Mr. FoESTER on the Molecular Formation of Crystals. 495 



planes will appear replacing the edges tangentially (vide Fig. 11). This can 

 be readily demonstrated. 



The face thus formed will belong to the cube. In Fig. 12 we have a repre- 

 sentation of the crystal so produced; it is the same as that in Fig. 11. We 

 have now seen that the union of the cube with the tetrahedron follows as a 

 natural consequence from our hypothesis of twelve poles ; but it would not be 

 at all so easy to see, on the hypothesis of six poles, how four corners of a cube 

 can be modified, and not the remaining four ; — in fact, it would be directly 

 contradictory to the laws of symmetry. We have already seen that the hypo- 

 thesis of six poles explains the cubical cleavage ; but it is a fact that no crystal 

 possessed of cubical cleavage exhibits the tetrahedral form, or that of any of its de- 

 rivatives. We have, hence, a striking confii-mation of our hypothesis. In 

 precisely analogous ways we can deduce tlie octahedron, the dodecahedron, the 

 ikositetrahedron — in fact, every form of the first system, except hemihedral 

 forms with parallel faces, namely, the pentagonal dodecahedron and the hemi- 

 octakishexahedron ; but it is well known that these two forms are never foujul 

 in combination with the tetrahedron, or any other hemihedral form icithout parallel 

 faces. Here is another fact which gives the strongest support to our theory. 

 We thus meet with two remarkable and isolated exceptions, which, viewed as 

 matters of experiment, are sufficiently singular, but which follow as a beautiful 

 and natural result of theory. 



We have already seen that in case of the cubical formation, where the 

 poles are six in number, the cleavage is parallel to the faces. AVe have shown 

 that there will always be some plane of cleavage, and that it will always be 

 the direction of least resistance. But in the case of twelve poles, in the 

 tetrahedral formation before us, it is easily seen that the cleavage is again 

 parallel to the faces ; for there arc only two possible planes of cleavage, namely, 

 parallel to the faces and tangential to the edges ; but cleavage parallel to the 

 faces evidently separates each molecule from thi'ee others ; while cleavage 

 tangential to the edges separates each from four others : the former is, of 

 course, that of least resistance. 



We will now show how spherical molecules may assume the form of the 

 dodecahedron, and will then proceed to prove that this formation will give 

 rise to the dodecahedral cleavage. 



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