78 



the symmetry-principle come so strongly to the fore, as where 

 crystalline matter is considered: even in early times the typical 

 polyhedral forms of the crystals and their beautiful geometrical 

 shapes made so strong an impression on observers, that for a long 

 while this external form was considered the essential feature of 

 the crystalline state in general. 



Thus crystallographical research was developed primarily by 

 the intense and exclusive study of the polyhedral limiting forms 

 of the crystals; and it was by investigations of this kind that Hauy 

 more than a hundred years ago discovered the fundamental law which 



bears his name, and which gives the 

 key to the remarkable fact above 

 mentioned, that only symmetry- 

 axes with a rather small number of 

 quite determined periods are met 

 with in such crystal-polyhedra. 



2. The law of Hauy, which 

 became the very foundation of 

 modern crystallography, may be- 

 elucidated as follows. 



Let XOY, XOZ, and ZOY 

 (fig. 87) be three arbitrary faces 

 of a crystal, of which faces the 

 intersections are not parallel to 

 the same straight line in space; 

 their edges OX, OY, and OZ 

 intersect in 0. Let ABC be another 

 face of the crystal. The segments 

 a, b, and c, cut off by this plane ABC on the edges OX, OY, 

 and OZ chosen as coordinate-axes, shall fix its position entirely. 

 Now according to the law discovered by Hauy, any other possible 

 face of the 'crystal, let us say A'B'C' , must necessarily fulfil 

 the special condition, that the ratio of the segments OA ' : OB' : OC' 

 be always expressible in the form: ma: nb: pc, the numbers m, n, and 

 p being rational numbers, and in most cases even very simple ones. 

 These numbers m, n, and p are quite sufficient to fix the plane 

 A'B'C' with respect to its direction in space, as determined by the 

 perpendicular from O upon it; and thus, if OA' be taken equal to 

 ma, OB' and OC' will assume the values nb and pc at the same time. 

 In crystallography these numbers m, n, and p are not themselves 



Fig. 87. 



