Law of Definite Mathematical Ratio

It is to be noted that in general the ratio of the intercepts of a crystal face upon the crystallographic axes can be expressed by whole numbers or definite fractions. These numbers, or fractions, are commonly simple, such as 1, 2, 3, ½, 1/3, 2/8, etc., and in the great majority of cases are 1 or ∞. This law, that the axial intercepts of all crystal faces form a definite mathematical ratio, is an extremely important one. It is a necessary corollary to the theoretical considerations given on page 8 and following.

Indices. Various methods of notation have been devised to express the intercepts of any crystal axes, and several different ones are in common use. The most universally employed is the system of indices of Miller. While not as simple for a beginner, perhaps, as some one of the systems in which the parameters of the crystal faces are used, it adapts itself so much more readily to crystallographic calculations and consequently has so wide a use that it seems wise to introduce it here.

The indices of a face consist of a series of whole numbers which have been derived from its parameters by their inversion and, if necessary, the subsequent clearing of fractions. The indices of a and c axes respectively, and therefore ordinarily the letters which indicate the different axes are omitted.

Common use is made of what is known as the symbol of a form. A symbol of any form consists of the indices of the face having the simplest relations to the axes. This is used when it is desired to refer to some particular crystal form, and the symbol then stands for the whole form and not simply for the single face whose indices it is.