266 Peniield — Solution of ProUems in Crystallography 



subsequently traced with India-ink in order to adapt the work 

 to photo-engraving. The illustrations, therefore, are all from 

 the original sheets on which the plotting was done. 



IsoMETEic System. 



In this system the axes are of equal lengths, and it is neces- 

 sary to determine from measured angles the symbols of the 

 forms, or, having the symbols given, to find the interfacial 

 angles. The fii'st problem that will be presented is one which 

 might, for example, be given to a beginner for the purpose of 

 introducing the subject of mathematical crj^stallography and 

 showing the relations between crystal forms and their angles. 

 It may be supposed that the student is supplied either with 

 models or crystals from which measurements are derived. 



Starting with a simple combination, the poles of the cube 

 are first located at 100, 010 and 001, figure 16, and those of 

 the dodecahedron at distances of 45°, midway between them, 

 the positions of 101 and Oil being determined by the stereo- 

 graphic scale. By means of the small circle protractor it may 

 now be found that the distances between the poles of the 

 dodecahedron are 60°. The octahedron truncates the solid 

 angles of a cube, and is in tiie zones between cube and dodeca- 

 hedron (compare figure 5) ; its poles may therefore be located 

 by drawing the necessary great circles. The pole 11 1 being 

 thus located, it ma}^ be found by the stereographic scale that 

 the distance from 001 to 111 is 54|°, theory 54° 44', and from 

 111 to 110 is 35i°, while like results will be obtained by meas- 

 uring from 100 or 010 with the small circle protractor. Thus 

 a conception is quickly gained, not only of the relations of the 

 important forms, cube, octahedron and dodecahedron, but also 

 of the values of their interfacial angles. 



For simplicity of plotting, the tetrahexahedron is the next 

 best form to consider, and a fluorite crystal with bevelled edges 

 may be chosen as an example for study. The angle of one of 

 the bevelling face to cube is 18° 26', and the poles appear on 

 the stereographic projection in the zones between cube and 

 dodecahedron. To determine the symbol of one of the poles, 

 for example, 310, figure 16, it is only necessary to note the 

 distance from 100 (18° 26') and draw a line from 010 making 

 the same angle with the diameter, which line intersects the 

 front axis (radius) at -J ; the ratio on the first and second axes 

 is therefore |- a : cr, and the indices of the pole are 310. The 

 interfacial angle of the tetrahexahedron, the distance from 310 

 to 301, measured with the small circle protractor, was found 

 to be 25° 50', theory 25° 50'. 



