238 PRINCIPLES OF CRYSTALLOGRAPHY. 



Naumanu's iDethod is still more circuitous: first, Weiss's parameters 

 must be calculated, aud then tbe^- must be introduced into the equation — 



a W c" ^ be' a" c a' b" a b" & b c" a' c a" b' 



in which a b c, a' b' c', a" b" c", represent the parameters of the faces. If 

 these numbers contain two figures, as is frequently the case iu the hex- 

 agonal system, there must be twelve multiplications, six divisions, and 

 the addition made. The division must be carried out to four decimal 

 places, and sometimes farther; while in Miller's system the convenience 

 of a calculation with whole numbers is always secured. 



This circuitous course has caused the adherents of the schools of Nau- 

 mann and Weiss, to this day, to use Quenstedt's method ; and they are 

 contented with an approximative zone-verification, while, since the foun- 

 dation of Miller's method, even the beginner is both capable of and 

 accustomed to verify ^very zone by means of the exceedingly simple 

 calculation of zone-equations. In fact, Kohscharow/ iu the year 18CG, 

 again first called attention to the zone-verification calculation, which, 

 since the publication of Weiss, had been almost entirely forgotten ; von 

 Eath,2 Hessenberg,^ and C. Klein^ followed, replacing the construction 

 iu specially -complicated cases by calculation. 



The use of the angle of the normal to the faces, instead of the interior 

 angle of the solid, is also important : in the first place, with respect to 

 conveuience and coi^ciseness, while, as a rule, the interior angle is 

 greater than 100°, and therefore contains three figures, the angles of 

 the normals have, for the most i^art, two figures ; further, the angles 

 measured at present with the reflecting-goniometer are for the most 

 part angles of normals. In the simple evaluation of an angle with the 

 eye even the supplement is easier to estimate than the real angle, be- 

 cause it is generally smaller. 



The most important advantage of normal angles is, that they can be 



immediately introduced into the calculation. This is especially apparent 



p. in tautozonal faces, in which, from two angles of every 



two out of three tautozonal faces, the third can be had 



by simple addition or substraction, (Fig. 1,) as — 

 <^ab + ^b c = <^ac. <^ a e — <^ a b = <^b c 



which is not the case with the angles made by the 



faces themselves. 



In the determination of combinations a very quick 



orientation is furnished by this method. Lastly, only 

 normal angles are suitable for introduction in spherical projections, 

 where they themselves directly form the sides of the spherical triangle. 



1 Von Kohscbarow, Materialien zur Mineralogie Russlantls, 1866, p. 216. 



2 Von Rath, Pogg. Ann., cxssii, 1867, p. 393. 



3 Hessenberg, Min., Not. ix in Seucheub. Ges., Abh. vii, 1870, p. 259. 

 * Klein, loc. cit., p. 481. 



