in the Study of Crystallography 329 



circle No. 3 are in azimuths 240 and 60. The inclination 

 of the faces is equal, and we find that the altitude of circles 

 i, 2, and 3 is 70^. 



The edges (i, 2), (2, 3), and (3, i) are found by measure- 

 ment of the positions of nodes (i, 2), (2, 3), and (3, i); 

 node (i, 2) in (330 and 54^); node (2, 3) in (90, 54^); 

 and node (3, i) in (150, 54^). The theoretical figures are 

 here given because the observed ones have been mislaid. 

 The errors, which did not exceed i, occurred in the quarter 

 sphere of azimuth over 180. This is always the case, because 

 the metrosphere is never exactly true to the globes, and the 

 errors show in proportion as it is shifted from its original 

 position. 



If we consider, as we did in the case of the cube, what 

 form gives in radial projection the same diagram on the 

 globe as the central representation of the tetrahedron, we 

 find that it is the combination of the cube and octahedron "in 

 equilibrium." The radial projection of each of its edges is an 

 arc of 60, and its principal sections are regular hexagons. 



In the central representation of the tetrahedron let one of 

 the great circles be suppressed. There remain three, and 

 they constitute the central representation of the rhombo- 

 hedron, which consists of six rhombs with plane angles of 60 

 and 120. 



When this diagram is looked on as a radial projection, it 

 is that of an octahedron in rhombohedral position, the two 

 basal faces being equilateral triangles, and the six prismatic 

 faces being isosceles triangles in which each of the equal sides 

 is double of the base. 



Let us now consider the pentagonal dodecahedron as it 

 occurs in nature as pyrites, and is designated by Miller 

 7r{oi2}, and by Kopp \ (2a : a : ca). We shall consider 

 one face, which shall be represented by circle No. o, and the 

 five contiguous faces, Nos. i, 2, 3, 4, and 5, which by their 

 intersections give No. o its pentagonal shape. From the 

 specification or measurement of the crystal we have the 

 inclination of the normal of o to the normal of four of the 

 other faces 66 25-3' or 66|, and to the fifth 53 7' 8 ' or 53. 



