It will happen in all cases of higher symmetry, as in that of topaz, for instance, that the poles in the lower hemisphere will project into the same points as those representing the faces in the upper hemisphere; but in cases of lower symmetry, where they are differently situated, they are usually represented by miniature rings instead of dots. From the interfacial angles measured on the goniometer the relative lengths and angular inclinations (if other than 90°) of the crystal axes can readily be calculated, by means of the simple formulæ of spherical trigonometry; and the stereographic projection constructed from the measurements as just described proves an inestimable aid to these calculations, by affording a comprehensive diagram of all the spherical triangles required in making the calculations. [Illustration: FIG. 49.—Stereographic Projection of Topaz. ] The relative axial lengths _a_ : _b_ : _c_ (in which _b_ is always arranged to be = 1), and the axial angles α (between _b_ and _c_), β (between _a_ and _c_), and γ (between _a_ and _b_), form the “elements” of a crystal. These, together with a list of the “forms” observed, and a table of the interfacial angles, define the morphology of the crystal, and are included in every satisfactory description of a crystallographic investigation. They are preceded by a statement of the name and chemical composition and formula of the substance, the system and the class of symmetry, and the habit or various habits developed by crystals from a considerable number of crops. An example of the mode of setting out such a description will be found on pages 157 to 160. Having thus made ourselves acquainted with the real nature of the distribution of faces on a crystal, and learnt how the crystallographer measures the angles between the faces by means of the reflecting goniometer, plots them out graphically on a stereographic projection, and calculates therefrom the “elements” of the crystal, it will be convenient again to take up the historical development of the subject so far as it relates to crystal forms and angles, and their bearing on the chemical composition of the substance composing the crystal, by introducing the reader to the great work of Mitscherlich, whose influence in the domain of chemical crystallography was as profound as that of Haüy proved to be as regards structural crystallography. CHAPTER VII THE WORK OF EILHARDT MITSCHERLICH AND HIS DISCOVERY OF ISOMORPHISM.