462 WILLIAM HALLOWES MILLER. 



known fiicts under a definite system. Taking cleavage as his guide, 

 and assuming that the forms of cleavage were not only the primitive 

 forms of crystals as a whole, but also the forms of their integrant 

 molecules, he endeavored to show that all secondary forms might be 

 derived from a few primary forms, regarded as elements of nature, 

 by means of decrements of molecules at their edges. In like manner 

 he showed that all the forms of a given mineral, like fluor-sjiar or cal- 

 cite, might be built up from the integrant molecules by skilfully placing 

 together the primitive forms. Haiiy's dissection of crystals, in a man- 

 ner which appeared to lead to their ultimate crystalline elements, 

 gained for his system great popular attention and applause. The sys- 

 tem was developed with great perspicuity and completeness in a work 

 remarkable for the vivacity of its style and the felicity of its illus- 

 tration. Moreover, a simple mathematical expression was given to 

 the system, and the notation which Haiiy invented to express the 

 relation of the secondary to the primary forms, as modified and im- 

 proved by Levy, is still used by the French mineralogists. 



The system of Haiiy, however, was highly artificial, and only pre- 

 pared the way for a simpler and more general expression of the facts. 

 The German crystallographer, Weiss, seems to be the first to have 

 recognized the truth that the decrements of Haiiy were merely a me- 

 chtuiical mode of representing the fact that all the secondary faces of a 

 crystal make intercepts on the edges of the primitive form which are 

 simjile multii)les of each other; and this general conception, once 

 gained, it was soon seen that these ratios could be as simply measured 

 on the axes of symmetry of the crystal as on the edges of the fun- 

 damental forms ; and, moreover, that when crystal forms are viewed 

 in their relation to these axes a more general law becomes evident, 

 and the artificial distinction between primary and secondary forms 

 disappeai's. 



Thus became slowly evolved the conception of a crystal as a group 

 of similar planes symmetrically disposed around certain definite and 

 obvious systems of axes, and so placed that the intercepts, or parame- 

 ters, on these axes bore to each other a simple numerical ratio. Rep- 

 resenting hy a : b :c the ratio of the intercepts of a plane on the three 

 axes of a crystal of a given substance, then the intercepts of every other 

 plane of this, or of any other crystal of the same substance, conform to 

 the general proportion m a : n b : p c, in which ;», 71, p are three sim- 

 ple whole numbers. This simple notation, devised by "NVeiss, expressed 

 the fundamental law of crystallography, and the conception of a crys- 

 tal as u system of planes, symmetrically distributed according to this 



