hey arise from points in the interior edge of the border, and spread out nearly at equal distances from each other, in all directions.

In fig. 8, are represented the crystals of Glauber's salt (sulphate of soda), which assumes 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 swelling 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 Hauy, have been particularly directed this way. Bergman, in his 12th Dissertation*, treats of the variety of the forms of crystals, of the various figures derived from the spathaceous form, of the structure of the most minute parts, and of the different modes in which crystals are generated. Romé de L'Isle has arranged crystals into six species, derived from the varieties of form. 1. Tetrahedron. 2. Cube. 3. Octahedron. 4. Parallelopiped. 5. Rhomboïdal octahedron. 6. Dodecahedron. But the ingenious researches of Hauy 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†, Vol xvii. 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 organization, 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 readily 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 rhomboïd of the same mineral (B), whose large angle is about 101°, we

(b) The name of rhomboïd is given by the author to a parallelopiped \(a, e\) (fig. 12.) terminated by six equal and similar rhombuses. In every rhomboïd, 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 \(a, e\) is the axis. In any one of the rhombuses \(a, b, d, f\), 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 \(a, b, c, f\) are the superior edges, and the sides \(b, d, f\) the inferior edges; \(b, f\) is the horizontal diagonal, and \(a, d\), the oblique diagonal. The rhomboïd is obtuse or acute, according as the angles of the summits are obtuse or acute. The cube is the limit of the rhomboïds.

Vol. VI. Part II. Structure of we should be led to believe that each of these two forms is quite distinct from the other. But this point of relation, which escapes 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 \( p s u t \) (fig. 9), and which had an angle of \( 133^\circ \), both with the remainder \( a b c s p h \) of the base, and with the remainder \( n e f \) of the plane \( i n e f \). Observing that the cuneiform segment \( p s u t i n \) 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 \( p s u t \) 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 sixth 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, \( d e, v y, g f \). The trapezium \( l q y v \) shews the section made below the edge \( k r \). This section is obviously parallel to that of the trapezium \( p s u t \); 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 KNPQF, 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 six 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 confounded with the points \( Y, Z \), the points \( S, R \) with the points \( U, Y, &c. \) there remained no structures of these pentagons, but the triangles \( YIZ, UXV, \) &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 lenticular, 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 rhomboïdal 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 rhomboïdal 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 rhomboïdal bases; feldspar, an oblique-angled parallelopiped, but not rhomboïdal; apatite or beryl, a right six-sided prism; the adamantine spar a rhomboid, a little acute; blende, a dodecahedron, with rhomboïdal 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, Structure of forms, among those which belong to the same substance, frequently presented indications of their structure, and by reasoning from their analogy with other divisible crystals, we may determine this structure, at least with a good deal of probability.

All deviations from the primitive form are called by Haury, secondary forms. The number of these forms has certain limits, which can be determined by theory, according to the laws which regulate the structure of crystals.

The solid of the primitive form, which is obtained by means of the operation described above, may be farther subdivided in a direction parallel to its different faces. All the surrounding matter is equally divisible by sections parallel to the faces of the primitive form. Hence it follows, that the parts detached by the aid of all these sections are similar, and only differ in their volume, which continually decreases in proportion to the extent of the division. Those, however, must be excepted, which are near to the faces of the secondary solid; for these faces not being parallel to those of the primitive form, the fragments which have one of their facets taken in the same faces, cannot exactly resemble those which are detached towards the middle of the crystal. For instance, the fragments of the hexahedral prism (fig. 9.), whose external facets make part of the bases, or of the planes, have not, in this respect, the same figure with those which are situated nearer to the centre, all of whose facets are parallel to the sections \( p z u t, l q y v \); but the difficulty which presents itself at first sight, in consequence of that diversity, is removed by the help of the theory, and the whole are reduced to a unity of form.

But the division of the crystal into small, similar solids, has a certain limit, beyond which we should arrive at particles so small, that they are no longer divisible, without destroying the nature of the substance, or decomposing it. At this term, the investigation stops; and to the small solids, which we might insulate if our organs and instruments were sufficiently delicate, Haury has given the name of integrant or integral molecules. He thinks it probable, that these molecules are those which were suspended in the fluid in which the crystallization took place. In general it may be observed that, with the aid of these molecules, the theory reduces to simple laws the different forms of crystals, and arrives at results which exactly represent those of nature.

When the nucleus is a parallelopiped, that is, a solid having six parallel faces, two to two, like the cube, the rhomboid, &c., and this solid admits of no other divisions than those which are made in the direction of its faces; it is obvious that the molecules which result from the subdivision, whether of the nucleus or of the surrounding matter, are similar to this nucleus. In other cases, the form of the molecules is different from that of the nucleus. There are, besides, other crystals which afford, by means of mechanical division, particles of different figures combined together through the whole extent of these crystals. The ingenious author of the theory has thrown out some conjectures on the manner of resolving the difficulty which these kinds of mixed structures present; and at any rate he observes that it does not affect the stability of the theory.

II. Laws of Decrement.

1. Decrements at the Edges.

The primitive form, and that of the integrant molecules, being determined, after the dissection 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 so regularly, and from which resulted polyhedra so 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 subtractions 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, GF, ED 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 o n f g (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 l m p u (fig. 17.) represent its Structure of its inferior base, this base will correspond exactly with crystals. The square 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 \( v \times y = (fig. 18.) \) 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 six 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 \( ABCD, BGCH \), (fig. 15.) the faces of the pyramids will not form continued 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 subtractions 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 \( p a e o, k l q u, m n t s, &c. \) correspond to the squares \( ABCD, DCGF, CBHG, &c. \) (fig. 14.), and form the superior basis of as many pyramids, incomplete 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 \( a e C l k D, c o B n m C \) (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 solid would have been composed of 24 isosceles 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.

Martial 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 present 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 isosceles 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 cross 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 CNBK (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 sulphuret 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 $11^\circ$ 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 of 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 $a'd'b'$, to the rhombus $a'f'd'$, $134^\circ 25' 36''$. Angles of the rhombus $n a d b'$; $a' b' = 114^\circ 18' 56''$; $n d' = 65^\circ 41' 4''$.

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 $c o$, $e d$, $e x$, contiguous to the summits $a$, $c$, 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 intersect 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 $r t e y$, $130^\circ$; of the same plane, to $k r y z$, $124^\circ 26' 5$; of the plane $t m g c$, to $m l i g$, $93^\circ$.

The primitive form of the topaz is that of a right-angled, four-sided prism $h y$ (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 $r n$ 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 $t m g c$, $l m g c$ (fig. 33.) on one side, and $b k z p$, $b u d p$, on the other, arise from a decrement by three rows on each side of the edges, $n v$, $v q$ (fig. 32.), which decrement remains suspended at a certain point, and leaves four rectangles $t r y c$, $k r y z$, $l k z i$, $u k d$, (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 $x d$, $x z$, $n i$, $n e$, are... 3. Decrement on the Angles.

This position of the rhomboidal 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 (fig. 27, and 28.), and which might assume the form of the dodecahedron whose planes are rhombuses (fig. 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', b', c', d', e', f', g', h', i', (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, n, y, z, shows one, and the letters n, t, l, m, p, o, r, s, show 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 level, the cubes may be supposed to be so small 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 31 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, II., 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 supposition 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 laminae 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 laminae applied on CDFG 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 BMUS (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 c', f', g', h' (fig. 41.) may correspond with Structure of the points c, f, g, h (fig. 40). Fig. 42, 43, 44, and Crystals 45 represent the four laminae which ought to rise successively above the preceding, the same letters being made to correspond. The last lamina is reduced to one cube \( \alpha \) (fig. 47) which should correspond with the same letter (fig. 45).

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 LG' of the lamina A'GL'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'GL'K' (fig. 40), while there are six which follow it as far as the cube \( \alpha \) (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 ZN, 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 pzy to puy, 78° 27' 47"; and to ir \( \approx s \), 10° 32' 13". Angles of the rhombus, pzy, p or r = 75° 31' 20"; s or y = 104° 28' 40". Inclination of the oblique diagonal drawn from p to r with the edge pu, 71° 35' 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 pr, ur, and through the intermediate edges pu, ir, are the same as on the nucleus.

To conceive the structure of this rhomboid, suppose that abdf (fig. 52) represents the face of the nucleus marked with the same letters, (fig. 12.) subdivided into a multitude of partial rhombuses, 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 abd, af'd, in such a manner, that on the first the two rhombuses bkl, fm are uncovered; that on the second the uncovered rhombuses are those traversed by the diagonals ca, uq, on the third those traversed by the diagonals sa, qz, &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 bf, will rise in the form of a roof above the rhombus abdf, and will meet on a common edge situated immediately above the diagonal ad, and which will be parallel to it; and, as there are six rhombuses, 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, f, g, &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' (fig. 51.). In this rhomboid the edges px, py, pu, are situated each as the oblique diagonals of the nucleus, or those which would be drawn from a to d, from a to z, from a to e, &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 px, py, pu, &c. (fig. 51.), and which would pass at an equal distance from these edges.

Rhomboïdal Iron ore, (fig. 53.).

Geometric Character.—Inclination of BCRP to BCOA or OCRS, 146° 26' 33"; angles of the rhombus BCRP, C or P = 117° 2' 9"; B or R = 62° 57' 51".

The laminae composing this rhomboid decrease by two rows on the angles bcr, ocr, bce, &c. (fig. 54.) which concur to the formation of the two solid angles cn, of a cubic nucleus. The faces produced, instead of being on a level, three and three, around these 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 55, which is an instance of these subtractions, the molecules composing the raw represented by the figure, are so arranged as if, of two, only one 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 \(a b c d\), \(d f g h\), \(g i l\), &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, \(b e g o\), \(n g g o\) of the rising pyramids \(=135^\circ 34' 31''\); of the edges \(e g\), \(g q\), \(129^\circ 31' 16''\). Angles of the trapezium \(b c g o\), \(b o r c = 103^\circ 48' 35''\); \(o o r g = 76^\circ 11' 25''\).

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, classed 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 \(e\), \(n\) (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 \(e b p r\) (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 \(P G\), \(T I\), \(R p\), \(S p\), \(k z\), \(y z\), &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 \(e p\), 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 \(a b c d e u\) (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 \(\frac{3}{4}\) or \(\frac{1}{4}\). 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 \(A B\) 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, Hauy observes, to a variety of these substances, presented 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... Structure of main faces parallel to those of the nucleus, which converge with the faces produced by the decrement, to diversify the form of the crystal. If, amidst this diversity of laws, sometimes insulated, 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 six 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 2044 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 strike 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 strike 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 strike 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, shew 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 triangular 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 \(a f_1, g a f_1, b a g, d e x, d e c, c e x\) (fig. 12.) of the primitive form. The six planes result from a decrement by two rows on the angles \(b d f_1, f x g, b c g, d f x, d b c, c g x\), 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 will successively correspond with the lines \(k i, k l, &c.\) 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 \(a b c n i h\) (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 laminae applied on the two other faces \(f a g x, b a g c\), do not decrease, but to assist 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 \(b d f_1, d f x, f x g, &c.\) 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 laminae 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. shews 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, \(146^\circ 26' 33''\); of the lateral triangles \(b g u, b g q\), to the adjacent pentagons, such as \(g u t m n, 154^\circ 45' 39''\).

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 \(c, n\) (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 60.), and of a second decrement by three rows on the lateral angles \(c b q, c r p, c r s, &c.\) which produce the triangles \(m n r, r n k, u g b, g s b, &c.\). These two decrements stop at a certain term, so Structure of that there remain faces parallel to those of the nucleus.

The first law produces the eight equilateral triangles 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, N, 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 icosahedra 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 MPS, NPL, URL, (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.

Polynomous Petunze (Hauy), fig. 64.

Geometric Character.—Respective inclination of the narrow planes, o n k m, c f h g, to the adjacent planes on each side, 150°; of the planes c f P g, P o m N to those contiguous to them by the edges t F, PN, 120°; of the heptagon p G c l d e x to the enneagon B z e b n o P r s, 90° 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 t c d e x, 135°; of the facet d e a b, or AB z p, to the same pentagon, 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 \( A R \), (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 \( Po m N \), 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 \( AO, NR \), (fig. 66.). 4. That the plane \( cf g h \), and its opposite, (fig. 64.), result from a decrement by four rows parallel to the ridges \( GD, BH \), (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, arise to planes similarly situated, such as \( o n k m \) and \( cf g h \), (fig. 64.), which is a consequence of the particular figure of the molecules.

With regard to the faces of the summit, the heptagon \( pG t c d e z \), (fig. 64.), is situated parallel to the base \( BRGO \), (fig. 66.). The enneagon \( B s r P o n b c z \) (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 to its full extent, and leaves subsisting the neighbouring heptagon parallel to the base \( BRGO \). It may be conceived, from what has been said 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 \( da fc, Ap GC \), are the result of a decrement by one row on the ridges \( GO, BR \) (fig. 66.). The facet \( de b a \) (fig. 64.) arises from a decrement by two rows parallel to the ridge \( BO \) (fig. 66.). With regard to the other facet \( AB z P \), 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 \( bc l h, k l s u \) (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 \( p G t c dez, fa b n k l i h \), 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 \( PN, Ft \). 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 single crystals.

III. Number of Primitive Forms.

In the examples which have been given, the author of the theory has chosen the parallelopiped 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 parallelopiped in general, which comprehends the cube, the rhombohedron, 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 rhombohedral 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 parallelopiped may not be extracted, by making the limitation to six selections, parallel two and two. In a great number of substances, this parallelopiped is the last term of the mechanical division, and consequently the real nucleus; but, in some minerals this parallelopiped is divisible, as well as the rest of the crystal, by further 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 conducts to a parallelopiped, divisible only by sections parallel to its six 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 \(a \cdot h \cdot s \cdot n \cdot o\) (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 \(a \cdot h \cdot n\), passes through one of the summits \(a\), and by two oblique diagonals \(a \cdot h, a \cdot n\), contiguous to the summit. By this section the solid angle \(i\) will be detached; and as there are six solid angles, situated laterally, viz. \(i, h, c, r, o, n\), 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 \(a \cdot c \cdot h, c \cdot r \cdot h, h \cdot i \cdot n, s \cdot c.\) of the cube; but each of the angles at the summits \(b \cdot a \cdot g, d \cdot s \cdot f, p \cdot s \cdot f,\) &c. of the acute rhomboid, are \(=60^\circ\), from which it follows, that the lateral angles \(a \cdot b \cdot f, a \cdot g \cdot f,\) &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 \(a \cdot b, a \cdot g, a \cdot e, s \cdot d, s \cdot f, s \cdot p\). 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 \(b \cdot a \cdot g, e,\) or \(d \cdot s \cdot f \cdot p\) (fig. 70.), so that the rhomboid will be found converted into a regular octahedron \(f\) (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. If, 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 interior edges \(g \cdot f, e \cdot g, d \cdot e, d \cdot f,\) 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, interspersed 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 \(b \cdot d \cdot f, e \cdot g \cdot p,\) (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 \(d \cdot b \cdot e, f \cdot g \cdot p,\) of the octahedron; and fig. 74. represents the case in which they would rest on the faces \(b \cdot f \cdot g, d \cdot e \cdot p.\) 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 fluoate 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. Structure of Crystals which belong to these substances. It is observed, however, that the tetrahedron is always one of those solids which concur in the formation of small rhombohedral 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, blends, 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 generalise 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".

Netwithstanding 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 F, C, G, P, I, A, by making the tour of the crystal. A second section parallel to the two rhombuses GLPF, BEAR, will coincide with another hexagon shown 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 resuming 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 six tetrahedrons, which have for external faces the halves of the three rhombuses CEDL, CLGH, CEBH, these six tetrahedrons will form a rhomboïd 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 GHBANE; and the third through the hexagon BEDFPO. 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 rhomboïds; from which it follows, that the 24 tetrahedrons, considered six and six, form four rhomboïds; so that the dodecahedron may be conceived as being itself immediately composed of these four rhomboïds, 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 rhomboïds, 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 rhomboïds, one of which would have its Structure of its summit in C, and the other in G, and which would Crystals 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 solid 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 sides 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. Haury 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.).

Geomet. Character.—Respective inclination of the trapezoids, united three and three around the same solid angle D, C, G, &c. 13° 48' 35"; of the trapezoids united four and four around the same solid angle u, x, r, &c. 13° 48' 36". Angles of any one of the trapezoids m D u L, L = 78° 27' 46"; D = 117° 2' 8"; m or u = 32° 15' 39". 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 laminae 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 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 (D).

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 EmC, EsC of the two pyramids that belong to the rhombuses CLDE, CEBH, are on a level, and form a quadrilateral EmCs. 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 striæ, 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, CEBHs, &c. (fig. 79.), and sometimes the furrows are so deep that they produce a kind of stair, the steps of which have a more particular polish and brilliancy than those of the facets, which are parallel to the faces CEDL, CHEE, 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 Haury has given the name of intermediary 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 characterised, 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. 8o.)

Geometric Character.—Respective inclinations of the triangles IAS, IBS, $139^\circ 54'$. Angles of the triangle IAS, $A = 22^\circ 54'$. $I$ or $S = 78^\circ 47'$.

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 $h i l m n r$, $c k u x y v$, &c.; from which it follows that the subtractions take place by rows of small parallelepipeds 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 $74^\circ 36'$.

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. 8t.)

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

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 $c q i$, $l b f$, $v \approx m$, &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 $g i p$, which alone forms the first row; 2. the two rhombuses $a s t i$, $p \approx d i$; 3. 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 Expally, 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 I 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 a b e d is composed of only nine molecules, and bears upon each of its sides only one square Structure of square $c$, $n$, $f$, or $r$; and that afterwards, by means of crystals, the application of new squares arranged round the former, the assortment 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 assortment 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 small 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.