CRYSTALLIZATION. 277 



will be equal isoscles trani;i('s; and if the spheroids be oblate, and the 

 axis halt the liieatest diameter, the tliree angles at the apex of the pyra- 

 mid will be right angles. The erystal will Inive ^ubie syninietry. but the 

 relative eondensation in the faees of the cube, octahedron, and dodeca- 

 hedron, will be as i -.{)%)" A -A)-! oil. Tlie easiest cleavage would there- 

 fore be cubic, as in lock salt and galena. 



Again, if the spheroids ha\'c their axes and greatest diameters in the 

 ratio of 1 : -/-. iii'<^l ^ve i)lacefour, as in Fig. 7, with their axes perpen- 



Fic. 7- 



dicular to the plain' of the figure, then place one upon tliem in the 

 middle, and then four more ui)on it, in positions corresponding to those 

 of the first four, we get a cubical arrangement, the center of a spheroid 

 in each angle of a cube, and one in the center of the cube. Crystals 

 so formed will have cubic symmetry, but the concentration of molecules 

 will be greatest in the faces of the dodecahedron, and their easiest 

 cleavage will be, like that of blende, dodecahedral. 



If spheroids of any other dimensions be arranged, as in Figs. 1 and 

 2, with their axes perpendicular to the plane of Fig. 1, we shall get a 

 crj'stal with the symmetry of the pyramidal system. If the sjdieroids 

 be prolate, the fundamental octahedron will be elongated in the direc- 

 tion of the axis, and if sufficiently elongated, the greatest condensation 

 will be in planes perpendicular to the axis, and the easiest cleavage, as 

 in pi'ussiate of i)otash, in tliose planes. On the other hand, if the 

 spheroids ])e sufticiently oblate, the easiest cleavage will be parallel to 

 the axis. 



If spheroids be arranged, as in Fig. (!, with their axes ])erpendicular 

 to th(^ i)lane of the figure, they will, in general, ])rodnc(^ rhombohedral 

 symmetry, with the rhombs acnte or obtuse, according to the length 

 or shortness of the axes of the spheroids. The cubical form already 

 described is oidy a particular case of the rhombohedral. If the ratio 

 between the axis of the spheroids and their greatest diameters beonly 

 a little greater or a little less than 1 : 2, tlie condensation will be great- 

 est in the faces of the rhombohedron, and the easiest cleavage will be 

 rhombohedral, as in calcite. if the spheroids be prolate, the easiest 

 cleavage will l)e ])erpendicnlar to tlie axis of symmetry, as in beryl and 

 many other crystals. Such crystals have a tendency to assume hex- 

 agonal forms — e([uiangular six-sided prisms and pyramids. To explain 

 this, it may be seen in Fig, (5 that, in i)lacing- the next layer Tii)on the 

 spheroids represented in the figure, the three spheroids which touch that 

 marked a may oc<rui)y eithei- tlie three adjacent white tiiangles or the 

 three black ones. Either position is ci^ually probable. The layer oc- 



