On the Gnomonic Projection of the Sphere in Crystallography. 37 



I take the opportunity of noticing that the theorem in sphe- 

 rical trigonometry, which I gave in the February Number, is 

 not new, but, as pointed out by Prof. Chauvenet m the Mathe- 

 matical Monthly (Cambridge, U.S.), is to be found m Cagnoh's 

 'Trigonometry' (1808).— A. C. 



2 Stone Buildings, W.C., 

 June 9, 1S59. 



VIII. On the employment of the Gnomonic Projection of the 

 Sphere in Crystallography. By W. H. Miller, M.A., F.R.S., 

 Professor of Mineralogy in the University of Cambridge*. 

 I rpHE first great improvement in the methods of crystallo- 

 J. graphy, after its establishment as a science, was un- 

 doubtedly made by Mohs and Weiss, independently of each 

 other, in substituting axes for hypothetical integrant molecules, 

 in the enunciation of the geometric laws discovered by Hauy. 

 Next to this in importance was the graphic method mvented by 

 Neumann, and described in his Beitrdge zur Krystallonomie. 

 He indicates the position of any face of a crystal by the outer 

 extremity of a radius of a sphere, drawn perpendicular to the 

 face • and that of a zone, or assemblage of faces intersecting one 

 another in parallel Unes, by a great circle the plane of which is 

 perpendicular to the intersection of any two of the faces which 

 constitute the zone. The several faces of a crystal being in this 

 manner indicated by points upon the surface of a sphere, or by 

 their poles, as it will be convenient to call them, and the zones by 

 great circles passing through the poles of the faces of the zone, 

 the points or poles may be projected upon a plane surface by any 

 of the known methods of projection. Of these the stereographic 

 offers many advantages on account of the facihty and the accuracy 

 with which the distances between the originals of any two pomts 

 may be measured; or the points determined in the projection, 

 having given the mutual inclinations of the faces they represent. 

 In the gnomonic projection, the corresponding constructions are 

 less simple. This projection labours also under the disadvantage 

 that the half of a crystal cannot, as in the stereographic projec- 

 tion, be exhibited on a single surface of finite extent. On the 

 othe'r hand, great circles being projected into straight lines, the 

 zones to which a given face belongs can be very readily ascer- 

 tained ; and the situation of a face common to two zones can be 

 much more easily determined than in the stereographic projec- 

 tion. There are also constructions of great simphcity for find- 

 ing the symbols of points in tlie projection, or for laying down 



♦ Communicatetl by tlio Author. 



