S. L. Penfield — Crystal Drawing. 41 



extremities of the first, a 19 —a 1 axis by laying off distances equal 

 to xw of figure 4, or one-sixth a x x of figure 1, locating them 

 below and above the horizontal line hh. The line a l7 —a J is 

 thus the projection of the first, or front-to-back axis. In like 

 manner determine the extremities of the second axis, « 2 , —a„ 

 by laying off distances equal to one-third xw of figure 4, or 

 oue-sixth a 3 y of figure 1, plotted below and above the line hk. 

 The line a 2 , — a 2 is thus the projection of the second, or right- 

 to-left axis. It is important to keep in mind that in clino- 

 graphic projection there is no foreshortening of vertical dis- 

 tances. This is evident from figure 5, where c, — c is supposed 

 to represent a vertical axis and XY ^ 



the trace of a vertical plane on which 

 the projection is made. The parallel 

 lines of sight, indicated by the arrows, 

 project the axis c, — c to c' ', — c' without 

 change of length. In figure 3 the axis 

 tf 2 , — & 2 is somewhat, and a„ —a x much 

 foreshortened, yet both represent axes 

 of the same length as the vertical, 

 a 3 , — a s , and of the plan above, when 

 plotted in clinographic projection. The 

 completion of the cube about the clinographic axes, as indicated 

 by the construction lines, figure 3, is too simple to need special 

 comment. 



It is wholly a matter of choice that the angle of revolution 

 shown in figure 1 is 18° 26', and that the eye is raised so as to 

 look down upon a crystal at an angle of 9° 28' from the hori- 

 zontal, as indicated by figure 4. Also it is evident that these 

 angles may be varied to suit any special requirement. As a 

 matter of fact, however, the angles 18° 26' and 9° 28' have 

 been well chosen and are established by long usage, and prac- 

 tically all of the figures in clinographic projection, found in 

 modern treatises on crystallography and mineralogy, have been 

 drawn in accordance with them. The development of the axes 

 as indicated by figures 1 to 4 yields the same result as that 

 obtained from following the scheme found in almost all text- 

 books of crystallography, accredited to Naumann.* 



It will be observed that figures 1 and 3 are in vertical align- 

 ment, and one of the chief features of this communication will 

 be to emphasize the value and importance of two projections, 

 orthographic and clinographic. The object of the upper figure 

 or plan is twofold : (1) it may be employed as a help in the 

 construction of the more complex clinographic projection 

 below, and (2) it serves to make clear certain relations which 



* Lehrbuch der Krystallographie, 1830, Band II, p. 400. 



