CRYSTALLOGRAPHY MINERALOGY. 3 1 1 



To effect this a circle is drawn through c, and the arc intercepted 

 by A and B on this circle is compared with the whole circumference, 

 equal arcs being measured by their chords ; and an approximate 

 common measure of the arcs is found by dividing each successive 

 difference by the next subsequent one, and so determining as 

 nearly as may be desired, by means of a continued fraction the 

 ratio of the arcs ; half the arc A B then measures the angle between 

 the faces. 



It was to F. E. Neumann, in the first case (1823), that crystallo- 

 graphy was indebted for the~idea of substituting for the somewhat 

 complex geometry of a number of faces forming a polyhedron the 

 fiction of a sphere described round a point within the crystal, 

 while radii normal to the different faces meet the surface of the 

 sphere in points which are the poles of the faces. The recognition 

 of zones of planes, the edges of which are parallel, had already 

 been made by LeVy and by Weiss. Great circles on the sphere 

 of Neumann, along which the poles of tautozonal planes were dis- 

 tributed, could now take the place of zones; and the beautiful 

 work of Grassmann, and subsequently of our great English crys- 

 tallographer, the Professor of Mineralogy at Cambridge, W. H. 

 Miller, in treating crystallography from this point of view, showed 

 how fertile was this principle in results. Among these were the most 

 simple and comprehensive, yet most complete, notation for every 

 face upon a crystal (the notation of C. F. Naumann only desig- 

 nating forms, and these often elaborately) ; and the application of 

 spherical trigonometry to all the ordinary problems of crystal- 

 lography, and therewith the means of handling the relations of 

 planes in zones and of zones with one another in such a manner 

 as to give the fundamental law of crystallography a new form and 

 a new significance as the rationality of the anharmonic function of 

 any four planes lying in a zone. To Grailich we owe the assertion 

 of the principle of the permanency of zones, and therefore of the 

 type of symmetry of a crystal at all changes of temperature. 



Bravais, treating a crystal as a network of molecules, with their 



