jS. L. Penjield — Crystal Drawing. 45 



that both time and accuracy might be gained by constructing 

 a suitable protractor, which is shown one-third its natural size 

 in figure 11. At the top is a graduated circle, X^ two of the 

 diameters of which inclined at 18° 26' to the vertical and hori- 

 zontal, represent unit lengths of the a and h axes in orthographic 

 projection. The uses of the circle and its graduation will be 

 explained later. The three large ellipses are the clinographic 

 projections of three circles uniting the ends of the isometric 

 A, —A; B, —JB and (7, — (7 axes; they represent, therefore, 

 the paths which the extremities of the axes would follow if 

 the latter were revolved in the three axial planes. The ellipses 

 may also be regarded as the clinographic projection of three 

 great circles of a sphere ; an equator, crossed by two meridians 

 at 90° to one another. The ellipses and their graduation were 

 plotted with much care, and the engraving was skillfully exe- 

 cuted by Messrs. Bormay & Co. of New York. Each axis is 

 divided into thirds, and a scale giving decimal parts of the 

 vertical axis accompanies the protractor. The quadrant of a 

 small ellipse j^has a radius equal to one-third that of the large 

 ellipse. It is intended for getting one-third the length of an 

 inclined a, axis, but it has not proved to be of much value. 

 Printed on cardboard, the protractor may be used for a long 

 time, it being intended that the axes shall be transferred to a 

 sheet of drawing paper by superimposing the protractor and 

 puncturing the unit lengths of the axes with a needle point. 



The axial protractor has been in use in the writer's laboratory 

 for four years, and has been found very convenient, not only 

 for plotting axes of the monoclinic and triclinic systems, but, 

 also, for constructing the axes of twin crystals. It may be said 

 of the protractor and also of the engraved axes that they have 

 proved to be not only time-savers, but they have also helped to 

 make the work of crystal drawing more accurate and better 

 understood. Students frequently encounter difficulties in crys- 

 tal drawing because the axes with which they are working 

 have not been plotted with accuracy, but by the use of the 

 engraved axes this difficulty, at least, is eliminated. 



A few examples will serve to illustrate the methods of using 

 the axial protractor in plotting inclined axes. 



In both the monoclinic and triclinic systems the same method 

 is used for plotting the a axis at the inclination £, hence one 

 example in the triclinic system will serve for the two classes 

 of crystals. The example chosen is rhodonite, and the data 

 needed are as follows : 



a : b : c=l'073 : 1 : 0*621 

 a=103° 18'; £=108° 44'. 

 ci/^b, 100^010 = 94° 26'. 



