158 Mr. J. Y. Buchanan on the Use of the Globe 



By a well-known rule the excess of the sum of these angles 

 above two right angles divided by four right angles, gives the 

 area of the triangle as a fraction of the surface of a hemi- 

 sphere. Corners delimited by more than three edges can b6 

 specified in the same way by splitting up the polygons which 

 subtend them into triangles. The secondary figures thus 

 described on the surface of the sphere are always different 

 from the primary ones. Thus, the corners of the cube, when 

 collected at, and radiating from the centre of the sphere, 

 delineate the regular octahedron, which in its turn, when 

 similarly treated, delineates the cube. It is a form of 

 inversion. 



It has thus been shown how the globe can be profitably 

 used along with the goniometer. We have put no restrictions 

 on the form of the polyhedron under measurement, and we 

 have been able to render a complete account of the faces and 

 their inclinations to each other, the direction of the edges 

 produced by their meeting, and the plane angles produced by 

 the meeting of pairs of edges. We have also been able to 

 draw a complete and accurate representation of the poly- 

 hedron as projected on the surface of the sphere, which can 

 be studied in detail. A collateral advantage which the 

 student gains by using the globe in the study of crystallo- 

 graphy or of solid geometry generally, is the excellent mental 

 discipline which it affords. A \erj short training of this 

 kind develops enormously the sense of direction. The 

 greatest advantage is reaped by those who combine the use 

 of the globe with the analytical treatment such as given by 

 Miller in his treatise. 



The globe offers great advantages for demonstration. It 

 is very easy to draw correctly the projection of a crystal, 

 especially when it belongs to one of the more symmetrical 

 systems. In the regular system, for instance, which Las 

 the greatest number of simple forms, the edges between the 

 faces of any of these forms are found at once when the poles 

 of the faces have been laid down by drawing arcs of great 

 circles at right angles to the great circle containing any 

 pair of poles and midway between them. In this way an 

 accurate representation of the crystal is quickly obtained. The 

 curvature of the faces is not greater than is frequently met 

 with in nature. Combinations of the simple forms can then 

 be studied with advantage and in great detail. Further, 

 hemihedral forms are as easily drawn as holohedral ones ; 

 more especially, twins, according to any law, can be com- 

 posed and drawn on the globe as easily as simple forms. 

 In short there is no operation in the geometry of crystals 



