in the Study of Crystallography 317 



points which catalogue the edges just as the poles do the 

 faces of the crystal. One diameter of the sphere represents 

 all the edges of the crystal which are parallel to it, therefore 

 the number of the diameters which have been thus entered 

 on the globe is the number of the different or independent 

 edges of the crystal. 



We have here considered the edges formed by the meeting 

 of adjacent faces, or the actual edges occurring on the crystal 

 or polyhedron under measurement. But when every face is 

 marked on the globe by its pole, it is equally easy to determine 

 the direction of the edge which would be formed by the 

 meeting of two faces which are remote from one another ; 

 so that, just as we were able, by measuring the arc between 

 every pair of poles, to catalogue the inclination of every face 

 to every other whether adjacent or remote, so we are able to 

 lay down and catalogue the direction of every edge made by 

 the meeting of every face with every other however they may 

 be situated relatively to each other. 



Again, the diameters representing the edges parallel to 

 them all cut one another in the centre of the sphere. The 

 arc connecting the similar extremities of any pair of such 

 diameters measures the angle included between them, which 

 is equal to thefl/ane angle included by them as edges of the 

 face which they assist to delimit. In this way we obtain a 

 catalogue of all the plane angles occurring on the faces of the 

 crystal ; and they are derived by a simple graphical construc- 

 tion from the observed inclinations of the faces. Further, 

 the diameters representing the edges which bound one face 

 all lie in the same plane ; therefore the extremities of such 

 diameters lie in one great circle. By drawing great circles 

 through all the groups of diameters lying in the same plane, 

 we get a group of great circles, each of which represents the 

 intersection with the sphere of a plane which passes through 

 the centre of the sphere and is parallel to the face of the poly- 

 hedron, which is bounded by edges parallel to the diameters 

 which lie in the plane, and the extremities of which are 

 connected by the great circle. We have thus a catalogue of 

 the faces of the crystal which are inclined to one another, 



