It will have been clear from the facts related in the previous chapters that the salient property possessed by all crystals, when ideal development is permitted by the circumstances of their growth, and the substance is not one of unusual softness or liable to ready distortion, is that the exterior form consists of and is defined by truly plane faces inclined to each other at angles which are specific and characteristic for each definite chemical substance; and that these angles are in accordance with the symmetry of some particular one of the thirty-two classes of crystals, and are such as cause the indices of the faces concerned to be rational small numbers.

It will also be clear that, given the presence of any face other than the three axial planes, the symmetry of the class—supposing the crystal to exhibit some development of symmetry and not to belong to class 32, the general case possessing no symmetry—will require the repetition of this face a definite number of times on other parts of the crystal. Such a set of faces possessing the same symmetry value we have already learnt to call a “Form,” and the faces composing it will have the same Millerian index numbers in their symbols, but differently arranged and with negative signs over those which relate to the interception of the back part of the a axis, the left part of the b axis, or the lower part of the vertical c axis; that is, parts to the front and right, and above, the centre of intersection of the three crystal axes are considered as the positive parts of those axes.

A form, if of general character, that is, if composed of faces each of which is inclined to all three axes, will comprise more faces the higher the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the hexakis octahedron, comprises as many as forty-eight faces, all covered by the form symbol {321}; while in the rhombic system the highest number of faces in a form is eight, in the monoclinic only four, and in the triclinic system two. It will also have become clear that the law of rational indices limits the number of forms possible of any one type. For instance, very few hexakis octahedra are known, the most frequently occurring ones besides {321} being {421}, {531}, and {543}. Forms, of any class, possessing higher indices than these are very rare, especially in the systems of lower symmetry.

We next come to a further very interesting fact about crystals. Let us imagine a crystal, on which the faces are fairly evenly developed, to be placed in the middle of a sphere of jelly, as indicated in Fig. 47 (reproduced from a Memoir by the late Prof. Penfield), so that the centre or origin of the axial system of the crystal and the centre of the sphere coincide. Let us now further imagine that long needles are stuck through the jelly and the crystal, one perpendicular to each crystal face, and so as to reach the centre. The crystal represented in Fig. 47 is a combination of the cube a, octahedron o, and rhombic dodecahedron d. If such a thing as we have imagined were possible, we should find that the needles would emerge at the surface of the sphere in points which would lie on great circles, that is, on circles which represent the intersection of the sphere by planes passing through the centre. Moreover, the points would be distributed along these circles at regularly recurring angular positions, corresponding to the symmetry of the crystal. If the crystal belonged to one of the higher systems of symmetry, it would happen that four of the points on at least one of these great circles, and possibly on three of them, would be 90° apart, that is, would be at the ends of rectangular diameters, which would most likely be the axes of reference. The other points would be distributed symmetrically on each side of these four points.

The great circles on which the points are thus symmetrically distributed—and they may legitimately be taken to represent the faces, for tangent planes to the sphere at these points would be parallel to the faces—are known as “zone circles,” and the faces represented by the points on any one of them form a “zone.” Now a zone of faces has this practical property, that when the crystal is supported so as to be rotatable about the zone axis—which is parallel to the edges of intersection of all the faces composing the zone, and is the normal to the plane of the great circle representing the zone—and a telescope is directed towards the crystal perpendicularly to the zone axis, while a bright object such as an illuminated slit is arranged conveniently so as to be reflected from any face of the crystal into the telescope, an image of it being thus visible in the latter, then it will be found that on rotating the crystal a similar image will be seen reflected in the telescope from every face of the zone in turn. Moreover, when the crystal is mounted on a graduated circle, the angle of rotation between the positions of adjustment to the cross-wires of the telescope of any two successive images, reflected from adjacent faces of the crystal, is actually the angle between the two points representing the faces concerned on the zone circle, and is the supplement of the internal dihedral angle between the two crystal faces themselves. It is, in fact, the angle between the normals (perpendiculars) to the two faces, the angle which is measured on the goniometer.

This is, indeed, the very simple principle of the reflecting goniometer, invented by Wollaston in the year 1809, and which in its modern improved form is the all-important principal instrument of the crystallographer’s laboratory. The work with it consists largely in the measurement of the angles between the faces in all the principal zones developed on the crystal. The very fact, however, that crystal faces occur so absolutely accurately in zones immeasurably lightens the labours of the crystallographer, and is one of prime importance.

The most accurate and convenient modern form of reflecting goniometer, reading to half-minutes of arc, and provided with a delicate adjusting apparatus for the crystal, is shown in Fig. 48. It is constructed by Fuess of Berlin.

The graduated circle a is horizontal and is divided directly to 15′, the verniers enabling the readings to be carried further either to single minutes, which is all that is usually necessary, or to half-minutes in the cases of very perfect crystals. The divided circle is rotated by means of the ring b situated below, and the reading of the verniers is accomplished with the aid of the microscopes c. The circle which carries the verniers is not fixed, except when this is done deliberately by means of the clamping screw d, but rotates with the telescope e to which it is rigidly attached by means of an arm and a column f. A fine adjustment is provided with the clamping arrangement, so that the telescope can be adjusted delicately with respect to the divided circle. Both telescope and collimator are rigidly fixed at about 120° from each other during the actual measurements. The collimator g is carried on a column h definitely fixed to one of the legs (the back one in Fig. 48) of the main basal tripod of the instrument. The signal slit of the collimator is carried at the focus of the objective about the middle of the tube g, the outer half of the latter being an illumination tube carrying a condensing lens to concentrate the rays of light from the goniometer lamp on the slit. The latter is not of the usual rectilinear character, but composed of two circular-arc jaws, so that, while narrow in the middle part like an ordinary spectroscope slit, it is much broader at the two ends in order to be much more readily visible; the central part is narrow in order to enable fine adjustment to the vertical cross-wire of the telescope to be readily and accurately carried out. The shape of this signal-slit will be gathered from the images of the slit shown in Fig. 61 (page 126) in Chapter X. The telescope carries an additional lens k at its inner, objective, end, in order that when this lens is rotated into position the telescope may be converted into a low power microscope for viewing the crystal and thus enabling its adjustment to be readily carried out.

The crystal l is mounted on a little cone of goniometer wax (a mixture of pitch and beeswax) carried by the crystal holder. The latter fits in the top of the adjusting movements, which consist of a pair of rectangularly arranged centring motions, and a pair of cylindrical adjusting movements; the milled-headed manipulating centring screws of the former are indicated by the letters m and n in Fig. 48, and those which move the adjusting segments are marked o and p. The top screw fixes the crystal holder.

The crystal on its adjusting apparatus can be raised or lowered to the proper height, level with the axes of the telescope and collimator, by means of a milled head at the base of the instrument, there being an inner crystal axis moving (vertically only) independently of the circle. Moreover, a second axis outside this enables the crystal to be rotated independently of the circle, the conical axis of which is outside this again. The two can be locked together when desired, however, by a clamping screw provided with a fine adjustment q. Freedom of movement of the crystal axis, unimpeded by the weight of the circle, is thus permitted for all adjusting purposes, the circle being only brought into play when measurement is actually to occur. With this instrument the most accurate work can be readily carried out, and for ease of manipulation and general convenience it is the best goniometer yet constructed.

The idea of regarding the centre of the crystal as the centre of a sphere, within which the crystal is placed (Fig. 47, page 62), gives crystallographers a very convenient method of graphically representing a crystal on paper, by projecting the sphere on to the flat surface of the paper, the eye being supposed to be placed at either the north or south pole of the sphere, and the plane of projection to be that of the equatorial great circle. The faces in the upper hemisphere are represented by dots which are technically known as the “poles” of the faces, corresponding to the points where the needles normal to the faces emerge from the imaginary globe, and all these points or poles lie on a few arcs of great circles, which appear in the projection either also as circular arcs terminating at diametrically opposite points on the circumference of the equatorial circle, which forms the outer boundary of the figure and is termed the “primitive circle,” or else, when the planes of the great circles are at right angles to the equatorial primitive circle, they appear as diametral straight lines passing through the centre of the primitive circle.

Such a stereographic projection offers a comprehensive plan of the whole of the crystal faces, which at once informs us of the symmetry in all cases other than very complicated ones. A typical one, that of the rhombic crystal of topaz shown in Fig. 22 (page 40), is given in Fig. 49.

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.

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.