272 Penfield — Solution of Problems in Crystallography 



eter: accordingly 'p' was located at 4° 50' from 001 on the 

 horizontal diameter, and by means of the curved ruler a small 

 circle was drawn through 2^' ^'^^ points on the divided circle 

 4° 50' from 100 and 100. The intersections of the small circles 

 shown in figure 19 locate the poles of fonr of the s faces. 

 Through the poles thus found diameters were drawn, which 

 were found to intersect the divided circle at about 11° 10' from 

 100, and a chord from 010 at right angles to one of these 

 diameters intersects the front axis (radius) at just a trifle short 

 of 0-2, or 1. The distance 100 to 510 by calculation is 11° 18'. 

 Erecting a perpendicular at y' and plotting the angle c /\s^ 

 25° 30', from the center, the intersection on the perpendicular, 

 measured with scale ISTo. 4, was found to be 0'093 or y'gth of 

 the vertical axis found, 1*774^-19 = -0933. The relation of 

 the s faces on the axes is therefore \a '. a \ y'gC, giving 5*1'19 

 as the indices. 



Hexagonal System. 



The example chosen for illustrating this system is a crystal 

 of calcite from Egremont, England, shown in figure 20. The 

 angle r Sr^ 74° 55', was selected as the fundamental measure- 

 ment, and the poles of r were located as follows: Since r /\r 

 = 74° 55', af\r^ figure 21, must equal the complement of one- 

 half r t\r^ or 52° 32-J-' ; accordingly, on the radii Oa^ and — a^ 

 the points s were located at 52° 32', respectively, from a^ and 



— «^3, and, using scale ]^o. 2 of the printed sheets, small circles 

 sc were drawn through the points s^ which intersect at r. The 

 distance Or^ measured with the stereographic scale, was found 

 to be 44° 40', calculated 44° 37'. The inclination thus found, 

 plotted from (9, meets the perpendicular from x' at 0*854, cal- 

 culated 0*854. Having located the poles r, the great circles 

 uniting them were constructed. The two scalenohedrons v 

 and c are in the principal zones of the rhombohedron, between 

 r and «, and were located on the stereographic projection by 

 the measurements aAt'=23° 31' and «Ac=10° 34', the small 

 circles c' s' and c" s", respectively, 23° 31' and 10° 34' from 



— (^3, figure 21, serving the purpose. The radius through v 

 and the chord froui a^ at right angles to it establish the rela- 

 tion on the horizontal axes, ^a : a — -J<x, and also the length of 

 the base-line Oy. The slope of the face, the distance Ov^ 

 measured with the stereographic protractor, was found to be 

 69° 00', calculated 69° 2', and this angle plotted from the cen- 

 ter intersects the perpendicular from y' at just a trifle short of 

 0*854, or unity, the discrepancy being due to slight errors in 

 plotting ; for example, the radius through v should have inter- 

 sected the divided circle at 19° 6' from m, whereas, as plotted, 

 it fell just short of 19°, an error perhaps of 8'. The sym- 



