Crystallography, the science of form and structure in the inorganic kingdom of nature. In the organic kingdoms, the animal and vegetable, each species has a specific form and structure evolved from the germ according to a law of development or growth. In the inorganic kingdom also, which includes all inorganic substances, whether natural or artificial, a specific form and structure belong to each species, and the facts and principles involved therein constitute the science of crystallography. The forms are called crystals; so that animals, plants, and crystals are the three kinds of structures characterizing species in nature. As the qualities of crystals depend directly on the forces of the ultimate molecules or particles of matter, crystallography is one of the fundamental departments of molecular physics, and that particular branch which includes cohesive attraction. Cohesive attraction in solidification is nothing but crys-tallogenic attraction, for all solidification in inorganic nature is crystallization. The solidification of water, making ice, is a turning it into a mass of crystals; and the word crystal is appropriately derived from the Greek ice. The solidification of the vapors of the atmosphere fills the air with snow flakes, which are congeries of crystals or crystalline grains. Solid lava, granite, marble, iron, spermaceti, and indeed all the solid materials of the inorganic globe, are crystalline in grain; so that there is no exaggeration in the statement that the earth has crystal foundations. The elements and their inorganic compounds are, in their perfection, crystals. Carbon crystallized is the diamond. Boron is little less brilliant or hard; and could we reduce oxygen to the solid state, it would probably (as we may infer from its compounds) have no rival among nature's gems. Alumina is the constituent of the sapphire and ruby, and silica of quartz crystals. Magnesia also has its lustrous forms. The metals all crystallize. Silica and alumina combined, along with one or more of the alkalies or earths, make a large part of the mineral ingredients of the globe, its tourmaline, garnet, feldspar, and many other species, all splendent in their finer crystallizations; and limestone, one of the homeliest of all the earth's materials, as we ordinarily see it, occurs in a multitude of brilliant forms, exceeding in variety every other mineral species. - The general principles in the science of crystallography are the following: I. A crystal is bounded by plane surfaces, symmetrically arranged about certain imaginary lines, called axes.
II. A crystal has an internal structure which is directly related to the external form, and the axial lines or directions. This internal structure is most obviously exhibited in the property called cleavage. Crystals having this property split or cleave in certain directions, either parallel to one or more of the axial planes, or to diagonals to them; and these directions are fixed in each species. In some cases, cleavage may be effected by the fingers, as with mica and gypsum; in others, by means of a hammer with or without the aid of a knife blade, as in galena, calcite, fluor spar; in others it is indistinguishable, as in quartz and ice. In all species, whether there be cleavage or not, crystals often show a regular internal structure through the arrangement of impurities, or by internal lines, striations, or imperfections; and, when there has been a partial solution or erosion of the crystal, there is often a development of new lines and planes, indicating that the general symmetry of the exterior belongs to the whole interior. III. The various forms of crystals belong mathematically to six systems of crystallization: the isometric, tetragonal or dimetric, Orthorhombic or trimetric, mono-clinic, triclinic, and hexagonal.
The greater part of the crystalline forms may be regarded as based on four-sided prisms, square, rectangular, rhombic, or rhomboidal in base; and the rest on the regular six-sided prism. The four-sided prisms are either right prisms (erect) or oblique (inclined). Any such four-sided prism may have three fundamental axes crossing at the centre, one vertical axis connecting the centres of the opposite bases and two lateral, connecting the centres of either the opposite lateral faces, or the opposite lateral edges. The six-sided prism is right, and has four axes, one vertical and three lateral. In the right four-sided prisms, the intersections of the axes are all at right angles; in the oblique, one or all of them are oblique angles. A. Right or orthometric systems. 1. Isometric system: the three axes equal, and thus of one kind. The system is named from the Greek equal, and measure. The cube (fig. 1), contained under six equal square faces, the regular octahedron (fig. 2), under eight equal triangular faces, the dodecahedron (fig. 3), under twelve equal rhombic faces, are examples of the forms. The three axes in the cube connect the centres of the opposite faces; in the regular octahedron they connect the apices of the solid angles; in the dodecahedron, the apices of the acuter solid angles. Examples: garnet, diamond, gold, lead, alum. 2. Tetragonal or dimetric system: one axis, called the vertical, unequal to the other two, or lateral, and the lateral equal; the axes thus of two kinds. The term dimetric is from the Greek twice, andmeasure. The square prism (fig. 4) is an example. As the base is a square, the lateral axes, whether connecting the centres of opposite lateral faces or edges, are equal; while the vertical may be of any length, longer or shorter than the lateral.
Under this system there are square octahedrons (fig. 5), equilateral eight-sided prisms, and eight-sided double pyramids (fig. 6), besides other forms. Examples: idocrase, zircon, tin. 3. Orthorhombic or trimetric (Gr. three times, and ) system: the vertical axis unequal to the lateral, and the lateral also unequal, or in other words, the three unequal. In the rectangular prism (fig. 7, a right prism with a rectangular base), the three axes are lines connecting the centres of opposite faces, and are unequal. In the right rhombic prism (fig. 8) the vertical axis connects the centres of the bases, and the lateral, the centres of the opposite lateral edges. Fig. 9 represents a rhombic octahedron, another form under this system. Of the two lateral axes in this system, the longer is called the macrodi-agonal, and the shorter the brachydiagonal. Examples: sulphur, heavy spar, Epsom salt, topaz. B. Oblique or clinometric systems. 4. Monoclinic system: one only of the intersections oblique. This system is named from the Greek , one, and to incline. If we take a model with three unequal axes arranged as in the trimetric system, and then make the vertical axis oblique to one of the lateral, we change the system into the monoclinic. While the right rhombic prism belongs to the orthorhombic system, the oblique rhombic prism and the related forms belong to the monoclinic system. Fig. 10 represents an oblique rhombic prism with its axes, and fig. 11 an oblique prism on its rectangular base, which is another form of the same system. Examples: borax, Glauber salt, sugar, pyroxene. 5. Triclinic system: all the three intersections oblique and the axes unequal. The forms are oblique prisms contained under rhomboidal faces. Examples: blue vitriol, axinite. C. The axes four in number. 6. Hexagonal system. In the regular hexagonal prism (figs. 12, 13) the vertical axis connects the centres of the bases, and the three lateral the centres of the opposite lateral faces (fig. 12) or edges (fig. 18); another form is a double 6-sided pyramid (fig. 14), and another a double 12-sided pyramid. Examples: beryl or emerald, apatite. Besides the hexagonal prism, this system includes the rhombohedron and its derivative forms, inasmuch as the symmetry of these forms is hexagonal.
The rhom-. bohedron (fig. 15) is a solid, bounded like the cube by six equal faces equally inclined to one another, but those faces are rhombic, and the inclinations are oblique. The relations of the rhombohedron may be explained by comparison with a cube. If the cube be placed on one solid angle, with the diagonal from that angle to the opposite solid angle vertical, it will have three edges and three faces meeting at the top angle, and as many edges and faces, alternate in position, meeting at the opposite angle below; while the remaining six edges will form a zigzag around the vertical diagonal; these six edges in zigzag might be called the lateral edges, and the others the terminal. The cube, in this position, is in fact a rhombohedron of 90°. If the cube were elastic, so that the angles could be varied, a little pressure would make it a rhombohedron of an angle greater than 90°, that is, an obtuse rhombohedron (fig. 15); or by drawing it out, it would become a rhombohedron of an angle less than 90°, or an acute rhombohedron (fig. 16). The diagonal here taken as the vertical axis is the true vertical axis of the rhombohedron; and as there are six lateral edges situated symmetrically around it, there are three lateral axes crossing at angles of 60°, as in the regular hexagonal prism.
Fig. 17 shows that a hexagonal prism may be made from a rhombohedron by cutting off the edges by a plane parallel to the vertical axis; another may be made by truncating the lateral angles parallel to the same axis. Examples: calcite, sapphire, quartz. Fig. 17 represents a common form of quartz; the same with the lateral edges truncated so as to make a six-sided prism is more common. IV. The relative values of the axes in any species are constant, and these values may be ascertained from the angles of inclination of the planes on one another. In the isometric system the axes are equal (see figs. 1 to 3), and the axial ratio is therefore that of unity. Calling the three axes a, b, c, it is in all isometric species a: b: c = 1: 1: 1. In the dimetric system the vertical axis (a) is unequal to the lateral (b, c), and the lateral are equal. Calling the lateral 1, a: b: c = a: 1:1, a being of any length greater or less than 1, and whatever the value, it is constant for the species. The axes of the fundamental octahedron (fig. 5) of any species being thus a: 1: 1, the axes of all other octahedrons of the same species may be expressed by the ratio ma: 1: 1, in which m is any simple number or fraction; and the value of ma being known, the angles of the octahedron may be calculated, and conversely.
Which octahedron in a series occurring among the crystals of a species shall be taken as the fundamental octahedron, is generally decided on mathematical grounds, that being so regarded which is of most common occurrence, or is most convenient for exhibiting the mathematical relations of the planes. In zircon (fig. 25) the octahedron assumed to be the unit or fundamental one is that having for the value of the vertical axis 0.6407, that of the lateral being a unit; but it would be as correct mathematically, though less convenient, to make the octahedron 2a: 1: 1 the fundamental one, in which case a would equal 1.2814. In calomel, the assumed fundamental octahedron has the value 1.232; and it is beyond question that crystallogenically this octahedron in calomel corresponds to 2a of zircon. In the orthorhombic or trimetric system the three axes are unequal, but the ratio is constant for each species, as in the dimetric. Taking the shorter lateral axis (b) as unity, the ratio for sulphur is a: b: c = 2.344: 1: 1.23; for heavy spar, 1.6107: 1: 1.2276. In obtaining these numbers there is the same kind of assumption that is explained above with regard to which octahedron shall be taken as the fundamental one; and so under the other systems of crystallization.
In the monoclinic system the obliquity of the prism is a constant, as well as the relative values of the axes. In Glauber salt this inclination is 72° 15', and the ratio of the axes is a: b: c = 1.1089: 1: 0.8962. In the hexagonal system, as in the dimetric, the vertical (a) is the varying axis; but its value is constant for each species. In quartz, a: b: c: d = 1.0999: 1: 1: 1; in calcite, 0.8543: 1: 1: 1. In other words, taking the lateral axes at unity, the vertical (a) in calcite is 0.8543. Crystallography owes its mathematical basis to this law. Constancy of angle for each species is involved. But this constancy is not absolute, as explained below. V. Each species, while having a constant axial ratio, may still crystallize in a variety of forms. Thus the diamond, which is isometric, occurs in octahedrons, in dodecahedrons, and in solids that are like octahedrons in general form, but have low pyramids of three or six faces in place of each octahedral face (called tris-octa-hedrons and hex-octahedrons, the number of faces being either 3 x 8 = 24, or 6 x 8 = 48), and in various combinations of these forms.
So, dimetric species, as idocrase, may occur in simple square prisms, or in square prisms with the lateral edges truncated or bevelled, or with different planes on the basal edges or angles, or in eight-sided prisms, or in square octahedrons, etc. In the species calcite, the number of derivative forms amounts to several hundreds. This simple fact shows that while cohesive attraction in calcite, for example, sometimes produces the fundamental rhombohedron, it may undergo changes of condition so as to produce other forms, and as many such changes as are necessary to give rise to all the various occurring forms of the species, with only this limitation, that they are all based on the fundamental axial ratio, 0.8543: 1. VI. In all cases of derivative or secondary forms, either (1) all similar parts (parts similarly placed with reference to the axes) are modified alike, or (2) only half, alternate in position, are modified alike. This law may be explained by reference to a square prism. In this prism there are two sets of edges, the basal and lateral; the two sets are unlike, that is, are unequal, and included by different planes.
One set may therefore be modified by planes when the other is not; moreover, when one basal edge has a plane on it, all the others will have the same plane, that is, a plane inclined at the same angle to the base; or if one has a dozen different planes, all the others will have the same dozen. Again, if a lateral edge is replaced by one plane, that plane will be equally inclined to the lateral planes, because those planes (or, what is equivalent, the lateral axes) are equal; and in addition, all the lateral edges will have the same plane. In a cube, the 12 edges are all equal and similar; and hence, if one of them has a plane on it, as in fig. 18, there will be a similar plane on each of the 12. Hence, we may distinguish a cube, modified on the edges, however much it may be distorted, by finding the same planes on all the 12 edges of the solid. The eight angles of a cube are similar, and hence they will all have similar modifications, either one plane, as in fig. 19, or three planes, or six as in fig. 20. Again, the eight angles of a square prism are similar and therefore are modified alike.
The square prism and cube differ in this, that in the cube, when there is one plane on each angle, that plane will incline equally to each of the three faces adjoining, because these faces are equal; while in the square prism, the plane will incline equally to the two lateral planes and at a different angle to the base. This general law, "similar parts similarly modified," is in accordance with what complete symmetry woul'd require. The exception mentioned, of half the parts modified without the other half, is exemplified in boracite (fig. 21), in which half of the eight solid angles of the cube have planes unlike those of the other half - a mode of modification that gives rise to the tetrahedron (fig. 22) and related forms; in tourmaline, in which the planes at one end of the crystal differ from those at the other; and in pyrite, in which on each edge there is only one plane out of a pair of bevelling planes. Fig. 23 represents a cube with all the edges bevelled, that is, replaced by two similar planes - a holohedral form; while fig. 24 is that of a hemihedral form, only one of the two bevelling planes being present on each edge, a common form of pyrite.
All such forms are said to be hemihedral (Gr. half, and face), while the former are said to be holohedral ( , all, and ). Many hemihedral crystals, when undergoing a change of temperature, have opposite electrical poles developed in the parts dissimilarly modified. VII. The derivative forms, under any species, are related to one another by simple multiples of the axial ratios. In calcite the fundamental rhom-bohedron has the axial ratio just mentioned, 0.8543: 1, that is, a = 0.8543. There are a number of derivative rhombohedrons among the crystalline forms of this species; one has the vertical axis 1/2a; another 1/4a; others 5/4a, fa, 2a, 3a, 4a, and so on, by simple multiples of the vertical axis of the fundamental form. So in zircon, of the dimetric system, as implied above, while a (vertical axis)=0.6407, the lateral being unity, there is one derivative octahedron (1, fig. 25) with the axes a: 1: 1; another, 2a: 1: 1; another, 3a: 1: 1; also a diagonal pyramid, a: 8: 1 (1 i in fig.); and three other forms (eight-sided pyramids) whose axes are severally 3a: 3: 1 (33, fig. 25); 4a: 4: 1 (44, ib.); 5a: 5:1 (55 ib.); or writing out the value of a, they are 1.9221: 3: 1; 2.5628: 4: 1; 3.2035: 5: 1. Through these numerical ratios the planes or figures of crystals are conveniently lettered, as in this example of zircon, i being used in place of the sign for infinity.
The same numerical axial ratios run through all crystalline forms, and by means of them the values of the angles are calculated. These facts show that the modifications which cohesive attraction (or, what is the same, crystallogenic attraction) undergoes in order to produce the various derivative forms of any substance take place according to a law of simple ratios. VIII. The physical characters of crystals have a direct relation to the forms and axes. Cleavage, hardness, color, elasticity, expansibility, and conduction of heat differ in the direction of different axial lines, and are alike in the direction of like axes. The difference of color between light transmitted along the vertical and lateral axes of a prism is often very marked, and the name dichroism (Gr. twice, and color), or the more general term pleochroism, is applied to the property. The hardness often differs sensibly on the terminal and lateral planes of a prism, and also, though less sensibly, in other different directions. IX. The angles of the crystals of a species, though essentially constant, are subject to small variations. The unequal expansion of inequiaxial crystals along different axial directions, alluded to under the last head, occasions a change of angle with a change of temperature; other small variations arise from impurities, or isomorphous substitutions, or irregularities of crystallization. There are also many instances of curved crystallizations which are exceptions to the general rule. A familiar example of curving forms is afforded by ice or frost as it covers windows and pavements. Diamonds have usually convex instead of plane faces. Rhombohedrons of dolomite and spathic iron often have a curving twist; half the faces are concave and those opposite convex. Other imperfections arise from an oscillating tendency to the formation of two planes, ending in making a striated curving surface. Thus nine-sided prisms of tourmaline are reduced to three-sided prisms with the faces convex.
X. While simple crystals are the normal result in crystallization, twins or compound crystals are sometimes formed. The six-rayed stars of snow (fig. 26) and the arrow-head forms of gypsum are examples of compound crystals. In the stars of show there are three crystals crossing at middle; in the arrow-shaped crystal of gypsum two crystals are united so as to form a regular twin. Many of these twin crystals may be imitated by cutting a model of the form in two, through the middle, and then inverting one part and uniting again the cut surfaces. Fig. 27 represents an octahedron placed on one of its faces with a plane intersecting at middle, and fig. 28 is the same form with the upper half revolved 60°. To explain its formation, it is necessary to suppose that the nucleal or first particle of the crystal was a double molecule made up of two molecules, in which one was thus inverted or revolved on the other. Another example is shown in fig. 30. Fig. 29 is a common form of tin ore; the four-sided prism has a pyramid at each end. It is represented as intersected by a diagonal plane. Fig. 30 is the same form after one half is revolved 90°, and this also is very common in tin ore.
Such twins, as well as other facts, prove that molecules have a top and bottom, or, in more correct language, polarity, one end being positive and the other negative, this being the only kind of distinction of top and bottom which we can suppose. Axial lines or directions of attraction are in fact necessarily polar, if it be true, as is supposed, that molecular force of whatever kind is polar. In the case of the compound crystal of snow, the nucleal particle must have consisted of three or six molecules combined. Those prismatic substances are compounded in this way which have the angles of the prism near 60° and 120°, and for the reason that 3 times 120° or 6 times 60° equal 360°, or the complete circle. In a case where this angle is nearly one fifth of 360° (as in marcasite), the twins consist of five united crystals. In compound crystals of another kind, the composition is produced after the crystal has begun to form, instead of in the first or nucleal particle. A prism, as in rutile, after elongating for a while, takes a sudden bend at each extremity at a particular angle, depending on the values of the axes.
In another case, as albite, which is triclinic, a flat prism begins as a thin plate; then a reversed layer is added to either surface; then another like the first plate; then another reversed; and so on, until the crystal consists of a large number of lamellae, the alternate of them reversed in position, yet all as solidly united as if a simple crystal. Such a kind of composition may be indicated on the surface in a series of fine striations or furrows, each due to a new plane of composition; and they are frequently so fine as to be detected only by means of a magnifying glass. This mode of twin is additional proof of the polarity of the crystallogenic molecule. If there were not some inherent difference in the extremities or opposite sides of the molecules or their axes, which is equivalent to polarity, there could not be this series of reversions during the formation of the crystal. External electric or other influence may be the cause of the reversion. XI. While simple and twin crystals form when circumstances are favorable, in other cases the solidifying material becomes an aggregate of crystalline particles.
Regular crystals often require for their formation the nicest adjustment of circumstances as to supply of material, temperature, rate of cooling, or evaporation, etc.; and hence imperfect crystallizations are far the most common in nature. A weak solution spread over a surface may produce a deposit of minute crystals, which, if the solution continues to be gradually supplied, will slowly lengthen, and produce a fibrous or columnar structure. In other cases, whether crystallization take place from solution, or fusion, or otherwise, the result is only a confused aggregate of grains, or the granular structure. Under these circumstances, the tendency in force to exert influence radially from any centre where it is developed or begins action, often leads to concentric or radiated aggregations, or concretions. The point which first commences to solidify, or else a foreign body, as a fragment of wood or a shell, becomes such a centre; and aggregation goes on around it, until the concretion has reached its limits. Basalt and trap rocks which have been formed from fusion are often divided into columns, and the columns have concave and convex surfaces at the joints or cross fractures, proving that they are concretionary in origin.
The centre or axis of each column is the centre of the concretionary structure, and therefore it was the position of the first solidifying points in the cooling mass. The distance therefore between the initial solidifying points determines in any case the size of the columns; and as the columns are large the thicker the cooling mass, the distance is greater the slower the cooling. The cracks separating the columns are supposed to be owing to contraction on cooling. XII. The system of crystallization of a given substance sometimes undergoes a total change, owing to external causes. Carbonate of lime ordinarily crystallizes in rhombohedrons, and is then called calcite; but in certain cases it crystallizes in trimetric prisms, and it is then called aragonite. The aragonite appears to form when the solution has a higher than the ordinary temperature. This property of presenting two independent forms is called dimorphism. Besides difference of form, there is in all such cases a difference of hardness and specific gravity. Carbon crystallizes in one set of forms, which are isometric, in the diamond, and in another, hexagonal, in graphite.
Glass and stone are dimorphous states of the same substance, and the former may be changed into the latter by slow cooling. - Modes of Crystallization. Crystallization requires freedom of movement among the particles engaged in the process. It may take place: 1. From solution, where a solvent serves to disunite the molecules of a solid,.and give them the free movement required. The crystallization of sugar or alum from a concentrated solution is an example of this method. The alum solution is simply set away to cool, and the crystals slowly form and cover any object that may be placed in the solution. With many solutions evaporation cautiously carried on will throw down a crop of crystals. Sea water, on slow evaporation, first deposits gypsum, afterward common salt, and then its mag-nesian salts. 2. From a state of fusion or of vapor. Heat in this case is the dissevering agent, and the removal of heat permits resolid-ification. Thus water becomes ice, and aqueous vapor snow; and melted lead, sulphur, and other substances may come out in perfect crystals. If a mass of melted sulphur, or of bismuth, after it has crusted over, be tapped and the interior run out, the cavity within will be found lined with crystals.
Camphor, when sublimed by a gentle heat, condenses again in delicate crystallizations. 3. From long continued heat without fusion. The heat used for tempering steel is far short of fusion, and yet it allows of a change in the size of the grains throughout the mass. Heat has crystallized beds of earthy sediment, and thus changed them into gneiss and mica schist without fusing the rocks; and there is reason to believe that even a low degree of heat long continued is sufficient for these results. By this means statuary marble, one of the earth's crystalline rocks, has been made of fossiliferous limestones. The white marble of Berkshire, Mass., is probably of the same formation with either the Chazy or the Trenton limestone, rocks full of fossils, in central New York and elsewhere. Such altered rocks are termed in geology metamorphic rocks. Nearly all the gems, and far the larger part of the crystalline rocks of the world, were crystallized by some metamorphic process. 4. From any circumstances that favor the combination. of the elements of a compound.
Crystallizations often take place at the moment of the combination. - Origin of the Modifications of Crystals. The particular modifications of form presented by the crystals of any substance sometimes depend on the nature of the solution depositing the crystals, and sometimes on wider terrestrial conditions. Common salt, crystallizing from pure water, almost invariably takes a cubic form; but if boracic acid is present, the crystals are cubes with truncated angles; or if the solution contains urea, the crystals are octahedrons. Carbonate of copper, in course of deposition, has been observed to change the form of the crystals on the addition of a little ammonia, and again to a still different form on adding sulphuric acid. Sal ammoniac ordinarily crystallizes in octahedrons; but if urea is present, it forms cubes. A floating crystal forming in a solution has been seen to assume secondary planes on becoming attached to the sides of the vessel. There are many examples where a substance, as calcite, for a time crystallized under one form, and afterward began a new form around or on top of the first. At Bristol, Conn., six-sided prisms of calcite have been found surmounted by short, flattened calcite crystals of the variety called nail-head spar.
At Wheatley's mine, Phoenixville, Pa., the same species, under the form of the sca-lenohedron, has been found covered and altered to a six-sided prism. Such facts prove some change, and probably a change in the nature of the solution supplying the carbonate of lime, the ingredient of calcite. In nature the crystals of a substance over a wide region are often identical in form. The calcite of the Niagara limestone at Lockport, N. Y., in all cases has the form called dog-tooth spar, or the scaleno-hedron; that of Booneville, N. Y., the form of short hexagonal prisms; that of the Rossie lead mine, a combination of other more complex forms. This is a general fact with regard to the crystallizations in rocks. In massive aggregate crystalline rocks there is a tendency to parallelism in the crystals, and hence at a granite quarry it is easier to split the granite in one direction than in others, owing to an approximate parallelism in the cleavage planes of the feldspar. To obtain large crystals artificially from solutions, a large supply of material is of course necessary. The most successful mode is to select certain of the best crystals that have begun to form, and supply them from time to time with new portions of the solution.
They will thus continue to enlarge, the crystallizing material tending to aggregate about the ready formed crystals rather than commence a new crop. Cavities in rocks sometimes contain a Vast amount of large crystals. At Zinken in Germany, a single cavity was opened last century which afforded 1,000 cwt. of quartz crystals, one of which weighed 800 lbs. In all such cases the supply of material was gradually introduced; for so little silica is taken up by alkaline waters that the solution of silica filling the cavity at any one time could make but a thin lining over its interior. When water freezes, there is at first a sheet of ice made by the shooting of prisms over its surface. After this, as the cold continues, the crust increases in thickness by gradual additions to the under surface, thereby causing an elongation of prismatic crystallizations downward. The body of the ice is consequently columnar, although not distinctly-so when examined in its firm state. In the melting of the ice of some lakes in spring, as has been observed at Lake Champlain, this columnar structure usually becomes apparent; and it is sometimes so decided, that when the ice is even a foot thick and strong enough to bear a horse and sleigh, the horse's foot will occasionally strike through, driving down a portion of a half-united columnar mass, which may rise again to refill the place as the foot is withdrawn.
When in this condition, a gale at night sometimes leads to a disappearance of all the ice before morning. A fact like this illustrates what must be the condition of the earth's crust if it has slowly cooled from fusion. The crystallizing rock material below, as the crust slowly thickened, would not necessarily take columnar forms; but there would be some system of arrangement in the crystals which would be of a worldwide character; and as the cleavable species feldspar is a universal mineral among igneous rocks, the earth's crust would derive some kind of structure - a cleavage structure, it might be called - from these conditions. Crystallization thus pervades the globe, and has had much to do in determining its grander surface features, as well as making gems, solidifying sedimentary strata, and furnishing material for the statuary and architect. It also affords man one of his best avenues for searching into nature, opening to view facts on which are based some of the profoundest laws in cohesive attraction, heat, light, and chemistry. - There are two methods of applying mathematics to crystallography now much used.
One, in which ordinary analytical geometry is employed, is explained at length in the Anfangs-griinde der Krystallographie and Elemente der theoretischen Krystallographie of Dr. C. F. Neumann, and is briefly presented in English in the 1st and 4th editions of Dana's "Mineralogy." The other is explained in Brooke and Miller's "Mineralogy," and also at more length and with more clearness in the "Physical Mineralogy" of Schrauf, published in German at Vienna, Another system, much inferior in beauty, is employed by French crystallographers.
(SABA, a market town of Hungary, in the county of Bekes, situated in the great Hungarian plain beyond the Theiss, 63 m. N. of Temesvar, on the railway to Pesth; pop. in 1870, 30,022, mostly Protestant. It carries on a considerable trade in corn, fruit, hemp, flax, cattle, and wine. Prior to 1846, in which year it was made a market town, it was known as "the largest village of Hungary".