3 1 6 On the Use of the Globe 



of the arc I to 2. The " bearing" of the third face from the 

 first two decides at once which of the two points of intersec- 

 tion is the pole of No. 3 on the sphere. The poles of all the 

 other faces are obtained by an exactly similar construction. 

 When this has been done, we have on the globe a number of 

 points which form a complete catalogue of the faces of the 

 crystal. Similarly, the arcs connecting each pair of poles 

 furnish a catalogue of the inclination of every single face 

 to every other. Every face of the crystal cuts the sphere of 

 projection in a circle having the pole of the face as centre. 

 Let us take the poles of two adjacent faces, Nos. I and 2, and 

 with a pair of compasses draw a circle round each of them 

 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 



