stereographically projected 126° point to the right 

 and of the 144° point to the left, as determined by scale No. 3, 

 are added, and their sum divided by two gives the radius of 

 the great circle under consideration. Figure 6, A, page 12, 

 illustrates how scale No. 2, figure 3, was derived from scale 

 No. 3. From the distance (from the center) of the stereo- 

 graphically projected 126° point, that of the 36° point was sub- 

 tracted, and the difference divided by two gives the radius of 

 the small circle under consideration. 



Scale No. 4 is not employed in the stereographic projection, 

 but is used in crystallography. The data supplied for making 

 scales 1, 2, and 3 was employed in the construction of the pro- 

 tractors. Scales and protractors were laid out and engraved 

 by means of a dividing engine. The plates were made by the 

 so-called wax process. In this process, a metal plate, covered 

 with a wax preparation, is employed, and the engraving tools 

 cut through the wax to the metal. Engine work on wax plates 



