234 PRINCIPLES OF CRYSTALLOGRAPHY. 



troduction of WhcwellV method of notation of the faces of crystals 

 was an importaut element in Miller's system. 



Miller's symbols consist, as will be explained later, of three indices, 

 which are inversely proportional to the intersections of the faces on the 

 three axes ; while in Weiss's system they are directly proportional. 

 Naumann's and Levy's systems sometimes give sections of the axes and 

 sometimes the relations betweeen two sections. The advantages of Mil- 

 ler's notations are very numerous : Chiefly it allows of representing every 

 individual face; while Naumann's and Levy's symbols give only the 

 Ibrm, i. e., the re-union of all the faces which belong together. When it 

 is necessary to represent the whole form in Miller's system, the symbol 

 of the face is represented in i)arentheses ; it has, therefore, the advan- 

 tage, that, according as it is required, either the face or the form can be 

 exactly and concisely designated. 



Miller's symbol is, besides, exceeding simple and convenient. While 

 here three low whole numbers, 0, 1, and seldom 2, are xuliicient, in 

 Weiss's system three or four fractions, and three or four letters, in groups 

 of three or four, are required, separated by colons : 



i a: h: oo c 



or 



A a' : a' : 'J a' : c 



In Naumann's, two fractions and a letter, witli perhaps as many as four 

 accents, as : 



2 P CO or i V', : 2 

 Levy's symbols are, in mixuy cases, complicated, as in pyramids : 



hi d' dh 



where there are three letters and three fractions. 



Kaumanu's, and Levy's symbols are not symmetrical with regard to 

 the crystallograi^hic axes, /, e., while with Miller the first, second, and 

 third indexes refer invariably to the first, second, and third axes, it is 

 never the case with IsTaamann, and with Levy only in the most compli- 

 cated cases (the pyramid of the second order) that every axis is repre- 

 sented by an index, and even in this case the signs of the axes change 

 their position. This symmetry of axes is important, because it makes 

 both the transformation of the indices in changes of axes, as well as the 

 calculation of zone-equations, excee«lingly simple and demonstrative. 



Singularly enough, this side of ]\Iiller's symbols has been attacked be- 

 cause in Naumann's and Levy's symbols the difference between pyra- 

 mids, prisms, domes, and pinacoids is apparent. This is, however, 

 extremely unjust. In Miller's system, in the symbol of the pyramid, 

 there are three O's of difterent values. In the symbol of a prism or dome, 

 an index is equal to 0; a piuacoid has the syuibol (1 0), (0 1 0), or 

 (0 1), which contains twoO's, and is certainly a difference which strikes 

 the eye. 



As opposed to the notation of Weiss, Miller's method, besides the 



1 Whewell, Phil. Traus., 1^25, p. 87. 



