CRYSTALLIZATION.

CRYSTALLIZATION is the symmetrical arrangement of the particles of a body when it passes from the liquid to the solid form. This arrangement is determined by the mutual action of the small solids of which the body is composed; and these solids are separated from the liquid by their force of cohesion. Crystallization is one of the most remarkable effects of cohesion. The qualities of a solid in which the force of cohesion is more easily overcome in one direction than another; its brittleness, elasticity, and ductility, depend on this arrangement of its particles.

Solid bodies are found either in irregular masses, or exhibit certain determinate forms by the process of crystallization. Those substances which are capable of assuming regular figures, uniformly affect the same form; subject, however, to certain deviations, from the operation of particular circumstances. Those bodies only can assume the form of crystals which are susceptible of being reduced to the fluid state. This is the usual method of crystallizing saline substances. The substance to be crystallized is dissolved in a sufficient quantity of water to retain it in solution. This is slowly evaporated; and as the bulk of the fluid is diminished, the particles are brought nearer to each other; they combine together by the force of cohesion, and form crystals. Some saline bodies, which dissolve but in small proportion in cold water, are found to be very soluble in hot water. But when this water cools, it is no longer capable of holding them in solution. The particles then gradually approach each other, and arrange themselves into certain determinate forms; or they crystallize. Many of the saline bodies which crystallize in this manner, combine with a considerable portion of water. This is called the water of crystallization. Other saline substances are equally soluble in hot and cold water. These substances do not crystallize by cooling the fluid; they assume regular forms only by diminishing its quantity. This is effected by means of evaporation by the application of heat. In salts which are crystallized in these circumstances, the proportion of water which enters into combination is small.

There are some classes of bodies which assume regular forms, but are not soluble in any liquid. Such, for instance, are metallic substances, glaas, and some other bodies. Substances of this nature are crystallized, by being previously subjected to fusion; and thus having combined with caloric, they are reduced to the liquid state, and the particles being separated from each other are left at liberty to arrange themselves into regular forms, or to crystallize, as the body cools.

But what is the cause which operates in determining the regular arrangement of the particles of bodies in these circumstances? or what is the cause of the same bodies in the same circumstances assuming regular figures? The ancient philosophers supposed that the elements of bodies consisted of certain regular geometrical figures; but it does not appear that they applied this theory to explain crystallization. The schoolmen ascribed the regular figure of crystals to their substantial forms; and others supposed that it depended merely on the aggregation of the particles, but without explaining to what this aggregation was owing, or the reason of the regular figures thus produced. According to Sir Isaac Newton and the theory of Boltzovich, Newton's, the particles of bodies held in solution in a fluid, are arranged at regular distances, and in regular order; and when the force of cohesion between the particles and the fluid is diminished, it is increased between the particles themselves. Thus they separate from the fluid, and combine together in groups which are composed of the particles nearest to each other. If we suppose that the particles composing the same body have the same figure, the aggregation of any determinate number of such particles will produce similar figures. Bergman is of opinion that the particles of saline substances possess a double tendency: by the one they arrange themselves in the form of spiculae; and by the other, these spiculae arrange themselves at certain angles of inclination, and according to the difference of these angles, different forms of crystals are produced. These effects are ascribed by the ingenious author to the mutual attraction which exists between the particles, which, according to the peculiar figures of the atoms, at one time arranges them in the form of spiculae, and then combines the spiculae thus formed under different angles of inclination. But this arrangement of the particles, or tendency to arrangement, assigned by Bergman as a cause, is only explaining the phenomenon by itself; while the cause of the tendency is yet unexplained. Nor will Newton's hypothesis be more satisfactory; for if the particles of a body, after being equally diffused in a fluid, are brought together by a general attraction, it will follow that every saline body should crystallize in the same manner.

According to the ingenious theory which has been proposed by Hauy, the integrant particles always combine in the same body in the same way; the same faces and the same edges are always attracted towards each other. But these faces and edges are different in different crystals; and hence originates that variety of forms which different bodies assuming regular figures by crystallization exhibit. But why are the same edges and the same faces attracted in the same way? This still wants explanation. If it be ascribed, as some have supposed, to a certain degree of polarity existing among the particles, it might enable us to account for the regular figures of bodies produced by the process of crystallization. For by the effects of this agent we might suppose that different parts of the particles of bodies are endowed with different forces; one an attractive, and another a repulsive force; and by the action of these two forces, the same arrangement of the particles will uniformly take place; for when one part of a particle is attracted, the other will be invariably repelled; and thus the same faces and edges will always be disposed in the same way. But it ought to be observed that the existence of this power, however satisfactorily factorily it might account for the phenomena, has by no means been proved; and even if its existence were completely established, the difficulty still remains how this polarity is to be explained.

Without entering farther into these speculations, we propose, in the two following sections, to present our readers with a comprehensive view of the formation and structure of crystallized bodies. In the first section we shall treat of the phenomena of crystallization, the means of conducting this process to obtain the most perfect crystals, and the modifications of which each of the forms is susceptible. In the second we shall give a short view of the theory of the structure of crystals.

Sect. I. Of the Phenomena of Crystallization, and the modifications to which it is subject.

The most complete set of observations which has yet appeared on this branch of practical chemistry has been made by M. Leblanc; and to his ingenious memoir we must acknowledge ourselves indebted for what we now lay before our readers that is new or interesting on this subject. This art, he observes, of managing or conducting the crystallization of salts, is in a great measure new; for it has hitherto attracted little attention. To insure success in obtaining perfect crystals, the process must be conducted in flat-bottomed vessels; and vessels of glass or porcelain are found preferable to those of any other materials for this purpose. The salt employed should be in a state of purity; and to favour the increase and regular form of the crystals, they are to be placed at a distance from each other in the vessels containing the solution. To these necessary precautions, it may be added, that the vessels in which the evaporation goes on should be at perfect rest, and that it is requisite to observe the density, or specific gravity, at which the solution begins to yield crystals.

The particles of any saline body cannot come into contact and form crystals, as long as the force of affinity between these particles and the fluid in which they are held in solution is greater than the mutual affinity of the particles among themselves. A salt, for instance, which begins to crystallize at a certain specific gravity of its solution in water, will afford no crystals when that specific gravity is diminished; for then the particles of the salt are removed to a greater distance from each other; and while, by this distance, the force of their mutual attraction is diminished, the attraction between these particles and the water in which they are dissolved is increased by the increase of the quantity of the solvent. But, on the other hand, if a solution which begins to crystallize at a certain specific gravity, is more concentrated, the crystals which are thus obtained are greatly multiplied, but they are heaped together in confused masses, exhibiting no regular forms. Thus, a solution which has been scarcely reduced to that degree of concentration at which it begins to crystallize, being poured while it is hot into the proper vessel for carrying on the process, or left at rest in the same vessel in which the solution is made, to cool slowly, will yield a small number of crystals, which will have no other defects than what are occasioned by their contact with the vessel. Even perfect crystals will be sometimes found among the smaller ones. When the concentration of the solution has not been carried too far, or not farther than what is effected by slow cooling, not only have the embryo crystals less bulk, but the particles having come into contact slowly and without confusion, they possess a greater degree of transparency. After a certain period, which varies according to the species of salt which is subjected to the operation, small crystals may be distinctly observed. These are to be carefully detached from each other, and placed in a different position. Being placed by this management on a different side, the defects occasioned by their contact with the vessel are soon repaired. From the crystals treated in this way, the finest and most perfect are to be obtained. This operation of changing the position of the crystal from one side to the other, ought to be repeated at least once every day, if we wish to obtain the completed crystals.

At the end of a certain period, the small crystals are to be removed, that the fluid may be more concentrated, either by a new evaporation, or by dissolving a new portion of the same salt. After the new solution has cooled, and the crystals which have formed in it are separated, if it has been too much concentrated, or too great a portion of salt has been added; the crystals of the first solution are then to be introduced and treated in the same way as formerly.

When the crystals have acquired a sufficient volume to handle them, and to choose such as we should deem fit to increase to the largest size, either as simple or complete crystals, or as exhibiting varieties from position or particular circumstances, the individual crystals are then to be separated, and solutions are to be prepared for them and brought to such a degree of concentration as to afford crystals in a mass; which latter being removed, the single crystals are introduced into these solutions, which are now in a proper state to favour their increase. The crystals may be either previously disposed in the vessel, and then the solution may be poured on; or having first introduced the latter, they may be afterwards distributed on the bottom of the vessel. And thus by continuing the same process, by taking care to change the position of the crystal from one side to the other frequently, and by keeping up the solution to a proper degree of strength, we may obtain crystals of any bulk we choose.

When the quantity of particles, which in a certain state of concentration continue to be mutually attracted, ceases if not diminished in consequence of their accumulation, let too long in the stage of this diminution the crystals cease to enlarge or increase in bulk; it happens, on the contrary, if they are left in the fluid, that they begin to dissolve. It is usually on the corners and angles that this decrease takes place; and in some salts it seems to go on piecemeal, so as to prevent distinct layers of the particles; for in this case lines parallel to the sides may be observed, and these are disposed like steps of stairs. Should the accident which is here alluded to be allowed to go on too far, it may often require a long time to repair it; but it is in general easy to avoid this inconvenience, by watching the progress of the operation and the increase of the crystals. If their corners or angles are observed to become less sharp, they must be removed till the fluid is farther concentrated, or they they must be introduced into a new solution of the same salt of the proper degree of strength. To prepare the new solution for the increase of the crystals, a quantity of the same salt is to be dissolved in a given portion of water, so that it shall be fully saturated. It is then allowed to cool and crystallize. The crystals being separated, the remaining solution is to be employed in such quantity as may be judged necessary to replace that in which the diminution of the crystals had commenced.

Sometimes it happens, from want of necessary precaution, that the new solution in which the process is to be conducted, either being too much saturated, or being disturbed by pouring from one vessel to another, exhibits many other points of attraction before the crystals whose increase is proposed. In this case a great number of small crystals make their appearance, and cover the surface of the former with a kind of incrustation. The small crystals, provided they are taken in time, may be removed without injury to the others; if not, they will be unavoidably spoiled.

When the crystals have reached such a size as that they may be placed one by one, without being in contact with each other, we must still continue frequently to change their position. This may be done with a spatula, a glass rod, or any instrument which will communicate nothing to the fluid. In this way the sides of the crystal which are alternately in contact with the bottom of the vessel will increase in equal proportion, and it will always remain complete.

It is chiefly in salts which furnish elongated prisms that the influence of position may be most distinctly seen. If, for instance, a crystal before it has acquired much volume is found to rest on one of its bases as well as on one of its sides, it will be observed to be compressed in the direction from base to base; and it will appear to be only a regular segment of the crystal, which having been placed on one of its sides has obtained a great bulk. If we take a six-sided prism whose summits are obliquely truncated, and if it be placed on one of its sides, it will enlarge in a greater or less degree, but always in such a manner that the distance from one base to the other shall never be less than the distance between the sides. But if the position be on one of its bases, then its principal increase will be in the direction of the sides, and it will appear to be compressed between the bases. At first sight, a crystal treated in this way will seem different from the former. For the corners form the summit of apparent pyramids which are separated by a four-sided prism. This circumstance affords a sufficient explanation of one of the causes which produce varieties in the appearances of a crystal with regard to its relative extent; it shows that there is no foundation for the opinion of a supposed balance between the particles of the salt and that of the solvent; and it shows also, that if the force of attraction be the efficient cause of the saline particles coming into contact, the force of gravitation acts at the same time, and modifies in a greater or less degree the effects of the first.

According to these observations, and the different states in which crystallized substances are found, it has been supposed that we might conclude; that the force of adhesion between the particles of the salt and those of the solvent, varies according to circumstances, which depend on the degree of tendency to combination between the bodies, and the relative weight or bulk of the parts of which these bodies are composed. If a crystal in the incipient stage of its increase be placed on one of its bases, it enlarges in the direction of its sides; but if it be reversed and placed on one of its sides, it enlarges in the dimensions of an elongated prism.

An insulated crystal, placed on one of its sides on a smooth surface, and left undisturbed to enlarge in size, presents on this part a kind of hollow, which corresponds exactly with the side which it replaces. Here the saline particles which cannot reach this surface, are distributed on the neighbouring parts with which they come in contact, with this circumstance, that the edges of the surface on which the crystal rests increase in proportion, but without allowing the liquid to have access to this surface.

The hollows which are formed at the surface of liquids differ sometimes from each other even in the same salt. If we suppose that a particle forms the incipient point of the hollow, the latter will assume a configuration corresponding to the side of the particle presented to the surface of the liquid: but the part which it touches increases also; and if by any circumstance a change of position happens, the hollow, thus necessarily formed according to the arrangement of the part which corresponds exactly to the surface of the liquid, will change its form, because the new position of the side presented differs from the first.

When a neutral salt, in a state of purity, and after being crystallized, ceases to produce any effect on vegetable blues, it is not supposed that any of its constituent principles is in excess. But if in this state it is found to combine with other bodies, in such a manner as to produce solid and well-defined crystals, we must admit that there exists an affinity between the salt and the body with which it has combined.

This subject, Leblanc observes, of the supra-composition, or compound combination, as it might perhaps be called, of which several salts are susceptible, has hitherto much occupied the attention of chemical philosophers. Some indeed have been pointed out by Bergman and others; but it has been remarked that these affinities are probably much more extensive than has been supposed; and not only with regard to neutral salts with each other, but also neutral salts with other bodies. Of this kind of combination is not to be reckoned that of one of the constituent parts of a salt being in excess, which frequently takes place in some salts, and is found to be more or less permanent. This circumstance seems to prove that certain salts have two different points in the combination of their constituent parts. Let us see what has been observed in this respect of the sulphate of alumina, which will perhaps explain the reason that this salt is almost always found in nature in the acidulous state. It is found that the more that alum approaches to the state of saturation by an additional portion of base, the less fluid the new combination becomes; and in all cases, after a certain time, which is longer or shorter according to circumstances, the portion which was added separates. It will perhaps appear in the sequel, that this tendency to combination which is constantly in action, producing an immense multitude of different individuals, resides not only among the properties of the simple principles, Many of the sulphates are always found in the acidulous state; and all of them seem to be susceptible of combination with a new quantity of the same base, till they reach the point of saturation. For example, the sulphate of copper, in the state in which it is usually found, crystallizes in eight-sided, oblique prisms, terminated by sides according to the obliquity of the prism. But if another portion of base be added, the crystals assume the form of pyramids of several faces, separated by a four-sided prism. The acidulous sulphate of zinc gives crystals of six-sided prisms, which are often very regular; but an addition of base produces a great change, for then the crystals are in the rhombohedral form, very little different from the cube. Alum in its ordinary state of combination crystallizes in the form of a regular octahedron; but in the intermediate proportions between this state and that of saturation, it assumes the form of a cube.

Hauy, as will be afterwards noticed, has demonstrated that the form of the primitive molecules is the same in all crystals of the same salt, and he has shown by calculation that the variations arise from the laws of decrement in the layers which surround the nucleus; but that the order according to which the secondary forms are produced may be interrupted, whether this form be complete or not; and the crystal may then, according to circumstances, return to its primitive form, or to some of those which are derived from it. But from the experiments of Leblanc, he thinks that these changes always depend on new conditions in the state of the fluid, as a different proportion of the principles of which the salt is composed.

If a crystal of octahedral alum be placed in a solution which forms cubic crystals of the same salt, the former will assume the cubic form, by giving up a series of molecules from the summits of the solid angles, so that the layers continue to decrease on the triangular faces till the crystal has completed its new form. In this process, the change may be stopped at any period, and crystals of every modification of form may be obtained. From this it follows, that the centre of each of the faces of the octahedron corresponds to a solid angle of the cube in which it is inscribed. But if a cubical crystal be introduced into the solution which yields the octahedron, its return to this latter form proceeds in the same order, by the subtraction of a series of molecules from the solid angles of the cube. It often happens, however, at the same time, that the subtraction of the molecules extends to the corners of the crystal; so that the layers of superposition decrease all at once, according to the order of the formation of the octahedron, and the dodecahedron with rhomboidal surfaces. This circumstance seems to suggest the possibility of obtaining crystals of alum of this latter form; but it seems to depend on a particular proportion which is not easily determined.

Thus we learn from experiment that salts which exhibit different forms of crystals can be made to assume each of these at pleasure. This phenomenon, which has not been much attended to, seems to merit particular investigation. The transition from one form to another may be explained according to the laws of diminution, by the successive and regular subtraction of series of molecules; so that the form actually obtained, the restoration of the preceding form, is easily explicable on the principle of restitution alone. It may be observed, that during this kind of metamorphosis, both operations, namely, that by which the crystal receives on the one hand a new form, and that by which on the other hand it increases on all its sides, constantly take place.

The particles of a salt which are in solution in a fluid, are attracted by it, particle by particle, without any separation or decomposition; but it is necessary that there be a balance of the attracting forces between the salt and the solvent. This is demonstrated by the following experiment. A vessel two feet high and two inches in diameter was filled with a solution of a proper degree of concentration for the growth of crystals, which were suspended at different heights from the bottom of the vessel to the surface of the fluid; and it was observed that the increase of the crystal was in proportion to its depth in the vessel, that which was nearest the bottom increasing most rapidly. When the liquid was deprived of saline particles by their accumulation on the crystals, by refi, and sometimes even by the influence of the atmosphere, the crystals decreased by similar gradations to those of their increase; so that at last reached that state when the crystals near the surface of the liquid were dissolved, while those towards the bottom continued to increase; and sometimes it happened that the crystals at the bottom of the vessel continued to increase on the surface which was in contact with it, while the opposite upper surface was in a state of dilution.

All the experiments which were made on salts of different degrees of specific gravity accord with this observation; and the difference in the degrees of saturation of the waters of the ocean, which depends on the difference of depth, seems to be in favour of this opinion. It is confirmed by the analysis of sea-water by Bergman and others, which was taken up in different places and at different depths. It receives still farther confirmation from a practice of the inhabitants of Salles in Bearn in estimating the degree of strength of a salt spring. An egg is thrown into the waters of the spring, and the whole water which covers the surface of the egg is thrown away, as it is not of a sufficient degree of concentration.

It is well known that a cold temperature is most convenient for the crystallization of salts. But it is not at the period when the salt begins to crystallize that it is most convenient to carry on the process; for then it sometimes happens, from too great concentration of the fluid, that the crystallization is too rapid and confused.

Hitherto saline substances which are susceptible of regular crystallization have been divided into two classes, according to the peculiarities in the formation of their crystals. The one class comprehends those crystals which are formed by cooling the fluid in which the solution is made. The other class includes those which are produced only during the evaporation of the solution. This distinction is no doubt well founded; but there are some exceptions to it which are necessary to be attended to in conducting the process of crystallization. If a saline solution which is too much saturated, be cooled, it furnishes a mass of crystals which which are confused and irregular, and which present no determined form except on those sides which are in contact with the liquid. If in this state the remaining liquid is poured off, it will yield another set of crystals, but in very small number; and there are some salts which continue to form crystals after being several times successively treated in this way, the number of the crystals still diminishing from the first degree of concentration. It will be found too that this will take place whether the process is carried on in the open air or in close vessels. It follows from this that the increase or the formation of crystals, in this case, depends solely on the mutual attraction of the particles, or on the attraction between the particles and the crystal; an attraction or affinity which is not destroyed by the cooling of the fluid, but is probably regulated by the distance of the particles, and the degree of force or affinity which exists between the particles and the solvent. In some saline solutions the increase of the crystals goes on in this manner for a long time. It is only in the interval between the cooling of the liquid to the temperature of the atmosphere, and that period when its degree of concentration is so diminished that the increase of the crystals ceases, that the latter proceeds with that degree of perfection of which it is susceptible.

It is not a property peculiar to dry substances to absorb moisture from the atmosphere. Liquids saturated with certain salts seem also to possess this property; for in some saline solutions, the liquids assume a solvent power which never fails to attack the crystals, and not only to prevent their increase, but to diminish the bulk which they had acquired. This accident can only be obviated by regulating the state of the atmosphere in which the evaporating vessels are placed, and preserving it free from an excess of moisture. From causes which produce a contrary effect, the evaporation becomes too rapid; this circumstance also requires to be attended to, and properly regulated, to ensure the full success of the operation.

From the preceding observations it will appear, that solutions of salts which are susceptible of crystallization have certain degrees of concentration which are necessary for the formation of crystals; and that they must be reduced nearly to that degree in which they begin to yield crystals, before it can be expected that they afford proper results. It is therefore necessary to attend particularly to the degree of concentration which each salt requires for the regular formation of its crystals, and to obtain them with that degree of transparency of which they are susceptible. We have seen that in the formation of crystals they may be removed from one vessel to another, and from one solution to another; and that in proportion to the slowness of the process they become more beautiful and more perfect. These operations, it may be added, require much patience and attention, but at the same time the observer is fully compensated for his trouble, by perceiving the progress of the crystallization, and by the interest which is excited in all its stages.

It is essential to know that neither the crystals formed during the artificial evaporation, nor those which are produced during the cooling of the solution, are proper to be made choice of for being increased and brought forward to the most perfect crystals. When a solution has become cold, that is to say, when it has acquired the temperature of the atmosphere, and it is deprived of the excess of saline particles which it held in combination during its increase of temperature, it is still in a condition to yield crystals, and as long as the distances between the particles are not too great to allow of mutual attraction. A solution saturated to excess affords on cooling a confused mass of crystals; but after the fluid is poured off, it will still produce more crystals, but in smaller number. The degree of concentration of the solution before it yielded the last product may be considered as the term of saturation most proper to be employed for the species of salt which is thus treated. But by the repetition of these operations, and the observation of their progress, it will not be difficult to discover the proper proportions between the salt and the solvent.

It seems to be a mistake to suppose, with some, that the crystals which are placed in favourable circumstances to become larger and more perfect, are injured by coming into contact with each other during their increase. It is undoubtedly better that they should be kept separate; but it does not appear that they are hurt by touching each other, if the number in the vessel be not too great, and they are not heaped or pressed together. In that crystallization which results from the cooling of a solution too much saturated, the crystals are always confused and interlaced with each other; and the molecules which are arranged in this kind of disorder experience a kind of irregular distribution; and it may be observed, that in this case the summits only of the crystals which are elevated from the kind of cake which is formed on the surfaces of the vessel containing the solution, present regular and determined forms. The mass in which these crystals are implanted is a confused heap.

No cavities have been observed on the faces of crystals, excepting those which are formed on the surface of fluids. Those which are produced on that side of a crystal which rests on the bottom of the vessel are more common in other salts. This phenomenon seems to merit more attention than has yet been bestowed upon it; as it explains easily the introduction of extraneous bodies which are sometimes detected in the interior of crystals. For when a cavity of this kind has acquired a certain depth, it is capable of receiving part of any foreign substance, and to be filled up by the change of position of the same crystal, retaining at the same time the extraneous matter. By a little art and dexterity, these fortuitous circumstances may be favoured, so that phenomena exhibited by such occurrences may be traced and observed at the pleasure of the operator. Experiments have been made with the view of ascertaining whether an extraneous substance could be substituted as the nucleus of a crystal; but from the result of these experiments, it does not appear that the particles of any salt have a tendency to combine with any foreign matter, and to form regular crystals. The portions of the salt which were attached to the extraneous substance were always separate and independent crystals.

There are some saline substances which retain in their solution an excess of particles even after cooling, and which being strongly agitated instantly deposit a great number of small crystals which render the solu-

The introduction of crystals of the same salt, it is well known, as in the case of a solution of Glauber's salt, promotes this sudden crystallization or separation of the excess of the salt. If, in this state of the solution, crystals are immersed with the view of having them large and regular, they are certain of being spoiled by the accumulation of a great number of small crystals on their surface, unless the precaution of immediately washing them with pure water when this happens is observed.

It may be remarked also that when the solution is diminished below a certain degree of saturation, the crystals not only cease to increase, but are also again in some measure dissolved; the corners and angles reduced and rounded. And if the crystals in this state be introduced into a solution of sufficient strength to promote their increase, supernumerary faces and truncatures, as they are denominated in technical language, are formed on the rounded corners and angles. But these faces always disappear as the increase of the crystals proceeds, and are replaced by corners and angles, which become at last sharp and distinct.

By attention to preserve the solutions of salt in perfect purity, we shall be more certain of obtaining the most beautiful and transparent crystals. Some fluids, after a certain time, are observed to deposit substances which are foreign to the salt held in solution, and were dissolved along with it. These substances sometimes appear in the form of earthy matters, which precipitate to the bottom of the vessel; in other cases they are diffused in the form of flakes, and sometimes they rise and swim on the surface. In all these cases, the crystals whose formation and increase are going forward must be removed, and the liquor must be filtrated before they are replaced.

A saline substance, which is capable of crystallization, possesses, in the state of minute division in which it is in solution, or in the condition of the molecules which compose it, a determined property which is uniform and constant, in which resides essentially the power of uniting in a certain symmetrical manner, and thus constructing regular solids. The results also are uniform and constant when the process is carefully conducted; but it is necessary to distinguish with accuracy the circumstances which accompany the operation, and may occasion a deviation from this uniformity. The sulphate of iron, for instance, usually crystallizes in the form of rhomboids; but sometimes it has been found to assume that of an irregular octahedron. And although it may be true that an elongated octahedron may be clasped with prismatic crystals, it does not on that account belong less to the octahedral form; but it seems probable that these different varieties, in the forms of crystals, depend on some changes which take place in the solutions themselves. The iron in the present case is constantly receiving new portions of oxygen from the atmosphere, and in this new combination it is precipitated in the fluid: this, therefore, occasions a change in the constituents of the salt.

Several sulphates are found to combine readily with each other: those of iron and copper are of this description; and the result of this compound crystal is always a rhomboid. It seems to be doubtful whether this should be considered as a case of simple interposition of one salt with the other.

When a liquid, which holds saline bodies in solution, is evaporated to a certain degree, a crust forms on the surface, acquires a certain thickness, and when this is removed, it is renewed. The point at which the liquid exhibits this appearance is known in chemistry, by the appellation of evaporation to a pellicle. When it has reached this point, the solution is in a state of complete saturation; and the final effect addition of dendritic quantity of fluid cannot be withdrawn without a corresponding quantity of salt assuming the solid form. On this principle Robinet has attempted to account for the formation of dendrites, or the arborecent appearance and efflorescence of some salts. Almost all the different species of fucus or sea-weed, he observes, are covered, in drying, with an efflorescence of white matter. In some species, this white matter was observed to possess a saccharine quality. A number of large roots of the fucus palmatus was hung up in the shade, and ten days had elapsed without the appearance of any thing on the surface. After that period it became white, and it was soon covered with a light downy substance, the filaments of which gradually increased to a considerable length. When this downy matter was brushed off with a feather, it was renewed till the plants were completely dry. This substance, on examination, was of a saccharine nature, mixed with a small portion of common salt, and a great quantity of mucilaginous matter. By solution and crystallization, the sugar was separated from the other substances.

In comparing the circumstances of this efflorescence with those of the formation of the pellicle, in the process of evaporation, the former seems to be a modification of the latter. In a vessel which contains a liquid saturated with a salt, the surface subjected to evaporation has no sooner assumed a solid form, than the surface immediately inferior is exposed to the action of the same causes, and produces the same effect; and this effect continues till this crust has become so thick, or so compact, as to prevent the contact of air, and then the evaporation ceases. But, on the contrary, in the fucus, the air acting only on the surface of the plant, the liquid which it contains cannot undergo the process of evaporation, without coming to the surface. The attraction of the matter of the plant tends to promote this motion; for as the liquid is equally diffused through its whole mass, it rises constantly to the surface, in proportion as this surface is dried by the surrounding air; and it would appear that this is the process in the desiccation of all thick and malleable bodies. Now, the saline matter which, in the present case, is in the state of efflorescence, having the same power of attraction on the liquid, the rudiments of each filament constitute, at the instant of their formation, part of the whole mass or body of the plant. They participate, therefore, of the same degree of moisture as that of the plant, and it is on their surface that the evaporation and crystallization of saline matter chiefly take place.

The mechanism of the dendritical or arborecent form of saline bodies seems to be in this way capable of explanation. The whole saline mass, which extends to the edges of the vessel, and even redescends externally, is constantly in the humid state, as long as any liquid remains in the vessel. It may be supposed, that the matter of the sides of the vessel determines, by its attraction, the external circle of the surface of the liquid to rise above the surface; a phenomenon which is sufficiently obvious, but especially in narrow vessels. This portion of liquid, which is more completely subjected to evaporation, gives origin to a circle of saline matter, which appears thus raised above the surface of the liquid, and which, being the first rudiments of the dendrites, contributes afterwards to its increase, in the way which has been already explained. Thus the vegetation of salts bears a striking resemblance to the process of efflorescence, or the formation of the downy matter on the surface of the focus.

There is yet another kind of crystallization which seems to depend on the same cause. This is the saline efflorescence, which occurs in different places on the surface of the globe, and is frequently in such quantity as to become an important object of manufacture. Without extending our observations to the efflorescence of soda on the surface of the soil in Egypt, or that of nitre in Asiatic countries, we may refer to the production of muriate of soda, or common salt, in different parts of Europe, in those places which are covered with the waters of the ocean during high tides. The waters of the sea with which the sandy shores are twice periodically moistened in the course of the month, are far distant from the point of saturation which determines crystallization. They rarely contain more than 3 parts of salt in 100; and the sand at the degree of moisture, in which it is left by the sea, is not impregnated with a sufficient quantity of saline matter to be worthy the labour of manufacturing; but, during the interval between the tides, these circumstances are greatly changed. The dry air of summer, by evaporating the moisture on the surface, allows the matter of the sand to attract towards the surface a similar portion of water, which was in the lower part of the soil, and which always tends to diffuse itself equally through the whole mass. This liquid, carrying with it the salt which it holds in solution, increases the quantity of saline matter which exists on the surface. This process continues without interruption, as long as there is no fall of rain. It reaches at last a certain point, at which the water subjected to evaporation is saturated with the salt; and this process cannot proceed farther without the deposition of crystals of the salt, which discover themselves by their shining appearance. After some days, the sand on the surface is collected, and about six times the quantity of saline matter is found in the same proportion of sand, when it was first moistened by the sea water (A).

Another phenomenon which takes place during the process of artificial evaporation, should not pass unnoticed. This is the formation of a saline crust at the bottom of the vessels in which the process is conducted. This seems to be the immediate effect of ebullition; for when the temperature of the liquid is kept under the boiling point, no such effect is produced. This crust is composed of all the saline substances which are held in solution in the liquid; and even these substances are found combined in the same proportion in which they actually exist in the solution. Whatever be the attraction of these substances for water, or even if they possess a deliquescent property, they are not less disposed to enter into combination during the formation of the solid crust on the bottom of vessels in which the process of evaporation is conducted with a temperature equal to the boiling point. A slight degree of attention will satisfy us, that the formation of this crust depends on the particular circumstances of the evaporation in the case of ebullition. It must be obvious, that in this case the stratum of liquid which is in immediate contact with the vessel, receives the caloric which penetrates its sides, is charged with it beyond its capacity, changes its state, and assumes the gaseous form, and by this change having entirely lost its solvent power, whatever saline matter is held in solution must assume the solid state in contact with the sides of the vessel, and consequently adhere to it. Thus it happens, according to a very judicious observation, that in different saline solutions, the results of which have been compared, these scales or crusts are more abundant in proportion as the degree of saturation is less.

To these observations we shall only add a short account of the phenomena of crystallization, as they were observed, with the assistance of a microscope, by Mr. Baker, and of the appearances of different saline bodies which he has described. This will not afford any valuable scientific information to the philosopher, but it may perhaps be the source of amusement to some of our readers, and the means, by a minute observation of the phenomena, of leading to some useful discoveries. The method which he followed in conducting these experiments, is the following. The substance to be examined is to be dissolved in a quantity of pure water, so as to be completely saturated. For salts of easy solubility, cold water may be employed; but for salts which are dissolved with more difficulty, hot or boiling water may be found necessary. In preparing the solution, the same rule may be observed as in preparing solutions for obtaining large crystals, which has been given in the former part of this section. The solution should be allowed to remain at rest for some hours, so that the first crystallization, if too much saline matter has been added to the liquid, may be allowed to take place. Thus the solution will be always of the same strength, and the same appearances may be uniformly expected.

When the solution is thus prepared, a drop of it may be taken up with the point of a quill, cut in the form of a pen, and placed on a flat slip of glass, spreading it on the glass with the quill till the liquid is so shallow as to rise very little above its surface. It is then

(A) Common salt is manufactured in this way on the sandy shores of the Solway Frith, in Annandale in Scotland. These flat shores are covered with the waters of the ocean during spring tides; and in the interval of these tides the evaporation by the heat of the sun and the action of the air is so considerable, as to leave the sand impregnated with a quantity of salt, sufficient to defray the expense and trouble of manufacturing it by filtration and boiling. then to be held over the clear part of a moderate fire, or the flame of a candle, and such a degree of heat applied as is found from experience to produce the necessary evaporation. This will be known by observing the formation of saline particles at the edges of the drop of fluid. The microscope being previously adjusted, and a magnifier of moderate power being fitted on, the slip of glass is to be placed immediately under the eye, and brought exactly to the focus of the magnifier. After running over the whole drop, the attention is to be directed to that side on which the process of crystallization first commences, and proceeds from the circumference towards the centre. The motion is at first slow, if too much heat has not been applied, but becomes quicker as the evaporation continues. In some crystallizations the configurations are produced towards the end of the process with great rapidity, and exhibit an elegance, order, and regularity, which imagination only can conceive. When this rapid action has once begun, the eye must be kept fixed on the object, till the whole process is completed, because new forms appear, quite different from those which were first produced, and which have been properly ascribed to a quantity of different salts mixed with the substance to be examined, when the precaution has not been used of having it in a state of purity. When the configurations are fully formed, and the water evaporated, such salts as are deliquescent, it is scarcely necessary to observe, are soon destroyed by attracting the moisture from the air; but those which are more permanent, and not disposed either to deliquesce or to be deprived of their water of crystallization, may be preserved, by being enclosed between glasses, for a long time, as amusing objects for the microscope. To make the liquid spread readily on the glass, the surface of it may be moistened with a little of it, and rubbed with the finger. In this way, the repulsion which sometimes is observed between the liquid and the glass is completely removed. During the evaporation, the object-glass of the microscope is sometimes obscured by the condensation of the water from the saline solution on the slip of glass, and the vision is thus rendered indistinct. When this happens, if the circumstance be recollected, the glass must be wiped and replaced. In examinations of saline solutions, and in observing the progress of crystallization, Mr Baker recommends the light of a candle in preference to the light of day, which latter being of a whiter colour and nearly the same with the transparent crystals, they are less distinctly seen than with the brown light of a candle.

Fig. 1. is a representation of the microscopical crystals of nitre or saltpetre. They begin to shoot out from the edges with very moderate heat into flat figures of different lengths, with straight parallel sides, and exceedingly transparent. They appear in different states of their progress at the letters, a, b, c, d, and e; a exhibits the appearance when they first begin to form. When a number of crystals have made their appearance they sometimes dissolve under the eye, and disappear entirely; but, by continuing to watch the changes which go on, the process is frequently observed to recommence, and new shoots put out. The first crystals sometimes become larger without undergoing any change of figure; and sometimes form in the way which is represented in the figure. When the heat is too great, as might be expected, the process goes on with great rapidity, and numerous ramifications are formed. This arises no doubt from the confused crystallization.

Fig. 2. shows the microscopical crystals of blue vitriol (fulphate of copper), which appear first round the edges, short at the beginning, but gradually increasing, as they are represented at the letters a, b, c, which denote their difference of form, and the progress of their growth. These crystals, which are transparent, assume a foiled regular form, and reflect the light from their polished sides and angles. As the evaporation proceeds, a great number of filaments as fine as hairs make their appearance, some crossing each other, as at d; and others exhibiting a stellated form with many radiations, as at e. The crystallization of this salt proceeds slowly. Towards the end of the process the regular crystals appear, and are finely branched as at f.

Fig. 3. is a view of the crystals of distilled verdigris, or acetate of copper. When it is immediately applied to the microscope, the regular figures 1, 2, 3, 4, 5, 6, 7, make their appearance; but if the solution is allowed to remain at rest for a few hours, and a drop of it is then heated on a slip of glass till it begins to congeal about the sides, sharp-pointed solid figures are formed, and shoot forwards. These crystals are often irritated obliquely, frequently arise in clutters, or shoot from a centre. Sometimes, towards the end of the process, and in the middle of the drop, they assume a foliated form, and have the appearance of four leaves of fern united by their stems.

Fig. 4. shows the microscopical crystals of alum. These are more or less perfect according to the strength of the solution, and the temperature employed. To prepare this salt for examination, the saturated solution may remain for some days. In that time crystals will form, and if what remains liquid should be found too weak, heat may be applied, which will again dissolve the crystals.

In fig. 5. is a view of the crystals of borax, or the subborate of soda. The drop of this solution should not be held too long over the fire, as it hardens on the slip of glass, and no crystals appear. A brisk heat for about a second is recommended as the best method. It is then applied to the microscope, and the crystals will form as in the figure.

Fig. 6. shows the microscopical crystals of sal ammoniac, or muriate of ammonia. Great numbers of thick, sharp, and broad spicules shoot from the edges, and from their sides are protruded others of the same form, which are parallel to each other, but perpendicular to the main stem. The formation of these crystals, unless the heat employed be very moderate, is very rapid.

Fig. 7. exhibits the appearance of the crystals of acetate of lead (sugar of lead). After a little of this salt is dissolved in hot water, and allowed to remain at rest for a short time, it is fit for being examined with the microscope. A drop of it put on a slip of glass, and heat being applied, will be seen forming round the edge, a regular border of a clear and transparent substance, which with a strong heat runs over the whole of the drop, and hardens on the glass; but when the heat employed is moderate, bundles of lines, arranged In fig. 8, are represented the crystals of Glauber's salt (fulphate of soda), which assume the form of ramifications, proceeding from the side of the drop, like the growth of minute plants. Other appearances present themselves in different periods of the process. It is indeed but of short duration, for when the crystallization has once begun, it goes on with great rapidity.

The examples which we have now given will, we apprehend, be sufficient to enable those who are curious in microscopical observations, to prosecute researches of this kind. Many more might have been given from the same author; but as experiments on crystallization, conducted in this way, are little susceptible of accuracy or precision, we wish to avoid dwelling out the article without conveying some useful information. Our chemical readers will readily perceive, that very different appearances will be the result of a slower or more rapid crystallization, greater or less purity of the salt, and different degrees of strength of the solution. In compound bodies, for instance, modifications in the form of the crystals are produced by a difference in the proportion of the constituent parts. The crystals of alum, which is a triple salt, viz. a sulphate of alumina and potash, are in the form of octahedrons. The addition of a quantity of alumina changes the form of the crystals to that of cubes; and if a cubic crystal of alum be introduced into a solution, the proportions of which afford octahedral crystals, the cubic crystal will assume the form of an octahedron, and the octahedral crystal put into a solution which affords cubic crystals, passes into that of the cube. The nature of the solvent also, in which the crystallization takes place, produces certain deviations in the form of the crystals. The solution of common salt in water affords cubical crystals, but in urine it crystallizes in the form of octahedrons. Muriate of ammonia dissolved in water, crystallizes in the form of an octahedron, but in urine it affords crystals in the form of cubes. But we now proceed to consider the theory of the structure of crystals, which will be the subject of the next section.

Sect. II. Of the Theory of the Structure of Crystals.

In the former section we have given a view of the phenomena of crystallization. The regular forms which bodies assume by means of this process, have occupied no small share of the attention of naturalists and chemical philosophers. The researches and investigations of Bergman, Romé de L'Isle, and Haüy, have been particularly directed this way. Bergman, in his 12th Dissertation*, treats of the variety of the *physic* forms of crystals, of the various figures derived from *Géométrie*, the spatheaceous form, of the structure of the most minute *Fissure*, and of the different modes in which crystals are generated. Romé de L'Isle has arranged crystals into five species, derived from the varieties of form. 1. Tetrahedron. 2. Cube. 3. Octahedron. 4. Parallelopiped. 5. Rhomboidal octahedron. 6. Dodecahedron. But the ingenious researches of Haüy on this subject have been followed by the completest and most successful investigation of the theory of the structure of crystals which has yet appeared. Of this theory, an account of which the reader will find in the Annales de Chimie †, and in his Traité de Minéralogie ‡, we now propose to give a comprehensive view.

This theory, the author observes, cannot be fully understood without the aid of analytical calculations. For beside the convenience of analysis, including in the same formula a great number of different problems, it is by means of it alone, that the theory can assume the character of absolute certainty in arriving at the same results which are obtained by observation. But notwithstanding these considerations, it seemed to be better for those who had not a competent knowledge of the science of calculation to prefer the method of simple reasoning, but accompanied with geometrical figures, which are so useful in giving a distinct conception of the arrangement of the small solids which combine together to form a crystal. This arrangement is denominated *structure*, in opposition to the term *organisation*, which expresses the more complicated mechanism of vegetables and animals. This method may perhaps be less direct, and less precise and expeditious, and it may require attention to those details which are passed over in the analytical method to reach its object more speedily; it has, however, this advantage, that the mind by its means perceives better the connexion of the different parts under consideration, and can more easily comprehend the facts with which it is furnished.

I. Mechanical Division of Crystals.

The same mineral substance, it is known, is susceptible of several different forms, well defined, some of which do not appear, at first sight, to have any common point of resemblance to indicate their relation. If, for instance, we compare the regular hexahedral prism of calcareous spar with the rhomboid of the same mineral (b), whose large angle is about $101^\circ$, we

(b) The name of rhomboid is given by the author to a parallelopiped $a, e$ (fig. 12.) terminated by six equal and similar rhombuses. In every rhomboid, two of the solid angles, such as $a, e$ opposed to each other, are formed by the junction of three equal plane angles; each of the six solid angles is formed by a plane angle equal to each of the three preceding, and by two other angles of a different measure, but equal to each other. The points $a, e$ are the summits, the line $ae$ is the axis. In any one of the rhombuses $ab, df$, which compose the surface, the angle $a$ contiguous to the summit, is called the superior angle; the angle $d$ the inferior angle; and the angles $b$ and $f$ are the lateral angles. The sides $ab, af$ are the superior edges, and the sides $bd, df$ the inferior edges: $bf$ is the horizontal diagonal, and $ad$, the oblique diagonal. The rhomboid is obtuse or acute, according as the angles of the summits are obtuse or acute. The cube is the limit of the rhomboids. Structure of we should be led to believe that each of these two crystals forms is quite distinct from the other. But this point of relation, which eludes notice, when we consider only the external form, becomes sensible when we attend to the intimate mechanism of the structure. Here the author gives a historical view of the progress of his researches, and traces the steps which led him to the discovery of what became as it were the key of his whole theory.

He had in his hand a hexahedral prism of calcareous spar, similar to that mentioned above, and which had been detached from a group of the same crystals. The fracture presented a very smooth surface, situated obliquely, like the trapezium \( p/u/t \) (fig. 9), and which had an angle of 135°, both with the remainder \( l/u/e/f \) of the plane \( i/n/e/f \). Observing that the concave segment \( p/s/u/t \) in which this fracture separated from the crystal, had for its vertex one of the edges of the base, namely the edge \( i/n \), he attempted to separate a second segment in that part to which the contiguous edge \( c/n \) belonged. For this purpose he employed the blade of a knife, directed with the same degree of obliquity as the trapezium \( p/u/t \) and aided by the stroke of a hammer. This attempt failed; but having tried the same operation towards the next edge \( b/c \), a new trapezium similar to the first came into view. The fourth edge \( a/b \) resisted the instrument, but the following, \( a/h \), readily yielded to mechanical division, and presented a third trapezium, having as fine a polish as the other two. The fifth edge \( i/h \), it is scarcely necessary to observe, could not be divided, more than the fourth and the second.

Proceeding then to the inferior base \( d/e/f/g/k/r \), it was soon found that the edges of this base, which admitted of divisions similar to the preceding, were not the edges \( e/f, d/r, g/k \), which corresponded to those which could be divided towards the upper part, but the intermediate edges \( d/e, u/y, g/f \). The trapezium \( q/y/v \) shows the section made below the edge \( k/r \). This section is obviously parallel to that of the trapezium \( p/u/t \) and the four other sections are in like manner parallel, two and two. Now, these different sections being in the direction of the natural joints of the laminae, it was easy to obtain others parallel to each of them, but it was found impossible to divide the crystal in any other direction. Following this mechanical division according to the parallelism stated above, new sections were obtained, always nearer to the axis of the prism; and when the sections were carried so far as to make the remainder of the two bases disappear, the prism was transformed into a solid \( O/X \) (fig. 10.) terminated by 12 pentagons, parallel two to two, of which those of the extremities, namely, SAOIR, GIODE, BAODC, on the one side, and KNPOF, MNPXU, ZQPXY, on the other, were the results of the mechanical division, and had their common vertices \( O, P \); situated in the centres of the bases of the prism, fig. 9. The five lateral pentagons RSUXY, ZYRIG, &c. (fig. 10.) were the remainders of the planes of the same prism.

In proportion as the sections were multiplied, always parallel to the preceding, the lateral pentagons diminished in height, and at a certain term the points \( R, G \) being compounded with the points \( Y, Z \), the points \( S, R \) with the points \( U, Y, &c. \) there remained no structure more of these pentagons, but the triangles \( YIZ, UX/Y, &c. \) (fig. 11.) Beyond that term the sections coming to pass over the surface of these triangles, diminished gradually in extent, till at last the same triangles were lost, and then the solid obtained from the hexahedral prism, appeared to be a rhomboid \( a/e \) (fig. 12.) exactly similar to that which is commonly denominated Iceland spar.

So unexpected a result led the ingenious author to the examination of other calcareous crystals in a similar manner, all of which yielded to mechanical division in such a way, as, when all the external surfaces had disappeared, the nucleus which remained was always a rhomboid, of the same form as the first. All that was necessary was to discover the direction of the sections which conducted to the central rhomboid.

To extract, for instance, this rhomboid from the spar which is usually denominated tectonic, and which is itself a much more obtuse rhomboid, having its large plane angle equal to \( 114^\circ 18' 56'' \), it was necessary to begin with the two vertices, and to make the sections pass through the small diagonals of the faces. But if it is wished, on the contrary, to get at the nucleus of the rhomboidal spar with acute vertices, the direction of the sections of the planes must be parallel to the edges contiguous to the summits, and in such a manner that each of them shall be equally inclined to the faces which it cuts.

These results are the more worthy of attention, as it would seem at first, that in the process of crystallization, after the rhomboid has been once adopted with regard to a determined species of mineral, it ought always to re-produce it with the same angles. But the paradox which arises from this diversity of appearance, is explained by the double use of the rhomboidal form, which serves here to disguise itself, and conceals fixed and constant characters under a variable external appearance.

If we take a crystal of a different nature, such as a cube of fluor spar, the nucleus will have a different form. This will be, in the present case, an octahedron, which we shall obtain by taking off the eight solid angles of the cube. Heavy spar will produce for a nucleus a right prism with rhombohedral bases; seld spar, an oblique-angled parallelepiped, but not rhombohedral; apatite or beryl, a right fixed-sided prism; the adamantine spar a rhomboid, a little acute; blende, a dodecahedron, with rhombohedral planes; iron of the island of Elba, a cube, &c.; and each of these forms will be constant in relation to the whole species, so that its angles will undergo no variation which is appreciable: and if we attempt to divide the crystal in any other direction, we shall not be able to find any joint; we shall only obtain indeterminate fragments; it will rather be broken than divided.

These solids inscribed each in all the crystals of the same species, ought to be regarded as the true primitive forms on which all the other forms depend. All minerals, it is true, are not susceptible of mechanical division, but the number is greater than what appeared at first sight; and with regard to those crystals in which the attempts to discover the natural joints have failed, it has been remarked that their surface striated in a certain direction, or the relation of their different forms, II. LAWS OF DECREMENT.

1. Decrements at the Edges.

The primitive form, and that of the integrant molecules, being determined, after the diffusion of the crystals, we must investigate the laws according to which these molecules were combined, to produce around the primitive form those kind of coverings which terminated in regularly, and from which resulted polyhedra different from each other, although originally of the same substance. Now, such is the mechanism of the structure subject to these laws, that all the parts of the secondary crystal superadded to the nucleus, are formed of laminae, which decrease regularly by subdivisions of one or more ranges of integral molecules, so that theory determines the number of these rows, and by a necessary consequence the exact form of the secondary crystal.

To have a distinct idea of these laws, let us take a very simple and elementary example. Conceive EP (fig. 13.) to represent a dodecahedron whose faces are equal and similar rhombuses, and that this dodecahedron is a secondary form, having a cube for its nucleus or primitive form. By the inspection of fig. 14., the position of this cube in the crystal may be easily conceived. The small diagonals DC, CG, GP, FD of the four faces of the dodecahedron, being united round the same solid angle, form a square CDFG. Now there are six solid angles composed of the four planes, namely the angles L, O, E, N, R, P (fig. 13.), and consequently, if sections are made to pass through the small diagonals of the faces which compose the solid angles, six squares will be successively uncovered. These squares will be the faces of the primitive cube, of which three are represented at fig. 14., namely CDFG, ABCD, BCGH.

This cube would evidently be an assemblage of cubic integral molecules, and it would be necessary that each of the pyramids, such as LDCGF (fig. 14.), which rest on the faces, should be itself composed of cubes equal to each other, and to those which form the nucleus. To have a more distinct conception of this arrangement, let us compose an artificial dodecahedron of a certain number of small cubes, the arrangement of which will be an imitation of the process of nature in disposing the molecules in the formation of the dodecahedron.

Let ABGF (fig. 15.) be a cube composed of 729 small cubes equal to each other, in which case each face of the whole cube will include 81 squares; that is, 9 on each side, which will be the external faces of as many partial cubes representing the molecules. This cube will be the nucleus of the dodecahedron which is to be constructed. On one of the faces, as ABCD, of the cube apply a square lamina, composed of cubes equal to those which form the nucleus, but having towards each a row of cubes less than if it were on a level with the contiguous faces BCGH, DCGF, &c. This lamina will be composed of 49 cubes, that is, 7 on each side, so that if the inferior base be on fig. (fig. 16.) this base will fall exactly on the square marked with the same letters in fig. 15. Above this first lamina let a second be applied, composed of 25 cubes, 5 on each side, so that if Impu (fig. 17.) represent its Structure of its inferior base, this base will correspond exactly with crystals marked with the same letters in fig. 15. If in like manner a third lamina be applied to the second, which is composed only of 9 cubes, that is, 3 on each side, so that \( v = y \approx (Fig. 18.) \) being the inferior base, shall correspond with the square marked with the same letters in fig. 15.; and if on the middle square \( r \) of the preceding lamina the small cube \( r \) (fig. 19.) be placed, this will represent the last lamina.

When this operation is completed, it will appear that there is formed on the face \( ABCD \) (fig. 15.) a four-sided pyramid, of which this face is the base, and the cube \( r \) (fig. 19.) is the summit. And if the same operation be continued on the other five sides of the cube, we shall have fixed four-sided pyramids, resting on the six faces of the nucleus, which is enveloped with them on all sides. But as the different rows of laminae composing these pyramids project beyond each other for a certain way, as appears on fig. 20., where the parts raised above the planes \( BCD, BCG \) represent the two pyramids which rest on the faces \( ABCD, BCGH \), (fig. 15.), the faces of the pyramids will not form continuous planes; for they will be alternately re-entering and salient, in some measure imitating a stair with four sides.

Let us now suppose that the nucleus is composed of a number of almost imperceptible cubes incomparably greater, and that the laminae applied on the different faces, which may be called the laminae of superposition, continue to increase towards their four edges by subdivisions of one range of cubes equal to those of the nucleus, the number of these laminae will be incomparably greater than in the preceding hypothesis; and at the same time the cavities or furrows which they form, as they alternately become salient or re-entering, will be almost imperceptible; and indeed it might be supposed that the cubes of which the crystal is composed are so small as to become quite imperceptible to our senses, and the faces of the pyramids to be perfectly smooth.

Now \( DCBE \) (fig. 20.) being the pyramid which rests on the face \( ABCD \) (fig. 15.), and \( CBOG \) (fig. 20.) the pyramid applied to the face \( BCGH \) (fig. 15.), if we consider that every thing is uniform from \( E \) to \( O \) (fig. 20.) in the manner in which the laminae of superposition mutually project beyond each other, we may readily conceive that the face \( CEB \) of the first pyramid ought to be exactly in the same plane as the face \( COB \) of the contiguous pyramid, so that the union of these two faces should form a rhombus \( ECOB \). But we have, for the 6 pyramids, 24 triangles similar to \( CEB \), which consequently will be reduced to 12 rhombuses, from which results a dodecahedron similar to what is represented in fig. 13. and 14.

The cube, before it arrives at the form of the dodecahedron, passes through a multitude of intermediate modifications, of which one is shown at fig. 21. The squares \( p a e o, k l q u, m n i s, &c. \) correspond to the squares \( ABCD, DCGF, CBHG, &c. \) (fig. 14.), and form the superior bases of as many pyramids, incomplete structures from the deficiency of the laminae with which they ought to terminate. The rhombuses \( EDLC, ECOB \) (fig. 13.), by a necessary consequence, are reduced to simple hexagons \( a c / k D, c o B n m C \) (fig. 21.), and the surface of the secondary crystal is composed of 12 of these hexagons and 6 squares. This is the case with the boracic spar (the borate of magnesia and lime), with the exception of some facets which surmount the solid angles, and which depend on a different law of decrement.

If the diminution of the laminae of superposition proceeded in a more rapid ratio; for example, if each lamina had had on its circumference, two, three, or four rows of cubes less than the inferior lamina, the pyramids produced on the nucleus by this diminution being more depressed, and their contiguous faces being no longer on a level, the surface of the secondary crystal would have been composed of 24 icosceles triangles, all inclined to each other. Decrement on the edges, is that which takes place parallel to the edges of the nucleus, and it ought to be distinguished from another kind of decrease to be afterwards mentioned.

2. Examples of Decrease on the Edges.

Marital Pyrites, or Dodecahedral Sulphuret of Iron.

Geometric Character.—Inclination of any one of the pentagons, as \( DPRFS \) (fig. 27.), to the pentagon \( CPRGL \), which has the same base \( PR, 126° 56' 8'' \).

Angles of the pentagon \( CPRGL, L = 121° 35' 17''; C \text{ or } G = 106° 35' 57'' 30''; P \text{ or } R = 102° 36' 19'' \).

Let us conceive again a cubic nucleus, whose different edges are lines of departure to the same number of decrements which take place at the same time in two different ways; that is, by the subtraction of two rows parallel to the edges \( AB, CD \) (fig. 15.), and of one row parallel to the edges \( AB, BC \). Let it be supposed also that each lamina being only equal in thickness to a small cube of the side \( AB \) and \( CD \), is on the contrary equal to double the thickness of the side \( AD \) and \( BC \). Fig. 22. represents this disposition with regard to the decrements which proceed from the lines \( DC, BC, \) (fig. 15.). It is plain that on account of the more rapid decrease in proceeding from \( DC \) or \( AB \), than from \( BC \) or \( AD \), the faces produced in the first case will be more inclined to the plane \( ABCD \), while the faces produced in the second will remain as it were behind, so that the pyramid will no longer be terminated by a single cube \( E \), as in fig. 20., which on account of its minuteness seems to be only a point, but by the row of cubes \( MNST \) (fig. 22.) which, supposing these cubes to be infinitely small, will prevent the appearance of a simple ridge. By a necessary consequence, the pyramid will have for its faces two trapeziums, such as \( DMNC \), resulting from the first decrement, and two icosceles triangles, such as \( CNB \), which will be the effect of the second decrement (c).

Let

(c) Here the face which corresponds to \( ABCD \) (fig. 15.) has 25 squares on each side, as may be seen in fig. 22. The structure of this pyramid may be imitated artificially, by regulating the arrangement and number of the cubes represented in the same figure. Let us suppose farther, that with regard to the laminae of superposition, which arise on the face BCGH (fig. 15.), the decrements follow the same laws, but in crooked directions; in such a way that the more rapid of the two may take place in proceeding from BC, or from GH, towards the vertex of the pyramid, and the slower decrement in proceeding from CG, or BH, towards the same vertex. The pyramid which results from these decrements will be placed in a direction opposite to that which rests on ABCD, and will have the position represented at fig. 25, where the edge KL, which terminates the pyramid, instead of being parallel to CD, like the edge MN, (fig. 22. and 23.) is on the contrary parallel to BC. We shall then conceive what is to be done, that the pyramid which will rest on DC, GF (fig. 15.) may be turned as it is represented in fig. 24., and may have its terminating edge PR parallel to CG (fig. 15.). The pyramids which will rest on three other faces of the cube, will stand like that which arises on the opposite face.

But as the decrements which produce the triangle CNB (fig. 23.) make a continuity with those from which results the trapezium CBKL (fig. 25.), these two figures will be in the same plane, and will form a pentagon CNBKL (fig. 26.). For the same reason the triangle DPC (fig. 24.) will be on a level with the trapezium DMNC (fig. 23.) and by applying the same reasoning to the other pyramids, it will be conceived that the six pyramids having for their whole faces 12 trapeziums and 12 triangles, the surface of the secondary solid will be composed of 12 pentagons, which will correspond to the 12 rhombuses of fig. 13., but with this difference, that they will have other inclinations.

This solid is represented at fig. 27., and with its cubic nucleus at fig. 28., where it may be seen how to proceed in the extraction of this nucleus. If, for example, a section be made passing through the points D, C, G, F, the pyramid which rests on the face DCGF of the nucleus will be detached, and by this section the latter will be uncovered.

Among the crystals belonging to the sulphurite of iron, or the arseniate of cobalt, there is found a dodecahedron, having the faces equal and similar pentagons, and having for its nucleus a cube in the position above described. But there are an infinite number of possible dodecahedra, which may have for faces equal and similar pentagons, and will differ from each other by the respective inclinations of their faces. Of all these dodecahedra, the one whose structure would be subjected to these laws, gives $126^\circ 56' 8''$ as the angle formed by the inclination of any two of its faces DPRFS, CPRGL (fig. 27.) at the edge of junction PR, as might be shewn by calculation. Some mineralogists, overlooking the use of geometry in the consideration of crystals, have confounded the dodecahedron of pyrites with the same regular geometrical figure in which all the sides and angles of each pentagon are equal; but there is a striking difference between these two dodecahedra. The regular dodecahedron gives only $116^\circ 33' 54''$, as the inclination of its respective pentagons, making a difference of nearly $11^\circ 22'$ between it and the other. And indeed the regular dodecahedron cannot be produced by any law of decrement whatever, however compound it may be supposed, in regard to a cubic nucleus; and, as may be demonstrated generally, for a nucleus of any form. There are then two kinds of dodecahedra, one whose faces are rhombuses, and another whose faces are pentagons, produced upon a cubical nucleus, in consequence of two simple and regular laws of decrement, in a direction parallel to the edges of the nucleus. By varying these laws in different other ways, a multitude of new polyhedra, having the same nucleus may be constructed.

**Obtuse or Lenticular Calcareous Spar**, (fig. 30.)

**Geometric Character.** Inclination of the rhombus n a d b', to the rhombus a i f' d', $134^\circ 25' 36''$. Angles of the rhombus n a d b'; a o b' = $114^\circ 18' 56''$; n o r d' = $63^\circ 41' 4''$.

This variety arises from a decrement by a single row on both sides of the edges a b, a g, a f' (fig. 31.) and e o, e d, e c, contiguous to the summits a, e, of the nucleus. An idea may be formed of its structure, by comparing it with that of the dodecahedron whose planes form rhombuses (fig. 13. and 20.), originating from the cube, (fig. 15.) and by supposing that the laminae, instead of decreasing at the same time on all the edges, decrease only to those contiguous, three by three, to the angle C and its opposite. The faces formed in that case will be reduced to six, which, by prolonging themselves, according to the law of continuity, so as to interfect each other, will compose the surface of a rhomboid analogous to the one which we are now treating of, excepting that it will have other angles, on account of the cubical form of its integral molecule.

From this it may be conceived, that the diagonals drawn from a to b' (fig. 30.), from a to g', from a to f', &c., on the secondary rhomboid, will be confounded with the edges a b', a g', a f' (fig. 31.) of the nucleus, which serve as lines of departure for the decrements; and hence to extract this nucleus, the planes of the sections must pass along these diagonals, as has been already remarked.

**Common Topaz**, (fig. 33.)

**Geometric Character.** The inclination of the trapezoid s r t m to the adjacent plane r t e y, $136^\circ$ of the same plane, to k r y z, $124^\circ 26'$; of the plane i n g e to m l i g, $90^\circ$.

The primitive form of this topaz is that of a right-angled, four-sided prism k y (fig. 32.), the bases of which are rhombuses, having the angle h o r = $124^\circ 26'$. According to theory, in regard to the integrant molecule, the height r y is to the side r a nearly in the ratio of 3 to 2. The pyramidal summit of the topaz results from a decrement by two rows of small prisms on the edges x r, r n, n h, h x of the superior base of the primitive form. The planes i n g e, l m g e (fig. 33.) on one side, and b k o p, b u d p, on the other, arise from a decrement by three rows on each side of the edges n v, a q (fig. 32.), which decrement remains suspended at a certain term, and leaves four rectangles t r y e, k r y z, l h i, u h c d, (fig. 33.), parallel to the planes of the primitive form. The effect of this decrement is shewn at fig. 34., where the rhombus h n r w is the same as fig. 32.; and all the small rhombs by which it is subdivided, or which are exterior to it, represent the bases of so many molecules. The lines r d', x z, n i, n e, are... 3. Decrement on the Angles.

This position of the rhombohedral nucleus inclosed in the regular hexahedral prism of the calcareous spar being discovered, did not directly lead to the determination of the laws of those decrements of secondary crystals. More simple intermediate steps were necessary. To conceive the method of investigating these new decrements, it may be remarked that the same substances which exhibit the dodecahedron with pentagonal planes originating from the cubes (figs. 27. and 28.), and which might assume the form of the dodecahedron whose planes are rhombuses (figs. 13. and 14.), are found also under that of the regular octahedron. But if the laminae of superposition decrease only on the edges of the two opposite faces of this cube, as on those of the superior base ABCD (fig. 14.) and of the inferior base, we shall in general have two pyramids applied on these bases. And if we suppose the effect of the law of decrements continued in the space situated between the bases of the cube, we shall arrive at an octahedron, whose angles will vary as there is a greater or smaller number of rows subtracted. But no law, however complicated, can give equilateral triangles as the faces of this octahedron.

On the other hand, by dividing a regular octahedron originating from a cube, the cubic nucleus will appear to be so situated in this octahedron that each of its six solid angles corresponds to the centre of one of the faces of the octahedron; but this could not be the case by supposing a decrement on the edges. The law of decrement accomplishes its ends; in such cases, by a different progress from that which conduces to the forms already described.

Let ABCD (fig. 35.) be the superior or inferior surface of a lamina composed of small cubes, whose bases are represented by the squares which subdivide the whole square. The series of cubes to which the squares a, b, c, d, e, f, g, h, i, belong, are on the diagonal drawn from A to C; and they form one string, (fig. 36.) which will not differ from the string of the cubes a', n, g, r', s', t', u', v', w', x', y', z', (fig. 35.), lying in the direction of the edge AD, excepting that in the former the cubes touch only by one of their edges, and in the latter by one of their faces. There are also, throughout the whole extent of the lamina, strings of cubes parallel to the diagonal. The series of letters q, v, k, u, x, y, z, flows one, and the letters n, t, l, m, p, o, r, s, flows another string.

The laminae of superposition, it may be conceived, project beyond each other one or more rows of cubes in a direction parallel to the diagonal. In like manner may be constructed around the cubic nucleus, solids of different figures, by placing successively above the different faces of this nucleus laminae which may arise in the form of pyramids, and which will experience this kind of decrement. The faces of these solids will be roughened by an infinite number of salient angles formed by the exterior points of the composing cubes. This follows from the angular figure which is continually presented by the edges of the laminae of superposition. But these points being on a structure of level, the cubes may be supposed to be so joined that the faces of the solid will appear smooth and continued planes.

Around the cube ABGF (fig. 37.), as a nucleus, let a secondary solid be constructed, in which the laminae of superposition shall decrease on all sides by a single row of cubes, in a direction parallel to the diagonals; and let ABCD (fig. 38.), the superior base of the nucleus, be subdivided in 81 small squares, representing the exterior faces of an equal number of molecules. Fig. 39. represents the superior surface of the first lamina which ought to be placed above ABCD (fig. 38.) in such a manner that the point a' may correspond to the point a, the point b' to the point b, the point c' to the point c, and the point d' to the point d. By this disposition the squares Aa, Bb, Cc, Dd (fig. 38.) remain uncovered, which will fulfill the above law of decrement; and the borders QV, ON, IL, GF (fig. 39.) project by one row beyond the borders AB, AD, CD, BC (fig. 37.), which is necessary that the nucleus may be enveloped towards these edges. For if the edges of the lamina represented (fig. 39.), as well as the following, coincided with the lines ST, EZ, YX, MU, on which superposition they would be on a level with AD, AB, CD, BC (fig. 38.), they would form re-entering angles towards the analogous parts of the crystal. Thus in the lamina applied on ABCD (fig. 37.) all the edges answering to CD would be on a level with CDFG, of which they would form a prolongation; and in the lamina applied on DCFG all the edges analogous to the same ridge, CD would be on a level with ABCD, from which necessarily results a re-entering angle opposite to the salient angle formed by the two faces ABCD and CDFG. But by the laws which determine the formation of simple crystals, re-entering angles appear to be excluded. The solid will then increase in those parts to which the decrement does not extend. But this decrement alone being sufficient to determine the form of the secondary crystal, all the other variations which intervene only in a subsidiary manner may be set aside, excepting in the construction of artificial crystals, and in exhibiting the details relating to the structure.

The superior face of the second lamina will be like A'G'L'K' (fig. 40.), and this lamina must be placed above the preceding, in such a manner that the points a'', b'', c'', d'', may correspond with the points a', b', c', d' (fig. 39.), which will leave uncovered the squares having their exterior angles situated in Q, S, E, O, V, T, M, G, &c., and continuing to produce the decrement by one row. The solid increases towards the analogous edges at AB, BC, CD, AD (fig. 38.) since between A' and L', for instance, (fig. 40.) there are 13 squares, but between QV and LI (fig. 39.) there are only eleven.

The large faces of the laminae of superposition which were hitherto octagons QVGFILO (fig. 39.) having arrived at the figure of the square A'G'L'K' (fig. 40.), will after passing that term, decrease on all sides at the same time; and the following lamina will have for its superior face the square BM'VS (fig. 41.), less in every direction by one row than the square A'G'L'K' (fig. 40.). Let this square be disposed above the preceding, so that the points a'', f'', g'', h'' (fig. 41.) may correspond with the Thus it follows, that the laminae of superposition applied on the base ABCD (fig. 37. and 38.) produce, by the total of their decreasing edges, four faces, which in proceeding from the points A, B, C, D, incline one to another in the form of a pyramidal summit. These edges, it may be remarked, have lengths which begin by increasing as in fig. 39. and 40. and which then proceed decreasing. Thus the faces produced by the same edges increase at first, and afterwards decrease in breadth, so that they become quadrilaterals. One of these is represented at fig. 47., in which the inferior angle C is confounded with the angle C (fig. 37.) of the nucleus; and the diagonal LQ represents the edge L'G' of the lamina A'G'L'K' (fig. 40.), which is the most extended in the direction of that edge. And the number of laminae of superposition producing the triangle LCQ (fig. 47.) being less than that of the laminae producing the triangle LZQ, since there is here only one lamina preceding the lamina A'G'L'K' (fig. 40.), while there are six which follow it as far as the cube Z (fig. 46.) inclusively, the triangle LZQ (fig. 47.) composed of the sum of the edges of these latter laminae, will have a much greater height than the inferior triangle LCQ, as it is expressed in the figure.

The surface of the secondary solid, then, will be formed of 24 quadrilaterals, disposed three and three around each solid angle of the nucleus. But decreasing by one row, the three quadrilaterals belonging to each solid angle, such as C (fig. 37.) will be in the same plane, forming an equilateral triangle ZIN (fig. 48.). The 24 quadrilaterals, then, will produce eight equilateral triangles. One of these is represented at fig. 49. shewing the arrangement of the cubes that concur to form it; and the secondary solid will be a regular octahedron. This octahedron is represented at fig. 50., enclosing the cubic nucleus, so that each of its solid angles corresponds to the centre of one of the triangles IZN, IPN, PIS, SIZ, &c. of the octahedron. To extract this nucleus, it would be necessary to divide the octahedron in its eight solid angles, by sections parallel to the opposite edges. This is the structure of octahedral sulphuret of lead or galena.

Such then is an example of decrements on the angles which take place in a direction, parallel to the diagonals. By this denomination may be expressed precisely the result of each decrement, by denoting the angle which serves it as a point of departure.

Acute calcareous Spar, (fig. 51.)

Geometric Character.—Inclination of \( p \approx r \) to \( p \approx y \); \( 78^\circ 27' 47'' \), and to \( i \approx s \); \( 101^\circ 32' 13'' \). Angles of the rhombus \( p \approx r \); \( p \approx r = 75^\circ 31' 20'' \); \( z \approx y = 104^\circ 28' 40'' \). Inclination of the oblique diagonal drawn from \( p \) to \( r \) with the edge \( p \approx u \); \( 71^\circ 33' 54'' \).

Geomet. Propert.—The angles of the rhombus are equal to the respective inclinations of the faces of the Structure of nucleus, and reciprocally. The angles of the principal quadrilateral, or that which passes through two opposite oblique diagonals \( p \approx r \); \( u \approx r \), and through the intermediate edges \( p \approx u \); \( i \approx r \), are the same as on the nucleus.

To conceive the structure of this rhomboid, suppose that \( a \approx b \approx f \) (fig. 52.) represents the face of the nucleus marked with the same letters (fig. 12.) subdivided into a multitude of partial rhombules, which are the exterior faces of so many molecules. Suppose farther, that the laminae of superposition, applied on this face, decrease by one row towards the lateral angles \( a \approx d \); \( a \approx d \), in such a manner, that on the first the two rhombules \( b \approx h \); \( l \approx f \approx m \approx n \) are uncovered; that on the second the uncovered rhombules are those traversed by the diagonals \( c \approx u \); \( y \approx x \), on the third those traversed by the diagonals \( s \approx q \); \( z \approx c \); &c.; in which case the decreasing edges will successively correspond with these diagonals. By this law of decrement two faces will be produced; which, proceeding from the angles \( b \approx f \), will rise in the form of a roof above the rhombus \( a \approx d \), and will meet on a common edge situated immediately above the diagonal \( a \approx d \), and which will be parallel to it; and, as there are six rhombules, which undergo like decrements on the primitive form, the faces produced will be 12 in number. But, by the law of decrement by one row, the two faces which have the same angle \( b \approx f \); \( g \approx c \) (fig. 12.) for the point of departure, will be in the same plane; thus reducing the 12 faces to six, and transforming the secondary crystal into an acute rhomboid \( p \approx i \) (fig. 51.). In this rhomboid the edges \( p \approx s \); \( p \approx y \); \( p \approx u \), are situated each as the oblique diagonals of the nucleus, or those which would be drawn from \( a \) to \( d \), from \( a \) to \( x \), from \( a \) to \( c \); &c. (fig. 12.).

Crystals of this variety are found near Lyons in France; and the freestone of Fontainbleau, commonly called crystallized freestone, which is nothing else than calcareous spar, mixed with particles of quartz, exhibits the same form. The crystals of this stone yield to mechanical division, and have their natural joinings like those of pure spar, situated in the planes parallel to the edges \( p \approx s \); \( p \approx y \); \( p \approx u \); &c. (fig. 51.), and which would pass at an equal distance from these edges.

Rhomboidal Iron ore, (fig. 53.)

Geometric Character.—Inclination of BCRP to BCOA or OCRS, \( 146^\circ 26' 33'' \); angles of the rhombus BCRP, C or \( P = 119^\circ 2' 9'' \); B or \( R = 62^\circ 57' 51'' \).

The lamina composing this rhomboid decreases by two rows on the angles \( b \approx c \); \( o \approx r \); \( b \approx c \); &c. (fig. 54.) which concur to the formation of the two solid angles \( c \approx n \) of a cubic nucleus. The faces produced, instead of being on a level, three and three, around their angles, as in the case of decrement by a single row, incline one to the other, and extend above the faces of the nucleus in such a manner, that their diagonals are parallel to the horizontal diagonals of the same faces. The cube here answers the purpose of a rhomboid, which should have its summits in \( c \) and \( n \), in which case there would be only one axis passing through the summits. In the dodecahedron, on the other hand, with pentagonal planes... Structure of planes (fig. 27.) the cube performs the functions of a rectangular parallelopipedon, and then three different axes may be conceived, each of which passes through the middle of the two opposite faces. When the cube begins to perform the one or the other, in regard to one species of mineral, it is observed to continue that function in all the varieties of that species.

The crystals of rhomboidal iron are found among those of the iron ore of the island of Elba. It is uncommon, however, for the law of decrement to attain to its boundary, and for the rhomboid not to be modified by facets parallel to the faces of the nucleus. If the decrement which produces the rhomboid took place at the same time on the eight solid angles of the cube, there would result a polyhedron of 24 facets, of which nature are the crystals found at the Calton hill, Edinburgh, which have been considered as zeolites.

4. Intermediate Decrements.

In some crystals the decrements on the angles do not take place in lines parallel to the diagonals, but parallel to lines situated between the diagonals and the edges. This happens when the subtractions are made by double, triple, &c. rows of molecules. In figure 53, which is an instance of these subtractions, the molecules composing the row represented by the figure, are so arranged as if, of two, one only was formed. To reduce this case under that of the common decrements on the angles, we have only to conceive the crystal composed of parallelopipeds, having their bases equal to the small rectangles \(a b c d\), \(e f g h\), \(h i j l\), &c. The name of intermediate decrement is given to this kind of diminution.

Syntactic Iron Ore, (fig. 50.)

Geometric Character.—Respective inclination of the trapeziums \(b e g o\), \(n q g o\) of the rising pyramids \(=135° 34' 31''\); of the edges \(c g\), \(g q\), \(129° 31' 16''\). Angles of the trapezium \(b e g o\), \(b o = 103° 48' 35''\); \(o g = 76° 11' 25''\).

This variety of iron ore is found at Framont in Les Vosges. It commonly appears under the form of two opposite pyramids, and some groups reflect from the surface the prismatic colours. These crystals, classified by De L'Isle among the modifications of the dodecahedron with isosceles triangular planes, have for nucleus a cube performing the functions of the rhomboid. The two regular hexagons by which they are terminated, arise from a decrement by a single row of cubic molecules on the angles \(c\), \(n\), (fig. 54.) of the nucleus.

To comprehend the effect of this law, combined with the preceding, and which produces the lateral trapeziums, let it be supposed that \(c b p r\) (fig. 57.) represents the same square as fig. 54., subdivided into small squares, which are the external faces of so many molecules. Taking these molecules by pairs, so that they form rectangular parallelopipeds, having for bases the oblong squares \(b n g h\), \(h g m G\), &c. and imagine, that the subtractions are made by two rows of these double molecules, the edges of the laminae of superposition will be successively ranged in lines, as \(PG\), \(TL\), \(R P\), \(S P\), \(k z\), \(y z\), &c. and the sum of all these edges will produce two faces, which departing from the angles \(b\), \(r\), will converge, the one towards the other, and will unite themselves on a common ridge, situated above the diagonal \(c b\), but inclined to that diagonal.

The complete result of this decrement, then, is 12 faces; and it is shown by calculation, that the six superior faces being prolonged to the point where they meet the six lower faces, will form with them the surface of a dodecahedron, composed of two right pyramids united at their bases. By the effect of the first law, these pyramids are here incomplete, which gives the hexagon \(a b c d r u\) (fig. 56.) and its opposite.

5. Mixed Decrements.

The decrements in other crystals, either on the edges, or on the angles, vary according to laws, the proportion of which can only be expressed by the fraction \(\frac{1}{4}\) or \(\frac{1}{2}\). It may happen, for instance, that each lamina exceeds the following by two rows parallel to the edges, and that it may, at the same time, have an altitude triple that of a simple molecule. A vertical, geometrical section of one of the kinds of pyramids, resulting from this decrement, is represented at fig. 62.

The effects of this decrement may be readily conceived by considering that \(AB\) is a horizontal line taken on the upper base of the nucleus \(b a z r\), the section of the first lamina of superposition, \(g f e n\) that of the second. These are called mixed decrements, which exhibit this new kind of exception from the simplest laws. They, as well as the intermediary ones, rarely exist anywhere else, and they have been particularly discovered in certain metallic substances. The application of the ordinary laws, Haüy observes, to a variety of these substances, prevented such errors in the value of the angles, as led him to believe that they were inconsistent with theory. But extending his theory, he arrived at results so correct as removed every doubt of the existence of the laws on which these results depended.

All the changes to which crystals are subjected depend on the laws of structure which have been explained, and others of a similar kind. The decrements sometimes take place at the same time on all the edges, as in the dodecahedron having rhombuses for its planes, or on all the angles, as in the octahedron originating from a cube. Sometimes they take place only on certain edges of certain angles. There is sometimes a uniformity between them, so that it is one single law by one, two, three rows, &c. which acts on the different edges, or the different angles. Sometimes the law varies from one edge to the other, or from one angle to the other. This happens particularly, when the form of the nucleus is not symmetrical, as, for instance, when it is a parallelopiped, whose faces differ by their respective inclinations, or the measure of their angles. In some cases there is a concurrence of the decrements on the edges, with those on the angles, to produce the same form; and sometimes the same edge or the same angle is subjected to several laws of decrement succeeding each other. The secondary crystal, in some cases, has faces parallel to those of the primitive form, and which combine with the faces produced by the decrements to modify the figure of the crystal. Simple secondary forms, are those which arise from a single law of decrement, the effect of which entirely conceals the nucleus. Compound secondary forms arise from several simultaneous laws of decrement, or from one single law not having attained to its extent; so that there remain... Crystalization.

The structure of main faces parallel to those of the nucleus, which cut with the faces produced by the decrement, to diversify the form of the crystal. If, amidst this diversity of laws, sometimes inflated, sometimes united by more or less complicated combinations, the number of the rows subtracted were itself extremely variable; if, for instance, these decrements were by 12, 20, or 30 rows, or more, which is possible, the number of forms which might exist in each kind of mineral would be immense. But the power by which the subtractions are effected, seems to be very limited in its action. Its extent rarely exceeds 1 or 2 rows of molecules. Beyond four rows, only one variety of calcareous spar has been discovered. The structure of this variety depends on a decrement by five rows; but this seems to be a rare occurrence in nature. Yet, although the laws of crystallization are limited to two of the simplest, that is, those which produce subtractions by one or two rows, calcareous spar is susceptible of 2544 different forms, a number exceeding more than 50 times that of the forms at present known; and, admitting into the combination decrements by 3 and 4 rows, calculation will give 8,388,604 possible forms of the same substance, and by the operation of either mixed or intermediate decrements, this number will be greatly augmented.

The striae observed on the surface of many crystals is another proof in favour of the theory; for they always have directions parallel to the projecting edges of the laminae of superposition, which mutually go beyond each other, if the regularity of the process has not been disturbed. It must not, however, be supposed, that the inequalities arising from the decrements must be always sensible, if the form of the crystals be complete; for the molecules being extremely minute, the surface will appear finely polished, and no striae would be perceptible. In some secondary crystals, therefore, they are not to be seen, while they are quite distinct in others of the same nature and form. In the latter case, the action of the causes producing crystallization, has not enjoyed all the necessary conditions; the operation has been interrupted; and the law of continuity not having been observed, there have remained on the surface of the crystal, perceptible vacancies. These deviations have this advantage, that they point out the direction, according to which the striae are arranged in lines, and thus contribute to discover the real mechanism of the structure.

The small vacuities which the edges of the laminae of superposition leave on the surface of even the most perfect secondary crystals, by their re-entering and salient angles, show that the fragments obtained by division, whose external facets form part of the faces of the secondary crystal, are not like those drawn from the interior part. For this apparent diversity arises from these facets being composed of a multitude of small planes, really inclined to each other, but which being very minute, present the appearance of one plane. And if the division could reach its utmost bounds, these fragments would be resolved into molecules similar to each other, and to those situated towards the centre. It happens, too, that molecules of different figures arrange themselves in such a manner, as to produce similar polyhedra in different kinds of minerals. Thus the dodecahedron with rhombuses for its planes, which is obtained by combining cubic molecules, exists in granite, with a structure composed of small tetrahedra, having isosceles triangles for its faces. It exists also in sparry fluor, where there is also an assemblage of tetrahedra, but regular; that is to say, the faces of which are equilateral triangles.

Examples of Compound Secondary Forms.

Prismatic Calcareous Spar, (fig. 9.)

The bases of this prism are produced in consequence of a decrement, by a single row on the angles of the summits b a f, g a f, b a g, d e x, c e c, c e x (fig. 12.) of the primitive form. The six planes result from a decrement by two rows on the angles b d f, f x g, b c g, d f x, d b c, c g x, opposite to the preceding. Let a b d f (fig. 58.) be the same face of the nucleus, as fig. 12. The decreasing edges situated towards the angle of the summit a, will successively correspond with the lines h i, k l, &c., and those which look towards the inferior angle d, will have the positions pointed out by m n, o p; but as the first decrement takes place by one row, it is proved, that the face which results from it is perpendicular to the axis; and calculation shows, in like manner, that the second decrement taking place by two rows, produces planes parallel to the axis, and thus the secondary solid is a regular hexahedral prism.

To develop farther the structure of this prism, it may be remarked, that in the production of any one a b c n h (fig. 9.) of the two bases, the effect of one only of the three decrements which take place around the solid angle a (fig. 12.) may be considered, for example, of that which takes place on the angle b a f, supposing that the lamina applied on the two other faces f a g x, b a g, do not decrease, but to affect the result of the principal decrement which takes place in regard to the angle b a f. Here these auxiliary decrements are quite similar to that whose effect they are supposed to prolong.

The case will be totally different by applying the same observation to the decrements which are affected by two rows on the inferior angles b d f, f x g, f x g, &c., and which produce the six planes of the prism. If, for example, we consider the effect of the decrement on the angle d f x, it is necessary also that the lamina applied on the faces a f d b, a f x g (fig. 12.) should experience, towards their lateral angles a f d, a f x, adjacent to the angle d f x, variations which second the effect of the generating decrement. Here, however, these variations are intermediary decrements by rows of double molecules.

Amphitrigonous Iron Ore. Fig. 59. shows this crystal in a horizontal projection, and fig. 60. in perspective.

Geometric Character.—Respective inclination of the triangles g c n, g c d, &c., from the same summit, 14° 26' 33"; of the lateral triangles b g u, b q g, to the adjacent pentagons, such as g u f m n, 15° 45' 39".

This is the common form of the iron ore of the island of Elba. It results from a decrement by two rows on the angles e, n (fig. 54.) to the summits of a cubic nucleus; which produces the isosceles triangles g c n, g c d, n c d (fig. 59. and 62.), and of a second decrement by three rows on the lateral angles c b p, c r p, c r s, &c., which produce the triangles m n r, r n k, u g b, g g b, &c. These two decrements stop at a certain term, so that Structure of that there remain faces parallel to those of the nucleus.

Crystals. viz., the pentagons g u t m n, h d n k l, &c. (fig. 59.)

The first decrement is similar to that which produces the rhomboidal iron ore. The second has this property, that if its effect were complete, it would give a dodecahedron of isosceles triangles, or composed of two right pyramids united at their bases. The triangles of the summits are frequently narrowed by lines parallel to the bases g n, d n, g d, of these triangles, and which point out the direction of the decrement.

Analogical Calcaceous Spar, (Haur), fig. 61.

Geometric Character.—Inclination of any one, imeh, of the trapezoids of the summits to the corresponding vertical trapezoid e c p g, 116° 33' 54"; angles of the same trapezoid i = 114° 18' 56"; e = 75° 31' 23"; m or h = 85° 4' 52". Angles of the trapezoid e h o g, e = 90°; o = 127° 25' 53"; g = 67° 47' 44"; l = 74° 46' 23"; of the trapezoid e c p g, c = 65°; p = 98° 12' 46"; c or g = 100° 53' 37".

Geomet. Propert.—1. In each vertical trapezoid, the triangle c e g is equilateral. 2. The height e x of this triangle is double the height p x of the opposite triangle c p g. 3. In the trapezoid e h o g, and the other similarly situated, the angle h e g is a right angle. 4. If the diagonal g h be drawn, the triangle h e g will be similar to any one a o f (fig. 12.) of those which would be produced by drawing in the primitive rhombus the two diagonals b f, a d. 5. If in the trapezoid e m i h, or any other situated at the summits, the diagonals e i, m h be drawn, the height e l of the inferior triangle m e h will be double the height i l of the superior triangle m i h. 6. The triangle m i h is similar to ½ of the rhombus of every obtuse spar, divided by the horizontal diagonal, and the triangle m e h is similar to ¼ of the rhombus of the acute spar divided in the same manner.

The numerous analogies connecting this variety with different crystalline forms, whether considering certain angles formed by planes, or certain triangles obtained by drawing the diagonals of the trapezoids, led the author of this theory to give it the name of analogical spar. It is derived from three other varieties, viz., very obtuse spar, by the trapezoids e m i h, f i h l, &c.; metaflatic spar, by the trapezoids e m d c, e h o g, o h t z, &c.; and the prismatic spar by the trapezoids b a c k, e c g p, &c., which are consequently parallel to the axis. The trapezoids i m e h, f i h l, &c., are often separated by an intermediary ridge from the vertical trapezoids c e g p, g o r, &c. In that case the trapezoids c d m e, g e h o, &c., are changed into pentagons.

Icosahedral Sulphuret of Iron, (fig. 63.)

Geometric Character.—Reflexive inclinations of the isosceles triangles P L R, P S R, 126° 52' 11" of any one P N L of the equilateral triangles to each adjacent isosceles triangle, P L R, or L N K, 145° 46' 17". Angles of the isosceles triangle P L R, L = 48° 11' 20"; P or R = 65° 54' 20".

This variety is the result of a combination of the law which produces the octahedron originating from a cube (fig. 50.), with that which takes place for the dodecahedron with pentagonal planes (fig. 27. and 28.). The first law produces the eight equilateral triangles structure of which correspond with the solid angles of the nucleus; and the second produces twelve isosceles triangles, situated two and two above the six faces of the same nucleus. If a dodecahedron similar to that of fig. 28. were converted geometrically into this icosahedron, it would be sufficient to make the planes of eight sections pass through it in the following manner; viz. one through the angles P, M, L, (fig. 27.), another through the angles P, M, S; a third through the angles L, R, U, &c. By comparing the figures 27. and 63., the relation between the polyhedra will be seen by the correspondence of the letters; but this is merely an artificial operation; for it may be observed, that the nucleus of the icosahedron which would be obtained, would be much smaller than that of the dodecahedron, since the solid angles of the latter nucleus would be confounded with the angles D, C, G, &c. (fig. 28.) of the dodecahedron; but the other nucleus would have its solid angles situated in the middle of the equilateral triangles M P S, N P I, U R L. (fig. 63.).

The icosahedron of the sulphuret of iron, which is not very common, has been confounded with the regular geometrical icosahedron which has all its angles equilateral. Theory shows that the existence of the latter icosahedron is equally impossible in mineralogy as the geometrical dodecahedron. Among the five regular polyhedra of geometry, viz., the cube, the tetrahedron, the octahedron, the dodecahedron, and the icosahedron, the three former can only exist among minerals according to the laws of crystallization.

Polymorous Petunze (Haur), fig. 64.

Geometric Character.—Reflexive inclination of the narrow planes, o n k m, e f h g, to the adjacent planes on each side 150°; of the planes c t F g, P o m N to those contiguous to them by the edges t F, P N, 120°; of the heptagon p G c l d e z to the enneagon B z e b n o P r s, 99° 41' 8"; of the trapezium d a f c both to the plane n b a f h i l k, and to the heptagon p G c l d e z, 133°; of the facet d e a b, or A B z p to the same heptagon, 124° 15' 15'.

Hauy had not observed the petunze crystallized under its primitive form. This form, such as it is given by the mechanical division of secondary crystals, is that of an oblique prism of four planes (fig. 66.), two of which, such as GOAD, RBHN, are perpendicular to the bases ADNH, OGRB; the other two, viz., BOAH, RGDN, make with the former, angles of 120° at the ridges OA, RN, and angles of 60° towards the opposite ridges BH, GD. These planes are inclined to the bases at that place of the ridges GO, BR, 111° 29' 43"; and at the opposite ridges 68° 30' 17". This form is at the same time that of the molecule. By theory, the two parallelograms GOAD, OGRB, as well as their parallels are equal in extent; and the parallelogram BOAH, or its opposite, RGDN, is double each of the preceding. This may serve to explain the roughness of the sections made in the direction BOAH, when compared with those in the directions of the small parallelograms, the latter being always smooth and brilliant. If, however, the diagonal OR, be drawn, it will be found perpendicular to OA and RN; or, it will be situated horizontally, This mineral exhibits the most complicated variety which the author has observed among this kind of crystals. To comprehend its structure, suppose that \( b \) \( p \) \( y \) \( r \), (fig. 65.), represents a section of the nucleus AR, (fig. 66.), made by a plane perpendicular to the parallelograms GOAD, BOAH, and subdivided into a multitude of small parallelograms, which are the analogous sections of so many molecules. Here the side \( y \) \( r \) (fig. 65.), which is the same section of the cutting plane as GOAD, is greater than it ought to be in regard to the side \( c \) \( r \) (fig. 65.), which is the same section as BOAH (fig. 66.). But these dimensions are suited to those of the secondary crystal, and here occasion no difficulty, because it may be supposed that the primitive form has been extended more in one direction than in another; for this form is to be considered only as a convenient datum for the explanation of the structure, and the crystal consists merely in an assemblage of similar molecules; so that it is the dimensions of these molecules, which remain invariable.

By comparing fig. 64. and 65., it will be found, 1. That the plane \( f \) \( a \) \( b \) \( n \) \( k \) \( l \) \( i \) \( h \) (fig. 64.) and its opposite which correspond to \( m \) \( n \), \( d \) \( g \) (fig. 65.), are parallel to two planes of the nucleus, viz. GOAD, BRNH (fig. 66.), and therefore do not result from any law of decrement. 2. That the plane \( P \) \( o \) \( m \) \( N \), and its opposite (fig. 64.) which correspond to \( a \) \( o \), \( e \) \( g \) (fig. 65.), are also parallel to two of the planes of the nucleus, viz. BOAH RGDN, (fig. 66.). 3. That the plane \( o \) \( n \) \( k \) \( m \), and its opposite (fig. 64.) which correspond to \( o \) \( n \), \( e \) \( g \) (fig. 65.) result from a decrement by two rows parallel to the ridges \( A \) \( O \), \( N \) \( R \), (fig. 66.). 4. That the plane \( c \) \( f \) \( g \) \( h \), and its opposite, (fig. 64.), result from a decrement by four rows parallel to the ridges \( G \) \( D \), \( B \) \( H \), (fig. 66.), which decrement takes place on the other side of these ridges. From this it may be seen, that decrements different in their measure, give rise to planes similarly situated, such as \( o \) \( n \) \( k \) \( m \) and \( c \) \( f \) \( g \) \( h \), (fig. 64.), which is a consequence of the particular figure of the molecules. With regard to the faces of the summit, the heptagon \( p \) \( G \) \( i \) \( c \) \( d \) \( e \) \( z \), (fig. 64.), is situated parallel to the base BRGO, (fig. 66.). The enneagon \( B \) \( s \) \( r \) \( P \) \( o \) \( n \) \( b \) \( e \) \( z \) (fig. 64.) is produced in consequence of a decrement by one row on the angle OBR (fig. 66.), or parallel to the diagonal OR; which decrement does not attain its full extent, and leaves subsisting the neighbouring heptagon parallel to the base BRGO. It may be conceived, from what has been laid on the position of the diagonal OR, why the line \( e \) \( z \) (fig. 64.), which separates the two large faces of the summit, is situated horizontally, by supposing that the planes have a vertical position.

The trapeziums \( d \) \( a \) \( f \) \( c \), \( A \) \( p \) \( G \) \( C \), are the result of a decrement by one row on the ridges GO, BR (fig. 66.). The facet \( d \) \( e \) \( b \) \( a \) (fig. 64.) arises from a decrement by two rows parallel to the ridge BO (fig. 65.). With regard to the other facet \( A \) \( B \) \( z \) \( p \), which has the same position as the preceding, in relation to the opposite part of the crystal, it results from an intermediary law, by a row of double molecules on the angle OBR (fig. 66.). The rhombuses \( b \) \( c \) \( l \) \( h \), \( k \) \( l \) \( s \) \( u \) (fig. 67.) represent the horizontal sections of two of these double molecules taken in the same row, and whose relation to the rest of the arrangement will become sensible by comparing these rhombuses with those marked with the same letters in fig. 65. This variety of crystals is subject to a change of dimensions; the faces \( p \) \( G \) \( i \) \( c \) \( d \) \( e \) \( z \), \( f \) \( a \) \( b \) \( n \) \( k \) \( l \) \( i \) \( h \), and their opposites, which are at right angles to each other, are elongated in the direction of their breadth, exhibiting the appearance of a quadrilateral, rectangular prism, the summits of which would be formed by the faces situated towards the ridges PN, FR. Crystals of this variety, which are opaque, and of a whitish, yellowish, and sometimes reddish colour, are found in granites; some are in groups, and some, but more rarely, are met with in flinty crystals.

III. Number of Primitive Forms.

In the examples which have been given, the author of the theory has chosen the paralleloped for a nucleus, on account of the simplicity of its form. He has hitherto found that all the primitive forms may be reduced to six:

1. The paralleloped in general, which comprehends the cube, the rhomboid, and all the solids terminated by six faces parallel two and two. 2. The regular tetrahedron. 3. The octahedron with triangular faces. 4. The hexagonal prism. 5. The dodecahedron with rhomboidal planes. 6. The dodecahedron with isosceles triangular faces.

Among these forms there are some found as nucleus, which have the measure of their angles the same in different kinds of minerals. It is to be considered that these nuclei are composed, in the first instance, of elementary molecules, and that it is possible that the same form of nucleus may be produced in one species by elements of a certain nature, and in another species by different elements combined in a different manner, as we see integrant molecules, some cubic, and some tetrahedral, produce similar secondary forms by the operation of different laws of decrement. But it may be observed, that all the forms which have hitherto occurred as nuclei, on the different species, are such as have a particular character of perfection and regularity, as the cube, the regular octahedron, and the dodecahedron with equal and similar rhombuses for its faces.

IV. Forms of the Integral Molecules.

The primitive form is that which is obtained by sections made on all the similar parts of the secondary crystals; and these sections, continued parallel to themselves, conduct to a determination of the form of the integral molecules, of which the whole crystal is the assemblage. There is no crystal from which a nucleus in the form of a paralleloped may not be extracted, by making the limitation to fix sections, parallel two and two. In a great number of substances, this paralleloped is the last term of the mechanical division, and consequently the real nucleus; but in some minerals this paralleloped is divisible, as well as the rest of the crystal, by farther sections made in the different directions of the faces, from which results a new solid, which will be the nucleus, if all the parts of the secondary crystal superadded to this nucleus are similarly situated. When the mechanical division conduces to a paralleloped, divisible only by sections parallel to its fix faces, the molecules are parallelopipeds similar... Structure of similar to the nucleus; but in all other cases their form differs from that of the nucleus. This may be illustrated by an example.

Let \(a c h i n o\) (fig. 68.) be a cube, having two of its solid angles \(a, s\), situated on the same vertical line; this line will be the axis of the cube, and the points \(a\) and \(s\) will be its summits. Let it be supposed that this cube is divisible by sections, each of which, such as \(a h n\), passes through one of the summits \(a\), and by two oblique diagonals \(a h, a n\), contiguous to the summit. By this section the solid angle \(i\) will be detached; and as there are six solid angles, situated laterally, viz. \(i, h, e, r, o, n\), the six sections will produce an acute rhomboid, the summits of which will be confounded with those of the cube. At fig. 69., this rhomboid is represented existing in the cube in such a manner, that its six lateral solid angles \(b, d, f, p, g, e\), correspond with the middle of the faces \(c h i, c r s h, h i n s, &c.\) of the cube; but each of the angles at the summits \(b a g, d s f, p s f, &c.\) of the acute rhomboid, are \(=62^\circ\); from which it follows, that the lateral angles \(a b f, a g f, &c.\) are \(=120^\circ\). Besides, it is proved by theory, that the cube is the result of a decrement which takes place by a single row of small rhomboids, similar to the acute rhomboid on the six oblique ridges \(a-b, a-g, a-e, s-d, s-f, s-p\). This decrement produces two faces, one on each side of each of these ridges, making in the whole 12 faces; but as the two faces, having the same line of ridge for their departure, are on the same plane; by the nature of the decrement, the 12 faces will be reduced to six, which are squares, so that the secondary solid is a cube.

Suppose that the cube (fig. 68.) admits, in regard to its summits \(a, s\), two new divisions similar to the preceding six, one of which passes through the points \(c, i, o\), and the other through the points \(h, n, r\). The first will also pass through the points \(b, g, e\), and the second through the points \(d, f, p\), (fig. 69. and 70.) of the rhomboid; from which it follows, that these two divisions will each detach a regular tetrahedron \(b a g e\), or \(d s f p\) (fig. 70.), so that the rhomboid will be found converted into a regular octahedron \(e f\) (fig. 71.), which will be the real nucleus of the cube; for it is produced by divisions similarly made in relation to the eight solid angles of the cube. If we suppose the same cube to be divisible throughout its whole extent by analogous sections, it is clear that each of the small rhomboids, of which it is the assemblage, will be found in like manner subdivided into an octahedron, and two regular tetrahedrons, applied on the two opposite faces of the octahedron. By taking the octahedron for a nucleus, a cube may be constructed round it, by regular subtractions of small complete rhomboids. It, for example, we suppose decrements, by a single row of these rhomboids, having \(b\) for the point of their departure, and made in a direction parallel to the inferior edges \(g f, e g, d e, d f\), of the four triangles, which unite to form the solid angle \(b\), there will result four faces, which will be found on a level, and like the octahedron, with six solid angles, similar decrements around the other five angles will produce twenty faces, which taken four and four will be equally on a level, making in the whole six distinct faces, situated as those of the cube (fig. 68.). The result will be exactly the same as in the case of the rhomboid, considered as nucleus.

In whatever way we proceed to subdivide, either the cube, the rhombus, or the octahedron, we shall always have solids of two forms, that is to say, octahedrons and tetrahedrons, without being able to reduce the result of the division to unity. But the molecules of a crystal being similar, Haug thinks it probable, that the structure was, as it were, interfered with a multitude of small vacuities, occupied either with the water of crystallization or some other substance; so that, if it were possible to carry the division to its limits, one of those two kinds of solids would disappear, and the whole crystal would be found composed only of molecules of the other form. This view is the more admissible, as each octahedron being enveloped with eight tetrahedrons, and each tetrahedron being in like manner enveloped with four octahedrons, whichever of these forms may be supposed to be suppressed, the remaining solids will join exactly by their edges; so that in this respect there will be continuity and uniformity throughout the whole extent of the mass. It may be readily conceived how each octahedron is enveloped with tetrahedrons. By attending to the division of the cube only by the six sections which give the rhomboid, we may depart at pleasure from any two, \(a, s; o, h; c, n; i, r\), of the eight solid angles, provided that these two angles be opposite to each other. But by departing from the angles \(a, s\), the rhomboid will be in the position shewn at fig. 70. If, on the contrary, we depart from the solid angles \(o, h\), these angles will become the summits of a new rhomboid (fig. 72.), composed of the same octahedron as that of fig. 71., with two new tetrahedrons applied on the faces \(b d f, e g p\), (fig. 72.), which were unoccupied on the rhomboid of fig. 70. Fig. 73. represents the case in which the two tetrahedrons repose on the faces \(d b e, f g p\), of the octahedron; and fig. 74. represents the case in which they would rest on the faces \(b f g, d e p\). Hence, whatever may be the two solid angles of the cube assumed for the points of departure, we shall always have the same octahedron, with two tetrahedrons contiguous by their summits to these two solid angles; and there being eight of these solid angles, the central octahedron will be circumscribed with eight tetrahedrons, which will rest on its faces. By continuing the division always parallel to the first sections, the same effect will always take place. Each face of the octahedron, however small it may be supposed to be, adheres to a face of the tetrahedron, and reciprocally; and each tetrahedron is enveloped with four octahedrons.

The structure which is here explained is that of fluorate of lime, or fluor spar. By dividing a cube of this substance, we may at pleasure extract rhomboids which have the angles formed by their planes equal to \(120^\circ\), or regular octahedrons, or tetrahedrons equally regular. In some other substances, as rock crystal, carbonate of lead, &c., which being mechanically divided beyond the term at which we should have a rhomboid or a parallelopiped, parts of various different forms are obtained, arranged together even in a more complicated manner than in fluor spar. In consequence of these mixed structures, there is some uncertainty respecting the real figure of the integral molecules. mixture of molecules which belong to these substances. It is observed, however, that the tetrahedron is always one of those solids which concur to the formation of small rhombohedra or parallelopipeds that would be extracted from the crystal by a first division. But, on the other hand, there are substances, which being divided in every possible direction, resolve themselves only into tetrahedrons. Garnet, biotite, and tourmaline, belong to this number.

Several minerals are divisible into right triangular prisms. Such is the apatite, whose primitive form is a regular right hexahedral prism, divisible parallel to its bases and its planes, from which necessarily result right prisms with three planes. Fig. 76 represents one of the bases of the hexahedral prism, divided into small equilateral triangles, which are the bases of so many molecules, and which being taken two and two, form quadrilateral prisms, with rhombuses for their bases.

By adopting then the tetrahedron, in the doubtful cases already mentioned, all the forms of integral molecules may be in general reduced to three, which are remarkable for their simplicity, viz., the parallelopiped, the simplest of all the solids, having parallel faces two and two; the triangular prism, the simplest of all prisms; and the tetrahedron, which is the simplest of pyramids. This simplicity may furnish a reason for the preference given to the tetrahedron in fluor spar, and the other substances which have been mentioned as examples. But the ingenious author of the theory cautiously declines to speak decisively on the subject, as the want of direct and precise observations, he observes, leaves to theory only conjectures and probabilities.

But the essential object is, that the different forms to which these mixed structures lead, are arranged in such a manner, that their assemblage is equivalent to a sum of small parallelopipeds, as has been seen to be the case in regard to fluor spar; and that the laminae of superposition applied on the nucleus, decrease by subtractions of one or more rows of these parallelopipeds. The basis of the theory exists, therefore, independently of the choice which might be made of any of the forms obtained by the mechanical division.

With the help of this result, the decrements to which crystals are subject, whatever be their primitive forms, are found reducible to those which take place in substances, where this form, as well as that of the molecules, are indivisible parallelopipeds; and the theory has this advantage of being able to generalize its object, by connecting with one fact, that multitude of facts which, on account of their diversity, seem to be little susceptible of being brought to one common point. But what has been said, will be still more illustrated by examples of the manner in which we may reduce to the theory of the parallelopiped, that of the forms which are different from that solid.

Crystals whose Molecules are Tetrahedrons, with Isosceles Triangular Faces.

Garnet.

1. Primitive Garnet (fig. 76).

Geometric Character.—Respective inclinations of any two of the faces of the dodecahedron, 120°. Angles of the rhombus CLGH, C or G = 109° 28' 16"; Structure of L or H = 78° 31' 44".

Notwithstanding the vitreous appearance in general exhibited on the fractures of garnets of the primitive form, laminae may be perceived on them, situated parallel to the rhombuses which compose their surface. Let us suppose the dodecahedron divided in the direction of its laminae, and for the greater simplicity, let us suppose the sections to pass through the centre. One of these sections, viz. that which will be parallel to the two rhombuses DLFN, BHOR, will concur with a hexagon, which would pass through the points E, C, G, P, I, A, by making the tour of the crystal. A second section parallel to the two rhombuses GLPP, BEAR, will coincide with another hexagon thrown by the points D, C, H, O, I, N. And if the division be continued parallel to the other eight rhombuses, taken two and two, it will be found that the planes of the sections will be confounded with four new hexagons analogous to the preceding. But by removing all these hexagons, it will appear that their sides correspond, some of them with the small diagonals of the rhombuses of the dodecahedron, viz. those which would be drawn from C to G, from A to I, from C to B, &c. and others would correspond with the different ridges EC, GP, PI, EA, &c.

1. The planes then of the sections passing through the sides and through the small diagonals of the twelve rhombuses, will subdivide the whole surface into 24 isosceles triangles, which will be the halves of these rhombuses. 2. Since the planes of the sections pass also through the centre of the crystal, they will detach 24 pyramids with three faces; the bases of which, if we choose, will be the external triangles that make part of the surface of the dodecahedron, and of which the summits will be united in the centre.

Besides, if we take, for example, the five tetrahedrons, which have for external faces the halves of the three rhombuses CEDL, CLGI, CEBH, these five tetrahedrons will form a rhomboid represented by fig. 77, and in which the three inferior rhombuses DLGS, GHBS, DEBS, result from three divisions which pass, one through the hexagon DLGORA, (fig. 76); the second through the hexagon GHBAFN; and the third through the hexagon REDFPO. Fig. 77 also represents the two tetrahedrons, the bases of which make part of the rhombus CLGH. One of these is marked with the letters L, C, G, S, and the other with the letters H, C, G, S. And by applying what has been said to the other nine rhombuses, which are united, three and three, around the points F, A, H, (fig. 77), we shall have three new rhomboids; from which it follows, that the 24 tetrahedrons, considered fix and fixed, form four rhomboids; so that the dodecahedron may be conceived as being itself immediately composed of these four rhomboids, and in the last analysis of 24 tetrahedrons.

It may be observed, that the dodecahedron having eight solid angles, each formed with three planes, they might have been considered as the assemblage of the four rhomboids, which would have for exterior summits the four angles G, B, D, A; from which it follows that any one of the faces, such as CLGO, is common to two rhomboids, one of which would have its Structure of its summit in C, and the other in G, and which would themselves have a common part in the interior of the crystal.

We may remark farther, that a line GS (fig. 77.) drawn from any one G (fig. 76.) of the fold angles composed of three planes, as far as the centre of the dodecahedron, is at the same time the axis of the rhomboid, which would have its summit in C (fig. 76. and 77.). The composing rhomboids then have this property, that their axis is equal to the fides of the rhombus. From which, with a little attention, we may conclude, that in each tetrahedron, such as CLGS (fig. 77.), all the faces are equal and similar isosceles triangles.

If the division of the dodecahedron be continued by sections passing between those which we have supposed to be directed towards the centre, and which should be parallel to them, we should obtain tetrahedrons always smaller, and arranged in such a manner, that taking them in groups of six, they would form rhomboids of a bulk proportioned to their own.

The tetrahedrons, which would be the term of the division, were it possible to reach it, ought to be considered as the real molecules of the garnet. But it will be seen, that in the passage to the secondary forms, the laminae of superposition, which envelop the nucleus, really decrease by rows of small rhomboids, each of which is the assemblage of these tetrahedrons.

The sulphuret of zinc, or blende, has the same structure as the garnet. Hauy informs us that he has divided fragments of this substance by very clean sections, in such a manner as to obtain successively the dodecahedron, the rhomboid and the tetrahedron.

2. Trapezoidal Garnet, (fig. 78.).

Geometrical Character.—Respective inclination of the trapezoids, united three and three around the same solid angle D, C, G, &c., \(131^\circ 48' 33''\); of the trapezoids united four and four around the same solid angle u, x, r, &c. \(131^\circ 48' 36''\). Angles of any one of the trapezoids mD u L, \(L = 78^\circ 27' 46''\); \(D = 117^\circ 2' 8''\); \(m\) or \(u = 82^\circ 15' 3''\). The value of the angle L is the same as that of the acute angle of the nucleus of calcareous spar.

This variety is the result of a series of laminae, decreasing at the four edges, on all the faces of the primitive dodecahedron. For the more simplicity, let us first consider the effect of this decrement in regard to the rhombus CLGH (fig. 76.). We have just seen that this rhombus was supposed to belong in common to two rhomboids, which should have for summits, one the point C, and the other the point G. Let us suppose that the lamina applied on this rhombus decrease towards their four edges by subtractions of a single row of small rhomboids, in such a manner that in regard to the two edges CL, CH, circumstances are the same as if the rhombus belonged to the rhomboid which has its summit in C; and that in regard to the other two edges GL, GH, the effect is the same as if the structure of rhombus belonged to the rhomboid, which has its summit in G. This disposition is admissible here in consequence of the particular structure of the dodecahedron, which permits us to obtain small rhomboids; some of which have their faces parallel to the faces of that with its summit in C, and the rest to that having its summit in G (v).

The results of the four decrements being thus quite similar to each other, the laminae of superposition, applied on the rhombus CLGH, and on each of the other rhombuses of the dodecahedron, will form as many right quadrangular pyramids, which will have for bases these same rhombuses. Fig. 79. represents the pyramids which rest on the three rhombuses CLDE, CEBH, CGHB (fig. 76.), and which have for summits the points m, e, s, (fig. 76.); but on account of the decrement by a simple row, the adjacent triangular faces, such as E m C, E s C of the two pyramids that belong to the rhombuses CLDE, CEBH, are on a level, and form a quadrilateral E m C s. But we had 12 pyramids, and consequently 48 triangles. These divided by two give 24 quadrilaterals, which will compose the surface of the secondary crystal. But because the rhomboidal bases of the two pyramids extend more, in proceeding from L to E, or from H to E, than in proceeding from D to C, or from B to C, the sides mE, Es of the quadrilateral will be longer than the sides Cm, Cs. And besides mE will be equal to Es, and Cm equal to Cs. Thus the quadrilaterals will be trapezoids which have their sides equal two and two. There is no crystalline form in which the strike, when they do exist, show in a more sensible manner the mechanism of the structure than in this variety of garnet. We may here see the series of decreasing rhombuses which form each of the pyramids CLDE m, CEBH s, &c. (fig. 79.), and sometimes the furrows are so deep that they produce a kind of hair, the steps of which have a more particular polish and brilliancy than those of the facets, which are parallel to the faces CEDL, CHBE, of the nucleus.

If the decrements stop abruptly at a certain term, so that the pyramids are not terminated, the 24 trapezoids will be reduced to elongated hexagons, which will intercept 12 rhombuses parallel to the faces of the nucleus. To this variety Hauy has given the name of intermediate garnet.

In the sulphuret of zinc the regular octahedron is the result of a decrement by a row around the eight solid angles, composed of three planes, viz. C, B, O, G, F, D, A, I, (fig. 76.). The same substance also assumes the form of a regular tetrahedron, by the help of a decrement by one row on four only of the eight solid angles before mentioned, such as C, O, F, A. The structure of this tetrahedron is remarkable, as it presents an assemblage of other tetrahedrons with isosceles faces.

(d) Theory, the author observes, has conducted him to another result, which is, that the sum of the nucleus and laminae of superposition, taken together in proportion as the latter are applied one upon the other, is always equal to a sum of rhomboids; though at first view it does not appear that this should be the case, according to the figure of these laminae, which represent rising pyramids. Crystals whose Molecules are Triangular Prisms.

Oriental.

Hauy has thus denominated the gem which is known under the different names of ruby, sapphire, oriental topaz, according as the colour is red, blue, or yellow. The different varieties of this gem have not been accurately described, and the nature of the particular angles of each has not been precisely indicated, on account of the rare occurrence of regularly formed crystals, or, when such have been found, on account of their being defaced by being water-worn, or otherwise injured. But from some crystals which were sufficiently characterized, Hauy obtained the following results.

1. Primitive Oriental.

This mineral crystallizes in the form of a regular hexahedral prism, which is divisible parallel to its bases. According to theory, which points out other joinings parallel to the planes, the molecule is an equilateral triangular prism. The height of this prism, calculated by theory, is a little less than three times the height of the triangle of the base.

2. Elongated Oriental, (fig. 80.).

Geometric Character.—Respective inclinations of the triangles IAS, IBS, $135^\circ 54'$. Angles of the triangle IAS, $\Delta = 22^\circ 54'$. 1 or $= 78^\circ 47'$.

This form is the result of a decrement by a simple row of small quadrangular prisms on all the edges of the bases of the nucleus. Let $q d$ (fig. 75.) be the superior base, subdivided into small triangles, which represent the analogous bases of so many molecules. The edges of the laminae of superposition will correspond successively to the hexagons $h i l m n r$, $e k u x y v$, &c.; from which it follows that the subtractions take place by rows of small parallelopipeds of quadrangular prisms composed each of two triangular prisms.

3. Minor Oriental.

Geometric Character.—Dodecahedron formed of two right pyramids less elongated than those of the preceding variety. The triangles corresponding to IAS, IBS, are inclined to each other $122^\circ 36'$. In each of these triangles the angle of the summit is $31^\circ$, and each of the angles at the base is $74^\circ 30'$.

The law of which this variety is the result, differs from that which produces the preceding, as it determines a mixed decrement by three rows in breadth and two rows in height.

4. Enneagonal Oriental, (fig. 81.).

Geometric Character.—Inclination of each small triangle, such as $c q i$, to the adjacent base $a c i p l b g e d$, $122^\circ 18'$.

This is the elongated oriental, whose summits are replaced by two faces, parallel to the bases of the nucleus, with the addition of six small isosceles triangles $c q i$, $l b f$, $v z m$, &c., the three superior of which are alternate in position with the three inferior. These triangles are the result of a decrement, by three rows of small quadrangular prisms on the three angles of the superior base of the nucleus, such as $b$, $d$, $g$ (fig. 75.), and on the intermediate angles of the inferior base. It may be readily conceived, that in the decrement which takes place, for example, on the angle $g$, the three rows which remain unoccupied between that angle and the corresponding edge of the first lamina of superposition, are, 1. the small rhombus $g o i p$, which alone forms the first row; 2. the two rhombuses $o s t i$, $i d i z$, the three rhombuses situated on the same line behind the two preceding.

Crystals of this gem are chiefly found in the kingdom of Pegu. Some have been found in France, which have received the name of sapphires of Puy. They have been also found at a little distance from Velay, on the banks of a rivulet near the village of Expaltia, where they are mixed with garnets and hyacinths. These have all the characters of the stone which is denominated oriental sapphire.

V. Difference between Structure and Increment.

In what has been said respecting the decrements to which the laminae of superposition are subjected, the author observes, that it was his view only to unfold the laws of structure; and he adds, that he is far from believing that in the formation of a dodecahedral crystal, or one of any other form, having a cube for a nucleus, the crystallization has originally produced that nucleus such as it is extracted from the dodecahedron, by the successive application of all the laminae of superposition with which it is covered. It seems proved, on the contrary, that from the first moment the crystal is already a very small dodecahedron, containing a cubical nucleus proportioned to its small size, and that the crystal afterwards increases by degrees without changing its form, by new layers which envelope it on all sides, so that the nucleus increases also, preserving always the same relation with the whole dodecahedron.

An example taken from a plane figure will make this more striking; and what is said respecting this figure may be easily applied to a solid, since a plane figure may be always conceived as a section of a solid. Let ERFN (fig. 82.) be an arrangement of small squares, in which the square ABCD, composed of 49 partial squares, represents a section of the nucleus, and the extreme squares R, S, G, A, I, L, &c., that of the kind of stair formed by the laminae of superposition. It may be readily conceived, that the arrangement began with the square ABCD; and that different files of small squares were afterwards applied on each of the sides of the central square: for example, on the side AB, first the five squares comprehended between I and M, next the three squares comprehended between L and O, and then the square E. This increment corresponds with that which would take place if the dodecahedron began by being a cube proportioned to its bulk, and which increased afterwards with the addition of continually decreasing laminae.

But on the other hand, the arrangement may be conceived to be such as is represented in fig. 84., in which the square abcd is composed of only nine molecules, and bears upon each of its sides only one square. Structure of square e, n, f, or r; and that afterwards by means of the application of new squares arranged round the former, the affortment has become that of fig. 83, where the central square a' b' c' d' is formed of 25 small squares, and bears on each side of its sides a file of three squares, plus a terminating square e', n', f', or r'; and that, in short, by a farther application, the affortment of fig. 83, is converted into that of fig. 82.

These different transitions will give some notion of the manner in which secondary crystals may increase in bulk, and yet retain their form; and from this it will appear, that the structure is combined with that augmentation of bulk, so that the law, according to which all the laminae applied in the nucleus of the crystal, when arrived at its greatest dimensions, successively decrease, in departing from this nucleus, existed already in the rising crystal.

Such is the ingenious theory of the structure of crystals, which the author observes, is in this similar to other theories, that it sets out from a principal fact, on which it makes all facts of the same kind to depend, and which are only as it were corollaries. This fact is the decrement of the laminae superadded to the primitive form; and it is by bringing back this decrement to simple and regular laws, susceptible of accurate calculation, that theory arrives at results, the truth of which is proved by the mechanical division of crystals, and by observation of their angles. But new researches are still wanting, in order to ascend a few steps farther towards the primitive laws by which crystallization is regulated. The object of one of these researches would be to explain how these final polyhedrons, which are as it were the rudiments of crystals of a sensible bulk, sometimes represent the primitive form, without modification; sometimes a secondary form produced in virtue of a law of decrement; and to determine the circumstances which produce decrements on the edges, as well as those which give rise to decrements on the angles.