302 REPORT — 1901. 



that they will most readily separate along zigzai? surfaces whose mean 

 transverse direction is that of normals to prism faces, e.g., AA' in fig. 3 ; 

 and, neglecting the corrugation of these cleavage surfaces, we have three 

 cleavage directions AA', BB', CC, making angles of 60° with each other. 

 Thus the hexagonal cleavage would result from a structure consisting of 

 hexagonal prisms just as well as from one consisting of triangular prisms. 

 The fact that most of the units which Haiiy obtains, whether molecules 

 integrantes or molecules soustractives, display holohedral symmetry 

 shows that there is room for some wider conception as to the ultimate 

 nature of the cleavage surfaces. 



2. Some of the figures to which cleavage leads are neither parallele- 

 pipeda which can be packed together as molecules soustractives, nor other 

 figures -^'hich can be packed together as molecules integrantes. The 

 octahedral cleavage of fluor spar, for example, leads either to octahedra 

 or tetrahedra not fitting closely together, but with spaces between them. 

 This incompatibility of the results of the partitioning with the conception 

 of uniform divisibility into identical plane-faced molecules indicates that 



Fig. 3. 



«> 



the molecules integrantes as well as the molecules soustractives are mere 

 geometrical abstractions ; indeed, such probably was the view of Haiiy 

 himself. 



3. Haiiy s method is not of universal application, since in some crystals 

 no cleavage planes are discoverable. In such cases supplementary hypo- 

 theses become requisite.^ 



Cleavage is, then, an uncertain guide to the determination of the form 

 of the ultimate particles of crystals. Nevertheless, cleavage led to the 

 discovery of the law of rational indices, and the conception of parallele- 

 pipedal units built up into a crystalline structure furnishes at any rate 

 an explanation of this law, and is in accordance with most of the properties 

 of crystals, whether it be derived from cleavage or not. Haiiy's moZecw/es 

 integrantes are even more suggestive, in the light of subsequent research, 

 than his molecules soustractives, since they reduce the problem of crystal 

 structure to a problem of partitioning space into similar polyhedra which 

 are not necessarily parallel. For example, the arrangement of triangular 

 prisms of fig. 1, which is suggested by cleavage parallel to the faces of an 



' Haiiy, Traite de Miniralogie, I. p. 27. 



