212 S. L. Penjield — Drawing of Crystals from, 



evident that, given the poles of a crystal plotted in the 

 gnomonic projection, it would be necessary to draw only one 

 line, the Zeitlinie, and to locate one point, the Winkelpttnkt, 

 W, in order to lind all possible directions for a plan and 

 parallel-perspective views, corresponding to figures 8 and 9. 



In any parallel-perspective drawing cor 

 responding to figures 3 and 9, it is impor- 

 tant to keep in mind that, since the projec- 

 tion is orthographic and made on an inclined 

 surface, there will be some fore-shortening 

 of vertical lengths. Thus, if one has in 

 mind a certain height of a crystal, or the 

 length of a vertical axis c, — c, figure 10, 

 and if XY is the trace of the plane on 

 which the projection is made, the length 

 c,— c would become fore-shortened toe', — cf* 

 The fore-shortening is best done graphically, 

 or it would be the length c, — c times the sine 

 of 10°, provided E, as in figures 1 and 7, is 

 10° from the center. 



The methods of drawing, as developed in the foregoing 

 pages, have been such as to yield parallel-perspective figures 

 essentially like the conventional ones found in treatises on 

 crystallography and mineralogy ; but, as already stated, the 

 plane on which a drawing is to be made may have any desired 

 position, and it may not be out of place to indicate briefly, by 

 an example, how easily the methods may be modified to suit 

 varying requirements. Figure 11 represents a stereographic 

 projection of augite, the forms being a, b, m and s (111), and 

 it is desired to draw a parallel- perspective on a plane parallel 

 to the pyramid face s, 111. Through s draw a diameter, and 

 on it locate E, 90° from s ; then draw the great circle SES' : 

 Under the conditions, the pole corresponding to P of figure 1 

 is s in figure 11. In the parallel-perspective, figure 12, such 

 directions as the edges a' \s and m" A «' are found by noting in 

 figure 11 where the great circles a' s and m" s' cross /SES' 

 (at x and y) and then following out the construction indicated 

 by the figure, as previously explained. 



In figure 12,. the plane angles of s are the same as those of 

 an actual crystal, and the angles, or their supplements, may 

 always be measured on the great circle SE/S' of the stereo- 

 graphic projection. To measure the angle made by the edges 

 a's and m's, figure 12, the great circles through the poles, 

 figure 11, are at right angles, respectively, to the two edges, 

 and their intersections with SE/S' are at x and w, hence the 

 angular distance x to w, measured with a stereographic pro- 

 tractor as 47°, is the supplement of the desired angle, or 133°. 



* Compare figure 5, page 41 ; this Journal (4), xix, 1905. 



