156 Mr. J. Y. Buchanan on the Use of th-e Globe 



with a radius which is greater than half the arc between the 

 two poles. These circles cut each other. Let the points of 

 intersection be joined by the arc of a great circle. That arc 

 is the projection on the sphere of the edge produced by the 

 intersection of the two particular faces, the normal radii of 

 which we have assumed to be equal. 



When the lengths of the radii are taken in any other ratio 

 than that of equality the position of the edge is shifted, but 

 its direction remains the same. It is always perpendicular to 

 the plane containing the normals to the faces, which form it 

 by their meeting. Therefore the great circle which is the 

 projection of the edge is at right angles to the great circle 

 drawn through the poles of the two faces forming the edge, 

 and the direction of the edge is the direction of the axis of 

 this great circle. If we imagine the edge to be carried 

 parallel to itself until it reaches the centre of the sphere, 

 it will coincide with the diameter which is the axis of the 

 great circle in which the poles of the faces lie. Let the 

 points where this axis pierces the surface be marked on the 

 globe. They fix the direction of the edge of the two faces, 

 and of all parallel edges. 



When this construction has been repeated for every pair of 

 adjacent poles on the sphere, we have the projections of all 

 the edges as arcs of great circles. And if they have been all 

 carried parallel to themselves to the centre of the sphere and 

 their extremities then marked, we have another series of 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 bv 

 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 



