in the Study of Crystallography 315 



are measured with the goniometer. Let us follow the process 

 as applied to a polyhedron of any number of plane faces, 

 arranged in any way so as to completely enclose the space. 

 The polyhedron or crystal is attached to the goniometer, by 

 means of which we can measure directly the angle included 

 between the normals to any two faces : alongside, we have 

 our globe with its metrosphere, on which we propose to enter 

 the results obtained with the goniometer. We imagine the 

 polyhedron situated inside the globe and concentrically with 

 it ; then the normals to its faces are radii of the sphere, the 

 surface of which they meet in points, which are called the 

 poles of the faces. The angle included between any two 

 such radii is measured by the great circle arc between the 

 two corresponding poles, and it is equal to the angle between 

 the two faces as given by the goniometer. Hence the angles 

 between the faces as measured on the goniometer can be 

 transferred directly to the globe, on which they are laid down 

 as arcs. 



We begin with the globe clean. Any point on it is chosen 

 as the pole of the first face. From this point a great circle 

 is drawn in any direction. When the angle between the first 

 and the second faces has been measured, it is laid down on 

 the globe as an arc of the same number of degrees of the graat 

 circle just drawn from the starting-point. The poles of the 

 two faces, Nos. I and 2, are in the extremities of this arc, 

 which is the base line of the survey of the crystal ; and, by 

 triangulation from it, the poles of all the other faces can be 

 laid down so soon as the angles between them have been 

 measured on the goniometer or otherwise determined. 



Consider a third face, No. 3, adjacent to Nos. I and 2, 

 but not parallel to either of them. Let angles be measured 

 between it and Nos. i and 2, and let these angles be trans- 

 ferred to the sphere as arcs. From pole No. I, as centre, with 

 the arc corresponding to the angles between Nos. I and 3 

 as radius, describe a circle on the sphere : similarly, with 

 No. 2 as centre, and arc equal to the angle between Nos. 2 

 and 3 as radius, describe another circle. These circles cut 

 each other in two points similarly situated on opposite sides 



