If a single ball were placed at rest upon such a stratum, it would be in contact with three of the lower balls; and the lines joining the centres of four balls, so in contact, or the planes touching their surfaces, would include a regular tetrahedron, having all its sides equilateral triangles. The construction of the acute rhombohedron, octohedron, and cube, on the same principle, is as simple as that of the octohedron; and any or all of these solids might be extracted from a sufficient number of such spheres, regularly piled up in the order which is adopted for shot in arsenals, without disturbing their original arrangement (34).

§ 121. Dr. Wollaston also showed that, by substituting for these molecules forms nearly allied to the sphere, such as oblate and oblong spheroids, many forms might be constructed which would not result from perfect spheres, but which are well known to crystallographers. Thus by grouping together oblate spheroids, the proportion of whose axes are as 1 to 2.87, in the same manner as the spheres in the formation of the acute rhombohedron, an oblate rhombohedron would result, whose angles would be those of calcareous spar. All solids thus composed would obviously be split by mechanical force, in directions parallel to their faces (35).

If the elementary spheroids, on the contrary, were oblong, instead of oblate, their centres would approach nearest to each

(34) These figures represent the structure of the tetrahedron, octohedron, acute rhombohedron, and cube, with spherical particles.

other, by mutual attraction, when their axes were parallel, and their shortest diameters in the same plane. The manifest consequence of such a structure would be, that a solid so formed would be liable to split into plates at right angles to its axis, and the plates would divide into prisms of three, or six sides, with equal angles; a structure and a cleavage, which are common to many well-known minerals, as the beril, phosphate of lime, &c.

§ 122. Amongst the physical phenomena which are calculated to lead to a knowledge of the intimate structure of crystallized bodies, we must not forget to mention the discovery of Mitscherlich, of the unequal expansion and contraction of certain classes of crystals by changes of temperature. Crystals belonging to the regular system, which we have imagined to be composed of perfectly spherical particles, expand equally in every direction by heating. Other crystals expand more in one direction than another, and show a tendency to approach to the nearest form of the regular system. Thus the angles in calcareous spar vary 8 between the temperatures of melting ice and boiling water, the obtuse angles diminishing, and the form approximating to the cube. The experiment by which Professor Mitscherlich established this important point is one of considerable delicacy; but common observation will be sufficient to prove it in several obvious instances. Melted litharge, allowed slowly to solidify and cool, when it reaches a particular point flies into minute fragments, from the irregularity of its contraction; and the double sulphate of potash and copper exhibits the same phenomenon in a very marked manner. If a little of this salt be melted in a spoon over a spirit-lamp, and the heat withdrawn, it congeals into a solid, of a brilliant green colour, and remains solid and coherent till the temperature sinks to nearly that of boiling-water, when all at once its cohesion is destroyed, and the whole is resolved into a heap of incoherent powder.

§ 123. As far as it has been examined, the hypothesis of the spherical and spheroidal molecules of crystals has not been found inconsistent with any of the established laws of the action of FORCE; either as regards the attraction which is conceived to group the particles together, or the antagonist forces by which they may be modified, cloven, or dissected; but, for the present,

crystallographers and mineralogists seem to have agreed to drop all speculations with regard to the internal structure and ultimate forms of the molecules of crystals, and confine themselves to the experimental determination, and the geometrical relations of their exterior forms.

In this point of view crystallography is founded upon our idea of symmetry, or a certain definite relation of the parts of a solid, which being no less rigorous and precise than other relations of number and position, is capable of becoming the sure basis of science*. Symmetry is the rule in all the kingdoms of nature, and a little reflection upon the observations of common experience will be sufficient to give such precision to the idea as is necessary for its scientific application.

The bodies of animals consist of two equal and similar sets of members on their right and left sides. Some flowers consist of three or five equal sets of organs, similarly and regularly disposed. Thus the Iris has three straight petals and three reflex ones alternately disposed. The rose has five petals of the corolla; and, alternate with these, as many sepals of the calyx.

This orderly and exactly similar distribution of two, three, and four, or any number of parts, is symmetry: and any departure from such order we look upon as a deformity. The idea is that of regularity; of completeness; of complex simplicity +.

Now all the symmetrical members of a natural product are, under like circumstances, alike affected, and crystallography rests upon this principle: that if one of the primary planes or axes of a crystal be modified in any manner, all the symmetrical planes and axes must be modified in the same manner.

§ 124. The axes of symmetry of a crystal are those lines in reference to which every face is accompanied by other faces, having the same positions and properties. Thus, a rhombohedron of calcareous spar may be placed with one of its obtuse corners uppermost, so that all the three faces which meet there are equally inclined to the vertical line. In this position every derivative face which is obtained by any modification of the faces or edges of the rhombohedron, implies either three or six such derivative faces; for no one of the three upper faces of the rhombohedron has any character or property different from the other two; and therefore there is no reason for the existence of

* WHEWELL'S Philosophy of the Inductive Sciences.

+ Ibid.

a derivative from one of the primitive faces, which does not equally hold for the others. Hence the derivative forms will, in all cases, contain none but faces connected by this kind of correspondence; the axis thus made vertical will be an axis of symmetry, and the crystal will consist of three divisions ranged round this axis, and exactly resembling each other.

§ 125. But this is only one of the kinds of symmetry which crystalline forms may assume. Instead of being uniaxal, they may have three axes of complete and equal symmetry at right angles to each other, as the cube, and the regular octohedron; or two axes of equal symmetry, perpendicular to each other and to a third axis, which is not affected with the same symmetry with which they are; such a figure is a square pyramid; or they may have three rectangular axes, all of unequal symmetry, the modifications referring to each axis separately from the other two; such as a right-rectangular prism. The law of crystalline symmetry is such, that if a face of a crystal be observed to bear a certain relation to one of the axes, other faces must fulfil the same condition with regard to the equal axes; hence, it follows that the forms which are allied to the cube or octohedron, all of whose axes are equal, are few, simple, and of perfect symmetry.

The introduction of this systematic arrangement of crystalline forms according to their degree of symmetry, is due to the concurrent labours of Weiss and Mohs. It is founded upon mathematical relations; but it has been remarkably confirmed by some striking physical properties of minerals, particularly with regard to their action upon light. A table is subjoined of the six systems of Weiss with their allied forms and examples; their relations to heat and light are also added, to the latter of which we shall have occasion to refer hereafter. The figures exhibit the relations of three of the allied forms of each system to the axes of symmetry, a, b; except the first, which only includes the cube and octohedron.