The most precise definition of a mineral would be—an inorganic body possessed of a definite chemical composition, and usually of a regular geometric form. Of these, the second is in one respect the direct outcome of the first; while many of the most important physical properties possessed by minerals are outcomes of the second.

Both the geometric form and the composition of minerals are produced and modified under the influence of general laws.

Mineral bodies occur in the three physical conditions of solid, liquid, and gas. Those now found in the last two states are few in number, and are of altogether inferior interest to those which occur as solids; but there is reason to believe that the minerals we know as solids once existed in the liquid or gaseous state, and that their present structure was determined in the process of solidification. All bodies thus formed may be divided into two great classes:—

1. Amorphous bodies, or such as do not possess a definite and characteristic geometrical form. These (when transparent) refract light singly in every direction (except when under stress); they are equally easy or equally difficult to break in all directions; when broken they exhibit a conchoidal or an earthy fracture; they are equally hard throughout all their parts; they are equally elastic in all directions; they conduct heat with equal rapidity and in equal amount in all directions.

2. Crystalline bodies, or such as occur in definite geometrical forms bounded by flat surfaces. These present greater facilities of separation of their particles, or “cleavage,” in certain directions lying in determinate planes than they do in others; most of them are neither equally hard nor equally elastic in all directions, conduct heat more rapidly in certain directions than they do in others, and, when transparent, refract light doubly except in certain directions.

Mineral bodies are found in both of the above classes; and the same mineral body may occur in both the amorphous and the crystalline condition. This is seen in the piece of gold shown in fig. 1, where the upper portion has a sharply angular and a well-defined shape, while the lower presents curvilinear and rugged outlines, similar to one another in no part. Under favouring circumstances, it is possible that every substance whose composition is capable of being represented by a definite chemical fornιula— i.e., which has an unvarying composition—may be capable of assuming a definite crystalline form.

Size and Form of Crystals.— They are of every size from over a yard in diameter to mere specks requiring a high power of the microscope to reveal their existence. Beryls have been obtained in America more than 4 feet in length by 2⅛ in thickness, weighing 2∣ tons. Equally large crystals of apatite have been found in Canada. There is a rock crystal at Milan 3} feet long by 5∣ in circumference, weighing 870 lb. The highest perfection of form, and hence of other properties, is only found, however, in crystals of moderate or of small size.

Variety of Form, and Constancy of Form.— The same mineral may be found in different localities, or sometimes in the same locality, exhibiting an almost endless variety of forms. Calc-spar occurs at a Scottish locality in acicular pyramidal crystals of which the length may be ten or more times as great as the width (fig.

2); in flat plates as thin as paper, in which the length is not the hundredth part of the width; also in prisms, pyramids, and rhombohedra, which at first sight (as in figs. 3, 4) seem destitute of any relationship to each other. This substance has elsewhere been noted in several hundred forms. The minerals fluorite, pyrite, and baryte have each been observed in over a hundred diverse forms. Nevertheless, however great the number, all the forms, in the case of each mineral, may be reduced or referred to a single type, by the simple process of examining its internal structure or the mode of arrangement of its molecules. This is accomplished in two ways—(l)by finding the weak joints in that arrangement, through splitting the crystal, and (2) by measuring the angular inclination of the outside surfaces which bound the form and, from these measurements, by simple mathematical laws, arriving at what has been termed its “primitive” or simplest form.

As regards the mere recognition of a substance, such measurement in itself suffices,—the angular inclination, if the same surfaces be measured, being unvarying in each species. It can, moreover, be shown that the possible range of external variety of form is governed by fixed mathematical laws, which determine precisely what crystalline forms are or may be produced for each species. Comparatively few of these actually occur in nature; but crystallographic laws can point out the range of those which can possibly occur, can delineate them even before they are found, and can in all cases show the relationship which subsists between them and the simple or fundamental form from which or out of which they all originate. It must be observed that in crystalline bodies the internal structure—that is, the arrangement of the molecules— is as regular in an outwardly shapeless mass as in the modelled crystal which presents itself as a perfect whole.

Definitions of Crystals, and their Members or Parts.— A crystal is a symmetrical solid, either opaque or transparent, contained within surfaces which theoretically are flat, and of a perfect polish, but which are actually frequently curved, striated, or pitted. These surfaces are called “planes,” or “faces.” The external planes of a crystal are called its “natural planes”; the flat surfaces obtained by splitting a crystal are called its “cleavage planes.” The intersections of the bounding planes are called “edges,” and planes are said to be similar when their corresponding edges are proportional and their corresponding angles equal. Crystals bounded by equal and similar faces are termed “simple forms.” The cube, bounded by six equal squares, the octahedron, bounded by eight equilateral triangles, and the rhombohedron, bounded by six equal rhombs, are thus simple forms. Crystals of which the faces are not all equal and similar are termed compound forms, or “combinations,” being regarded as produced by the union or combination of two or more simple forms. Edges are termed rectangular, obtuse, or acute, according as the angle at which the faces which form the edge meet is equal to, or greater or less than, a right angle. Edges are similar when the planes by the intersection of which they are formed are respectively equal and equally inclined to one another; otherwise they are unlike or dissimilar.

When a figure is bounded by only one set of planes, it is said to be “developed.” When an edge is cut off by a new plane, it is said to be “replaced”; when cut off by a plane which forms an equal angle with each of the original faces which formed the edge, it is said to be “truncated.” When an edge is cut off by two new faces equally inclined to the two original faces respectively, it is said to be “bevelled.” When a solid angle is cut off by a new face which forms equal angles with all the faces which went to form the solid angle, it is said to be truncated.

In classifying crystals and studying their properties, it is found convenient to introduce certain imaginary lines called “axes.” Axes are imaginary lines connecting points in the crystal which are diametrically opposite,— such as the centres of opposite faces, the apices of opposite solid angles, the centres of opposite edges. Different sets of axes may thus be drawn through the same crystal; but there is always one set, usually of three, but in one special class of crystals of four, axes, by reference to which the geometrical and physical properties of a crystal can be most simply explained. These axes intersect one another, either at right angles, producing “orthometric” forms, or at oblique angles, producing “clinometric” forms. The axes may be all equal, or only two equal, or all unequal.

There is a definite conventional position in which for purposes of description a crystal is always supposed to be held. With reference to this position one of the axes,— that which is erect or most erect,— is termed the “vertical,” and the others the “lateral.” The planes in which any two of the axes lie are called the “axial” or “diametral planes,”—sometimes “sections.” By these the space around the centre is divided into “sectants.” If there are, as is generally the case, only two lateral axes, the space is divided into eight sectants, or octants; but, if there are three lateral axes, it is divided into twelve sectants.

Primitive Forms of Crystals.— If we attempt to arrive, through a study of the internal structure of crystals, as evidenced by directions of weakness of cohesion, at the total number of primitive or parent forms which can exist, we find that there are thirteen such forms and no more.

Nine of these may be regarded as prisms standing upon a base, three as octahedra standing upon a solid angle; and tnere is one twelve-sided figure, or dodecahedron.

Prisms.— Of the prisms eight have a four-sided base

If the base is square and the prism stands erect—that is, if its sides or lateral planes, as they are called, are perpendicular to the base—the form is termed a “right square prism” (fig. 6). In this the four lateral planes are rectangular and equal; they may be either oblong or square; in the latter case the form is the “cube” (fig. δ). When the base is a rectangle instead of a square, the form is a “right rectangular prism’’ (fig. 7). In each of the above three forms the edges are twelve in number. In the cube all the edges are equal. In the square prism the lateral edges are all equal, but are different from the four equal edges of the base. In the rectangular prism, two at each base differ in length from the other two, while both differ from the lateral; hence there are here three sets of edges, four in each. In each of the three forms, however, the solid angles are eight in number, all equal, and each enclosed by three right angles.

When the base is a rhombus, and the prism stands erect, the form is a “right rhombic prism” (fig. 8). Two of the angles in the base being here acute and two obtuse, two of the solid angles corresponding each with each must differ from the others. So also must two of the lateral angles be acute and two obtuse.

The four lateral faces are equal.

When the base is a rhomboid, and the prism stands erect, it is only the opposite lateral faces that can be equal. The form is called a “right rhomboidal prism” (fig. 9).

When the base is a rhombus, but the prism stands obliquely on its base, the form is called an ‘ ‘ oblique rhombic prism” (fig. 10). Here the basal edges of the lateral planes are all equal in length, but on account of the inclination of the prism the angles which these edges form with the lateral edges of the lateral planes are two acute and two obtuse.

If all the edges of an oblique rhombic prism are equal in length to the breadth of the base, and if the lateral planes are rhombi equal in all respects to the basal, the form is called a“rhombohedron” (fig. 11). This is included within six equal planes, like the cube, but these planes have oblique angles. The rhombohedron thus bears the same relation to the oblique rhombic prism which the cube does to the right square prism. Of the eight solid angles of a rhombohedron only two are contained by three equal plane angles, and these two “apices,” as they may be called, are opposite one another. According as the apices are acute or obtuse, we have an acute or obtuse rhombohedron.

When the base of an oblique prism is a rhomboid, the prism becomes an “oblique rhomboidal prism” (fig. 12). In this form, only diagonally opposite edges are similar, as regards equality of length and the value of the included angle. Only opposite solid angles are equal, as are also the opposite and parallel faces.

A right prism may have an equilateral six-sided base; it is then called an “hexagonal prism.” This form may be developed in two positions relatively to each other, —one in which the transverse axes pass from the centres of opposite faces (fig. 13), the other in which they pass from the centres of opposite edges of the planes (fig. 14). The faces of the one set mutually truncate the edges of the other. If a rhombohedron be positioned so as to rest upon one of its apices, the faces of one hexagonal prism would truncate the lateral edges of the rhombohedron, while the faces of the other hexagonal prism would truncate its lateral solid angles. Hexagonal prisms may be longer or shorter than the width of their bases. The interfacial lateral angles are 120°. The angle between the lateral and terminal faces is 90°.

Octdhedra.— The planes of these eight-faced solids are triangular, and they may be regarded as made up of two four-sided pyramids applied to each other, base to base. They are always positioned so that they stand upon a solid angle with the “basal plane ”—that is, the plane which is the common base of the two pyramids—horizontal. In the primitive forms now under consideration the vertices of the two pyramids will in this position be vertically above and below the centre of the base. The upper and lower solid angles are then termed the “vertical solid angles,” and the four lateral solid angles are called the basal solid angles.

There are three octahedrons. In the “regular” octahedron (fig. 15) the base is a square, and the eight faces are equilateral triangles of equal size. There are twelve edges, which are all equal. The faces incline to each other at an angle of 109° 28' 16", and have the plane angles all 60°. There are six equal solid angles. When the base of the octahedron is square, but the other edges, although equal to one another, are either longer or shorter than the edges of the base, the form is a “right square octahedron” (fig. 16). In this the faces are isosceles triangles, the equal angles being at the basal edge of the planes. These basal edges are equal and similar, but differ in length and in angles from the eight equal pyramidal edges. When the base of an octahedron is a rhombus, it is called a “right rhombic octahedron” (fig. 17).

Dodecahedron.— This (fig. 18) has each of its twelve faces a rhombus. It is, like the cube and the octahedron, a solid which is symmetrical. The interfacial angles are all 120°, the plane angles are 109° 28' 16" and 70° 31' 44". The edges are twenty-four, and similar. There are fourteen solid angles, of which six are formed each by the meeting of four acute plane angles, and eight by the meeting of three obtuse plane angles.

It has been said that the above simple forms were arrived at through a study of the internal structure of crystals, chiefly as disclosed by cleavage. Inasmuch, however, as there are some minerals which cleave in only one direction, and many which cannot be cleaved in any direction, this method of investigation fails. Its employment, moreover, frequently led to conflicting or embarrassing results. A conflicting result is when a substance has more than one set of cleavages,—that is, splits up in directions which would result in the production of more than one of the above primary or simple forms. Thus the mineral fluorite occurs with much the greatest frequency in the form of the cube, and it might very consistently be held that its frequent occurrence in this form was a clear natural indication that the cube was the primary or simplest form of fluorite; but it splits up into an octahedron. Galena crystallizes frequently in the form of the octahedron; yet to cleavage galena yields a cubic primary form. It might be conceived that there had been, in each case, some special tendency to assume the cubic form and the octahedral form; but one and the same piece of rock may bear on its surface cubic crystals of fluor and octahedral crystals of galena, — each of the minerals having here assumed the primitive cleavage form of the other in preference to its own. The mineral blende crystallizes not unfrequently in octahedra, which yield the dodecahedron on cleavage. Fluor crystallizes in dodecahedra, yet yields the octahedron to cleavage. Argyrite crystallizes in cubes and in octahedra, but yields the dodecahedron on cleavage. Pyrite crystallizes in cubes, octahedra, and dodecahedra, and yields both the cube and the octahedron on cleavage.

These are most embarrassing results, but they clearly indicate so intimate a relationship to subsist between three of the above simple forms that it is obvious that one alone would serve as a type form for representing the others. The selection of that one should be based upon grounds of most eminent simplicity, and this again is to be arrived at by a consideration of the smallness of number of parts, i.e., of faces, edges, and solid angles. In such a consideration we find that the dodecahedron, with its higher number of each of these, at once gives place. The cube has six faces, the octahedron eight; simplicity here is in favour of the cube. The cube has twelve edges, the octahedron has twelve; in this respect they are equal. The cube has eight solid angles, the octahedron six; here the greater simplicity is on the side of the octahedron. So that this method of adjudicating by simplicity fails, and we are thrown back upon the relationships which may be unfolded through a consideration of the other elements of crystals,— their axes.

Systems of Crystals and Laws of Crystallization.

This consideration led, first, to the remarkable discovery that several of the above primary forms are mere modifications of each other, and ultimately showed that all crystals found in nature may be referred to six systems, based on certain relations of their axes, and that every face which could occur upon a crystal bears a definite and simple relation, in position and in angular inclination, to these axes.

As regards mere geometric measurement, there are several directions in which axes may with nearly equal advantage be projected. For example, in the cube (fig. 19) they may be drawn from the centres of opposite faces, as lettered O; or from opposite solid angles, as lettered C; or from the centres of opposite edges, as lettered D. There is abundance of evidence that each of these directions must be regarded as lines of dominant accretion of molecules.

But the accretion may be not only dominant but overwhelmingly so in one only of these directions in certain cases, or existent along one set of axes alone in certain others. In a specimen of native silver from Alva in Scotland (fig. 20), along O this is so much the case that the concreting molecules have done little more than delineate the form of an octahedron, and this they have only been able to do by aggregating themselves in lines of minute crystals of the very shape of which they were projecting the skeleton form. Moreover, a polar aggregation at...