in the Study of Crystallography . 165 



the crystal the augle of inclination of these two faces is a 

 right angle. Therefore in its original position the meridian 

 of the metrosphere coincides with great circle No. 1. Let it 

 he drawn. The third independent face, No. 2, which we 

 find on the crystal is inclined to each of the other two at a 

 right angle. With the metrosphere in its original position, 

 clamp the movable quadrant at 90° of azimuth, the quadrant 

 coincides with circle No. 2. It is generally more convenient 

 to bring the meridian to coincide with the great circles to be 

 drawn, because then the complete semicircle is drawn at 

 once, and both nodes are marked at the same time. For this 

 purpose the metrosphere is rotated from its original position 

 round the axis of the equator, which remains coincident 

 with circle No. 0, until division 90° of azimuth coincides with 

 node (0, ]/)• The meridian now corresponds with the position 

 of circle No. 2. Let it be drawn. Circle No. 2 obviously 

 cuts No. 1 at right angles, which can be at once verified by 

 measurement of the arcs between node (1, 2) and nodes (0, 2) 

 and (0, 2 ; ). If we attempt to place any of the three remain- 

 ing faces, we find that its place is already occupied by one of 

 the circles, 0, 1, or 2, to which it is parallel. If we now wish 

 to specify the crystal in terms of its central representation on 

 the sphere, we have, for the inclination of the faces to 1, — 90°; 

 to 2, -90°; and 1 to 2, -90° ; and, for the position and 

 direction of the edges : — edge No. (0, 1) azimuth 0°, alti- 

 tude 0°; edge (0, 2) - (90°, 0°) ; and edge (1, 2) azimuth 

 indeterminate, alt. 90°. The plane angle made on any face 

 by any other two is equal to the arc on the corresponding 

 circle contained between the nodes made by it on meeting the 

 two other corresponding circles. In the present case they are 

 found by measurement, as they are seen by inspection to be 

 right angles. 



In representing the cube by great circles parallel to its 

 faces, we have divided the hemisphere into four equal and 

 similar triangles. If we regard this diagram as the spherical 

 projection of a polyhedron where all the faces, whether 

 independent or not, are represented — where in fact the faces 

 have been delineated by drawing small circles round their 

 poles, and arcs of great circles through the intersections of 

 these circles to give the edges, we find that we have here the 

 radial projection of the regular octahedron. The trans- 

 ference of the faces of the cube parallel to themselves until 

 they coincide with planes of great circles, has the effect of 

 transferring the corners of the cube parallel to themselves to 

 the centre. If the extremities of each set of three edges 

 which go to form a corner are connected by arcs of great 



