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rotation to the parts supporting the crystal, independently, when necessary, of the movements of the graduated circle, a b.

To use the goniometer, it should first be placed on a pyramidal stand, and the stand on a small steady table, placed about six to ten or twelve feet from a flat window. The graduated circular plate, a b, should stand accurately perpendicular to the window, the pin, hi, being horizontal, with the slit end, í, nearest the eye. Place the crystal which is to be measured on the table, resting on one of the planes whose inclination is required, and with the edge at which those planes meet the farthest from you, and parallel to the window in your front. Attach a portion of wax to one side of the small brass plate, g; lay the plate on the table with one edge parallel to the window, the side to which the wax is attached being uppermost, and press the end of the wax against the crystal, k, till it adheres; then lift the plate, with its attached crystal, and place it in the slit of the pin, h i, with that side uppermost which rested on the table. 'Bring the eye now so near the crystal, as, without perceiving the crystal itself, to permit your observing distinctly the images of objects reflected from its planes; and raise or lower that end of the pin which has the small circular plate, h, attached to it, until one of the horizontal upper bars, m, of the window is seen reflected from the upper or first plane of the crystal, and till the image of the bar, n, is brought nearly to coincide with some line, l, below the window; as the edge of the skirting-board where it joins the floor. Turn the pin, h i, on its own axis, if necessary, until the reflected image of the bar of the window coincides accurately with the observed line below the window. Turn now the small circular plate, e, on its axis, and from you, until you observe the same bar of the window reflected from the second plane of the crystal, and nearly coincident with the line below; and having, in adjusting the first plane, turned the pin on its axis to bring the reflected image on the bar of the window to coincide accurately with the line below, now move the lower end of that pin laterally either towards or from the instrument, in order to make the image of the same bar reflected from the second plane coincide with the same line below.

'Having assured yourself, by looking repeatedly at both planes, that the image of the horizontal bar reflected successively from each, coincides with the same line below, the crystal may be considered as adjusted for measurement. Let the 180° on the graduated circle be now brought opposite the o° of the vernier, by turning the middle plate, d, and while the circle is maintained accurately in this position, bring the reflected image, n, of the bar, m, from the first plane, to coincide with the line, 7, below, by turning the smali circular plate, e. Now turn the graduated circle from you, by means of the middle plate, d, until the image of the bar reflected from the second plane is also observed to coincide with the same line. (Brooke's Crystallography, p. 30.) In this position, the reading of the vernier gives at once the inclination of the two planes to each other. It is almost superfluous to remark that the reflecting goniometer can only be applied in cases in which the surfaces of the crystal have sufficient polish and brilliancy to reflect the image of the line by means of which the angle is read off.

(81) Symmetry of Crystalline Form.-The study of the geometrical relations of different crystalline forms to each other belongs to the science of crystallography. It will be sufficient for the present purpose to indicate the general principle upon which the classification of crystals is founded. This principle is the symmetrical arrangement upon which every crystalline form is constructed.

Symmetry, or a complex uniformity of con

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SYMMETRY OF CRYSTALS.

[81.

figuration (that is, similarity in the arrangement of two or more corresponding forms round a common centre), is the general law of creation, both in the vegetable and animal kingdoms. It is exhibited in the correspondence in external form of the right and left side of the body in animals, in the similar arrangement of the leaf on either side of its midrib, in the two lobes of the dicotyledonous seed, and indeed it attracts the notice of every observer in numberless cases. The same law holds good still more rigidly, though not so obviously, in the constitution of every crystal. If one of the primary planes or axes of a crystal be modified in any manner by molecular forces acting within the liquid or the crystal, all the symmetrical planes must be modified in the same manner.

The imaginary line which thus governs the figure, and about which all the parts are similarly disposed, and with reference to which they correspond exactly, is termed the axis of symmetry in a crystal. If a rhombohedron of Iceland spar be held with one of its obtuse angles uppermost, the vertical line which joins that angle to the opposite obtuse angle is the axis of symmetry of the crystal. Each extremity of the axis is formed by the meeting of three planes, each similar to the others, and all inclined to the axis at an equal angle. If any internal molecular force produce the replacement of any of the edges of one of those faces, the same cause must act with similar intensity upon the corresponding edge of the other faces, and produce a corresponding modification. The variation thus introduced into the form of the crystal has a symmetrical character; and the alteration, which is experienced by each of the three divisions of which the crystal consists, is consequently similar in each case.

There are, however, crystals that possess more than one axis of symmetry; and an arrangement of crystalline form, first proposed by Weiss, and which is now universally adopted, is based upon the relation which these axes bear to each other. These axes, it must be remembered, are imaginary lines, which connect the opposite angles or faces of a crystal, and all of them intersect each other in the centre of the figure. In the regular system, to which the cube, the regular octohedron, and rhombic dodecahedron belong, there are three axes, which are all equal, and cross each other in the centre of the crystal at right angles. If one of the faces or edges upon any of these equal axes be modified, not only are all the faces or edges upon that axis similarly modified, but all the faces and edges of the entire crystal experience a similar modification; since the symmetry of all the

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axes is alike, and the molecular modifying force acts equally upon all. But this rule, though of very general application, is not without exception. If, for instance, a crystal rest upon one face during its formation, the mechanical obstacle to its symmetrical development is frequently the cause of considerable interference with the regular growth in this direction, but this interference does not operate upon the upper and exposed faces. This interference of causes external to the crystal is very generally observed in crystalline masses artificially obtained (75). The crystals of which the mass is composed cross each other in all directions, and form a confused structure, from the surface of which project isolated crystals, one extremity only of which is developed regularly.

Some minerals occur in forms termed pseudomorphous (from audos, a falsehood, uoppn, form); that is to say, they exhibit forms which are not truly related to their own crystalline system. Such pseudomorphous crystals are formed by deposition in cavities previously occupied by crystals of a different nature, but which have been slowly dissolved out of the mass in which they were included, leaving spaces corresponding to their form; and during the process of the solution of the original crystal, or after its completion, the new compound has gradually taken the place, and adapted itself to the form, of the crystal which has undergone removal, as when, for example, quartz is found in the form of heavy spar, fluor spar, cale spar, or spar, or plumbic sulphate.

(82) Classification of Crystals.-Crystals are subdivided into six classes or systems, founded upon the relation of their axes of symmetry to each other. These relations exert an influence not only upon the geometrical connexion of the forms of crystals, but also upon their optical and physical properties. It is necessary in studying crystalline forms, the relations of which are often very complicated, always to place the crystal in a definite position. It will be found most convenient to place the principal axis in a vertical direction. The observance of this rule greatly facilitates the comparison of the compound with the simple forms.

The six classes into which crystals are subdivided are the following:-1st, the Regular or Tessular system; 2nd, the Right square prismatic, or pyramidal; 3rd, the Rhombohedral; 4th, the Prismatic; 5th, the Oblique; 6th, the Doubly oblique.

1. The Regular, or Tessular, or Cubic System, is characterized by three equal axes, a a, a a, a a, figures 56, 57, 58, around which the crystals are symmetrically arranged: crystals of this class are hence often designated as monometric: the three axes

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REGULAR OR TESSULAR SYSTEM.

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cross each other at right angles. Crystals belonging to this system expand equally in all directions when heated, and refract light simply. The most important varieties of simple forms are the cube, as shown in fluor spar, common salt, and iron pyrites (fig. 56, 1): the octohedron (fig. 56, 5), exemplified by alum and FIG. 56.

magnetic iron ore; the tetrahedron (fig. 58, 3), sometimes seen in copper; and the rhombic dodecahedron (fig. 57, 3), as in the garnet and cobalt sulphide. Upon the geometrical relations of these forms, a single instance showing one of the simplest cases of such a connexion will suffice :—

From the cube may readily be deduced the three other allied forms of the regular system. By truncating each of the eight solid angles by planes equally inclined to the three adjacent faces of the cube, we obtain the octohedron, in which the three axes of the cube terminate in the six solid angles of the figure, one of which consequently corresponds to the centre of each side of the cube. (See fig. 56.) The faces marked o are those of the octohedron.

By replacing each of the twelve edges, d d d, of the cube, we arrive at last at the rhombic dodecahedron. (Fig. 57.)

By truncating the alternate angles, t t, we obtain the tetrahedron, as shown in fig. 58.

Homohedral, or Holohedral forms, are those which, like the

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cube and octohedron, possess the highest degree of symmetry of which the system admits. Hemihedral forms, on the other

hand, are those which may be derived from a holohedral form, as the tetrahedron is from the octohedron (fig. 50), or from

the cube (fig. 58), by supposing half the faces of the holohedral form omitted, or its alternate angles or edges replaced, according to a certain law. Again, if half the faces of a hemihedral crystal be omitted, a tetartohedral form is the result.

These relations will be readily traced, even by those unacquainted with geometry, by cutting out two or three cubes in soap, or some other sectile body, and paring down the angles or edges in the manner above described.

In a similar manner, by inserting wires into an apple (fig. 59), we may represent to the eye the direction assumed by each of the axes of a

FIG. 59.

crystal; and by winding a piece of thread round each point of the wires, and stretching the thread across from one wire to another, the outline of an octohedron belonging to any of the systems is readily obtained.