VII.] CRYSTALLIZATION. 73 



intercepts of sucli a face stand to each other in the ratios 

 of I ^ and °^^ 



In practice, the angles that the faces of a crystal make 

 on each other are measured by means of Wollaston's 

 reflecting goniometer, and the crystallographic elements 

 are then mathematically calculated from these. 



When the angles at which the axes of any species 

 intersect each other, and the numerical values of the para- 

 meters, are known, all the crystallographic elements ofCrystallo- 

 the species are known : and any form is a possible one iu fj^^ents 

 the species (though it may not have been actually found) 

 in which the angles of the faces on each other are such as 

 to fulfil those conditions. 



Consequently, two kinds of variation of form occur Variations 

 within the limits of the same species. The magnitude of ' 



the faces is subject to no law : and this variability, of 

 course, gives rise to great variability of form. If the 

 typical form of a species, for instance, is cubic, the actual 

 form of any specimen of the species may be that of a 

 square or rectangular prism ; and from such a specimen 

 it will be impossible to tell what the values of the para- 

 meters are. But if there are " secondary faces " cutting 

 off the edges or angles formed by the meeting of the 

 primary ones, the values of the parameters may be calcu- 

 lated from the angles made by the secondary faces on the 

 primary ones. 



The kind of variability of form just described may be 

 regarded as analogous to the irregular variability of the irregular 

 forms of such low organisms as lichens. But besides this 

 irregular variability, many species are regularly variable, f^j._ ^^°^' 

 presenting distinct geometrical forms, of which however 

 the crystallographic elements are the same. In crystals of 

 the cubic system, for instance, the three axes are all at 

 right angles, and the three parameters are all equal ; these 

 are the simplest of all possible crystallographic elements ; 

 and in this system the cube, the regular octohedron, the 

 rhombic dodecahedron, "and some others, are all found. 

 Combined forms are also found, such as cubes with angles 

 truncated by octahedral faces, octahedra with augles 



