318 On the Use of the Globe 



and the inclination of any two of the great circles measures 

 the inclination of the pair of faces. Also the diameters 

 marked as meeting the surface of the sphere in the great circle 

 supply the number and direction of the edges which bound 

 the particular faces (where it must be remembered that 

 parallel edges are represented by the same diameters) and the 

 plane angles of the face are given by the angles between the 

 diameters. The representation of the faces of the crystal by- 

 great circle planes and that of the edges by diameters, all of 

 which necessarily meet in the centre, facilitates the choice of 

 a suitable system of crystallographic axes. 



If we consider the poles of three adjacent faces, Nos. I, 2, 

 and 3, and draw small circles round them, the radii of which 

 are equal, and of such a length that each circle cuts the other 

 two, then, as before, we have the projection of each edge 

 represented by the arc of the great circle connecting the 

 intersection of each pair of circles. These three arcs cut 

 one another in a point inside of the triangle formed by the 

 intersection of all three small circles. This point is the pro- 

 jection or pole of the corner formed by the meeting of planes 

 i, 2, and 3. If this corner be carried parallel to itself to the 

 centre, its bounding edges will coincide with their parallel 

 diameters, which thus form representative parallel central 

 corners. These diameters meet the sphere in the extremities, 

 which have been already fixed. If these points be connected 

 by arcs of great circles, they determine a spherical triangle 

 whose area is a measure of the corner. 



By a well-known rule the excess of the sum of these angles 

 above two right angles divided by four right angles gives the 

 area of the triangle as a fraction of the surface of a hemi- 

 sphere. Corners delimited by more than three edges can be 

 specified in the same way by splitting up the polygons which 

 subtend them into triangles. The secondary figures thus de- 

 scribed on the surface of the sphere are always different from 

 the primary ones. Thus, the corners of the cube, when collected 

 at, and radiating from the centre of the sphere, delineate the 

 regular octahedron, which in its turn, when similarly treated, 

 delineates the cube. It is a form of inversion. 



