PRINCIPLES OF CRYSTALLOGRAPHY. 239 



This also shows one of the advantagfvs of the metliod of spherical pro- 

 jections, which is entirely wanting in Quenstedt's system. Since, further, 

 Miller's entire method of calculation is based upon spherical trigonometry, 

 the illustrating figure is shown on the projection, which, therefore, at 

 the same time represents the zone-connection of the form and the method 

 of the calculation of the crystal. 



Spherical projection has, finally, the great advantage of being limited, 

 so that the geometrical situation of all faces can be actually delineated, 

 and can be united iti a comprehensive representation, a characteristic 

 which is wanting both in the gnomonic method and that of Quenstedt. 

 In this way alone it is possible to use projection for the introduction of 

 all the i^hysical relations, which circumstance, on account of its increas- 

 ing use, is a very important one. 



A reproach, which, although perhaps not expressed, still is silently 

 made against this method of projection, is that in its construction trian- 

 gles and dividers are necessary, while with Quenstedt's method trian- 

 gles alone are used. This reproach is, however, entirely without foun- 

 dation ; for, in the first place, dividers are necessary for every exact 

 projection, even if only the convenient form provided with steel points; 

 but for general use both triangles and dividers are unnecessary, be- 

 cause on account of the extraordinary simplicity of zone-calculation 

 the adherents of Miller's system of spherical projection are accustomed 

 to use it only for representing and not for investigating existing zones, 

 and they may therefore save themselves the trouble of making an exact 

 drawing, unless they intend to publish. 



To the many advantages of Miller's method no one has yet been able 

 to oppose a disadvantage. If, in spite of this, it has not yet generally 

 found its way into Germany and France, it is owing solely to the fact 

 that in these countries Haiiy, Weiss, and jSTaumann have taught ; but 

 when such completely independent theories are offered, the learner 

 satisfies himself for the most part with the system which has been ex- 

 pounded ; or if he afterward goes beyond that, the system to which he 

 was first accustomed is easier, and his knowledge of it more funda- 

 mental, so that he does not become acquainted wdth many of the advan- 

 tages of the new system. 



The introduction of the Whewell-Miller principle was tried in Germany 

 by Frankenheim, and in France by Bravais and de Senarmont, without, 

 however, any permanent result. Eecently the young german school, on 

 account of the prominence to which the physical examination of crystals 

 has attained, begins to make itself master of detached parts of Miller's 

 method. 



The purpose of the following pages is summarily to develop what is 

 necessary for the solution of a combination, or for the knowledge of the 

 physical nature of a crystal. We shall, therefore, in the first section, 

 treat, according to Miller's method, the pure geometrical relations of a 

 crystal, so far as they are requisite for the determination of combinations. 



