I9I2.] THE LAW OF RATIONAL INDICES. 107 



These are selected because of the good agreement between the 

 measured and calculated angles. Outside of its position in certain 

 zones the only proof of a face lies in this agreement. Ordinarily an 

 agreement as close as ten to thirty minutes of arc is sufficient to 

 establish a face. For the common form-rich minerals, such as 

 orthoclase, tourmaline, fluorite, magnetite, pyrite, barite, anglesite, 

 calcite, aragonite, cerussite, stibnite, hematite, etc., it is certain that 

 some of the faces have complex indices. To be convinced of this 

 fact let one look over the list of forms of the above mentioned min- 

 erals in Goldschmidt's " Krystallographische Winkeltabellen."' For 

 calcite one half of the forms (162 out of 325) have indices greater 

 than 10. The law of simple mathematical ratio is hardly compatible 

 with this fact. 



Many crystals have what are called vicinal faces. These are 

 faces with very high indices which replace faces with very simple 

 indices. Thus apparent cubic crystals of fluorite from the north of 

 England are in reality bounded by faces of a tetrahexahedron with 

 the symbol (32-i-o). Here each cube face is replaced by a very low 

 four-faced pyramid. Vicinal faces are often regarded as accidental 

 or in some way irregular and are usually excluded from the law of 

 rational indices as they are of course inconsistent with the law of 

 simple mathematical ratio. As they lie in prominent zones and as 

 their arrangement conforms to the symmetry of the crystal on which 

 they occur, they can hardly be excluded from the list of faces, though 

 their origin is not clearly understood. The only possible argument 

 for excluding them is that the exact indices of such faces can not 

 always be determined, for the agreement between measured and cal- 

 culated angles must be exceptionally good to establish the face. 

 Miers* found that on alum very flat trisoctahedral faces replace the 

 octahedral faces. In one case the measurements indicated the sym- 

 bol (251 -251 -250). As Miers says, this form can not be regarded 

 as established. It may be some other form with a little different 



^ For recent additions to these lists see Whitlock, School of Mines Quar- 

 terly, Vol. 31, p. 320; Vol. 32, p. 51 (1910). 



^ Phi'.osophical Transactions of the Royal Society, A, Vol. 202, pp. 459- 

 523 (1903)- 



