It may be said then briefly that, with respect to external form, the crystals of any one sub stance will have the same symmetry, which will be one of the 32 types, that the angles be tween corresponding faces will be constant, and that the possible shapes will constitute a series in which the positions of the faces are not arbitrary, but fulfil certain conditions (the law of rational parameters).
The methods by which angles are measured and symmetry symbols and constants deter mined are described in brief in the article on CRYSTALLOGRAPHY.
The Dependence of Physical Behavior upon Direction in Crystals as Evidence of Regular Internal Structure.—The physical behavior of a crystal varies in the direction in which the test is applied. Like effects are ob tained in directions which are parallel in the crystal and, generally, unlike effects are pro duced in directions not parallel, and like effects are obtained in directions symmetrically related in the crystal form. All this may be interpreted as similar molecular arrangement in directions giving like effects and differing arrangement in directions giving unlike effect.
These facts were suggested toward the close of the 17th century by the results of the studies of calcite made by Bartholin and Huyghens, which may be summed briefly as follows: A ray of light transmitted through calcite in any direction, except one, is split into two rays usually following different paths so that an object viewed through calcite appears double.
The divergence of these paths (roughly shown by the distance apart of the images) varies with the direction. In one direction, which is also the direction of the principal axis of geometric symmetry, there is single refraction only, and in all directions equally inclined to this direction of single refraction the double refraction (shown by the distance apart of the images) is the same.
If polarized light, in which the vibration direction is known, is transmitted through crys tals, many proofs of the dependence of the effects upon the direction and upon symmetry can be obtained, the regularly varying indices of refraction, the vibration directions of the transmitted light, the differences in absorption and color — in fact, by these effects the sym metry can be accurately judged.
Even with very primitive apparatus Brewster and others obtained before 1820 the so-called °interference figures,° color rings of various shapes crossed by•dark bands, and Brewster was able to classify nearly all those substances to which Haiiy had assigned °primitive forms" and to correct several errors in classification made by the latter. Furthermore, these results were obtained in particular crystallographic directions and the distribution of color and shape of the figures were symmetrical to the planes and axes of geometrical symmetry of the crystals.
A striking proof of the dependence of opti cal behavior upon direction and of a relation between this and crystalline form is shown when polarized light is transmitted through quartz crystals in the direction of the prism axis. In this direction and in•this direction only in all such crystals the emerging ray is polarized in a different plane from the entering ray, or, as it is usually stated, quartz crystals rotate the plane Of polarization a definite number of degrees, dependent upon the thickness. But when crys
tals of quartz which show the less common faces are examined it is seen by the arrange ment of these faces that there are two sorts of crystals, the faces of which are relatively like the right and left hand, and Herschel and Brewster in 1821 showed that the plane of polar ization was turned to the right by the one sort of quartz crystal and to the left by the other.
The dependence of physical behavior upon direction in crystals is not limited to optical phenomena, for instance, spheres of quartz when uniformly heated expand with regularly varying inequality into ellipsoids of rotation, the direction of the principal symmetry axis of the crystal being the axis of rotation. A sphere of glass or opal would have expanded as a sphere.
A very remarkable instance of dependence upon direction and its connection with external form is shown by crystals in which there are different groupings of planes about the opposite ends of certain axes. These axes are not only "polar' in the sense mentioned, but when the crystal is heated, one end of the axis becomes negatively electrified and the other end posi tively. This fact was observed in tourmaline crystals as early as 1762 by dEpinus and the same property is shown by calamine, boracite, quartz and other minerals and always with re spect to an axis which shows different grouping of planes at the two ends.
That the cohesion of crystals varies with the direction and is equally strong parallel to all geometrically equivalent directions is indi cated by several characters. For instance, the directions of least cohesion are frequently shown by the "cleavage* already mentioned under Harry's discoveries and described in a separate article.
The inequality of the cohesion, except in par allel and similar directions, may also be shown by the variations in hardness and by the cracks developed by punching or pressing with a coni cal point, and it is especially shown in the sun building" of the crystal, which takes place under the action of a solvent. Solution proceeds with different velocities in different directions in the crystal and if stopped at an early stage the crys ftl faces are found to be pitted by little angular cavities, the faces of which belong to forms in the same series as the crystal, and not only that but the shapes vary on different faces of the crystal and conform in this and in their shape to the symmetry of the class to which the crystal belongs.
Other characters could be instanced, elas ticity, heat conductivity, electrical and magnetic characters and so on, but these are discussed more at length in the article PHYSICAL CRYSTAL LOGRAPHY. All observations to the same conclusions, that most of the physical characters are dependent upon direction and intimately connected with the external form, or a more exact statement is that the physical .characters and the external form are both results of the same cause, a 'regular internal structure,' which in turn is dependent upon the chemical composition as described in the article on