108 ROGERS— THE VALIDITY OF [yi^vch i, 



symbol. But according to Miers" these vicinal faces on alum prob- 

 ably have rational indices. 



If we decide to exclude vicinal faces where shall we draw the 

 line? In the zone of (hko) faces, for example, we have a large 

 series of possible faces. (^2-i-o) is undoubtedly vicinal and so 

 perhaps are (25-i-o) and (20-i-o). The faces (i2-i-o) and 

 (lO-i-o) are probably not vicinal, but what of the intermediate faces? 

 Such faces as (i2-9-i), (3-5-ii) and (ii-4-7) can hardly be con- 

 sidered as vicinal, yet they are comparatively complex. There is no 

 exact definition of a vicinal face. As a matter of fact there are all 

 gradations between very simple indices and very complex ones, the 

 limit in complexity apparently being determined by the limits of 

 measurement. There seems to be proof of indices at least as high 

 as 50 (see in the tabulated list apophyllite, danburite and calcite of 

 forms on page 106). 



Finding that the indices of crystal faces are often very large 

 numbers a few authors, notably Viola, ^^ express the opinion that the 

 law of rational indices has no meaning. For of course if we take 

 the indices large enough any plane can be expressed by whole num- 

 bers. It is manifestly impossible to prove by direct measurement 

 that the indices of all crystal faces are rational, for measurements 

 are subject to certain errors, the measured angle rarely ever coin- 

 ciding with the theoretical angle. 



But, as I shall show, there is indirect proof that the indices of 

 crystal faces are rational numbers. Since the time of Haiiy thou- 

 sands of crystals have been measured and among all these crystals, 

 which include both minerals and prepared compounds of the labo- 

 ratory, only axes of 2-fold, 3-fold, 4-fold, and 6-fold symmetry have 



" Philosophical Transactions of the Royal Society, A, Vol. 202, p. 476. 



" One reason why I am led to believe that they are really referable to 

 rational, although not to simple, indices is the following: During the growth 

 of the crystal, one set of vicinal faces is being continually replaced by another 

 along certain zones ; . . . but the change is not gradual, neither are the sur- 

 faces curved ; one plane reflecting surface is replaced by another plane ; and 

 although the images may for a time be multiple and confused, sharply defined 

 images emerge successively by the substitution of one image for another 

 per saltuni." 



^° Loc. cit.. p. 363, "Also kann das allgemeine Gesetz der rationalen Indices 

 keine Bedeutung fiir die Krystallographie haben." 



