686 THE PRINCIPLES OF SCIENCE. [CHAP. 



case each molecule will be similarly related to all those 

 which are next to it ; in a second case, it will be similarly 

 related to those in a certain plane, but differently related 

 to those not in that plane. In the simpler cases the arrange- 

 ment of molecules is rectangular; in the remaining cases 

 oblique either in one or two planes. 



In order to simplify the explanation and conception of 

 the complicated phenomena which crystals exhibit, an 

 hypothesis has been invented which is an excellent instance 

 of the Descriptive Hypotheses before mentioned (p. 522). 

 Crystallographers imagine that there are within each 

 crystal certain axes, or lines of direction, by the comparative 

 length and the mutual inclination of which the nature of 

 the crystal is determined. In one class of crystals there 

 are three such axes lying in one plane, and a fourth perpen- 

 dicular to that plane ; but in all the other classes there are 

 imagined to be only three axes. Now these axes can be 

 varied in three ways as regards length : they may be (i) all 

 equal, or (2) two equal and one unequal, or (3) all unequal. 

 They may also be varied in four ways as regards direction : 

 (i) they may be all at right angles to each other; (2) two 

 axes may be oblique to each other and at right angles to 

 the third ; (3) two axes may be at right angles to each other 

 and the third oblique to both ; (4) the three axes may be 

 all oblique. Now, if all the variations as regards length 

 were combined with those regarding direction, it would 

 seem to be possible to have twelve classes of crystals in all, 

 the enumeration being then logically and geometrically 

 complete. But as a matter of empirical observation, many 

 of these classes are not found to occur, oblique axes being 

 seldom or never equal. There remain seven recognised 

 classes of crystals, but even of these one class is not posi- 

 tively known to be represented in nature. 



The first class of crystals is defined by possessing three 

 equal rectangular axes, and equal elasticity in all directions. 

 The primary or simple form of the crystals is the cube, but 

 by the removal of the corners of the cube by planes vari- 

 ously inclined to the axes, we have the regular octohedron, 

 the dodecahedron, and various combinations of these forms. 

 Now it is a law of this class of crystals that as each axis is 

 exactly like each other axis, every modification of any 

 corner of a crystal must be repeated symmetrically with 



