hose 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, $120^\circ$. Angles of the rhombus CLGH, C or G = $109^\circ 28' 16''$; Structure of Crystals. L or H = $78^\circ 31' 44''$.

Notwithstanding the vitreous appearance in general exhibited on the fractures of garnets of the primitive form, laminae may be perceived on them, situated parallel to the rhombuses which compose their surface. Let us suppose the dodecahedron divided in the direction of its laminae, and for the greater simplicity, let us suppose the sections to pass through the centre. One of these sections, viz., that which will be parallel to the two rhombuses DLFN, BHOR, will concur with a hexagon, which would pass through the points E, C, G, P, I, A, by making the tour of the crystal. A second section parallel to the two rhombuses GLPF, BEAR, will coincide with another hexagon drawn 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 refusing 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 rhomboid represented by fig. 77, and in which the three inferior rhombuses DLGS, GHBS, DEBS, result from three divisions which pass, one through the hexagon DLGOR, (fig. 76); the second through the hexagon GHBAF; 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 rhomboids; from which it follows, that the 24 tetrahedrons, considered six and six, form four rhomboids; so that the dodecahedron may be conceived as being itself immediately composed of these four rhomboids, and in the last analysis of 24 tetrahedrons.

It may be observed, that the dodecahedron having eight solid angles, each formed with three planes, they might have been considered as the assemblage of the four rhomboids, which would have for exterior summits the four angles G, B, D, A; from which it follows that any one of the faces, such as CLGO, is common to two rhomboids, one of which would have... 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. Hauy informs us that he has divided fragments of this substance by very clean sections, in such a manner as to obtain successively the dodecahedron, the rhomboid and the tetrahedron.

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

Geometrical Character.—Respective inclination of the trapezoids, united three and three around the same solid angle D, C, G, &c. $146^\circ 26' 33''$; of the trapezoids united four and four around the same solid angle, u, v, r, &c. $131^\circ 48' 36''$. Angles of any one of the trapezoids mD u L, L = $78^\circ 27' 46''$; D = $117^\circ 2' 8''$; m or u = $82^\circ 15' 3''$. The value of the angle L is the same as that of the acute angle of the nucleus of calcareous spar.

This variety is the result of a series of laminae, decreasing at the four edges, on all the faces of the primitive dodecahedron. For the more simplicity, let us first consider the effect of this decrement in regard to the rhombus CLGH (fig. 76.). We have just seen that this rhombus was supposed to belong in common to two rhomboids, which should have for summits, one, the point C, and the other the point G. Let us suppose that the 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 the 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 Er, and CM equal to Cr. Thus the quadrilaterals will be trapezoids which have their sides equal two and two. There is no crystalline form in which the fissure, when they do exist, shew 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 CLDEm, 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, CHBE, of the nucleus.

If the decrements stop abruptly at a certain term, so that the pyramids are not terminated, the 24 trapezoids will be reduced to elongated hexagons, which will intercept 12 rhombuses parallel to the faces of the nucleus. To this variety Hauy has given the name of 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. Hauy has thus denominated the gem which is known under the different names of ruby, sapphire, oriental topaz, according as the colour is red, blue, or yellow. The different varieties of this gem have not been accurately described, and the nature of the particular angles of each has not been precisely indicated, on account of the rare occurrence of regularly formed crystals, or, when such have been found, on account of their being defaced by being water-worn, or otherwise injured. But from some crystals which were sufficiently characterized, Hauy obtained the following results.

1. Primitive Oriental.

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

2. Elongated Oriental, (fig. 80.)

Geometric Character.—Reflective 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 $b i l m n r$, $e k u x y v$, &c.; from which it follows that the subtractions take place by rows of small parallelopipeds of quadrangular prisms, composed each of two triangular prisms.

3. Minor Oriental.

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

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

4. Enneagonal Oriental, (fig. 81.)

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

This is the elongated oriental, whose summits are replaced by two faces, parallel to the bases of the nucleus, with the addition of six (small isosceles triangles $e g i$, $l b f$, $v z m$, &c. the three superior of which are alternate in position with the three inferior. These triangles are the result of a decrement, by three rows of small quadrangular prisms on the three angles of the superior base of the nucleus, such as $b$, $d$, $g$ (fig. 75), and on the intermediate angles of the inferior base. It may be readily conceived, that in the decrement which takes place, for example, on the angle $g$, the three rows which remain unoccupied between that angle and the corresponding edge of the first lamina of superposition, are, 1. the small rhombus $g o i p$, which alone forms the first row; 2. the two rhombuses $o s t i$, $p z d i$; 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 layer formed by the laminae of superposition. It may be readily conceived, that the arrangement began with the square ABCD; and that different files of small squares were afterwards applied on each of the sides of the central square: for example, on the side AB, first the five squares comprehended between I and M, next the three squares comprehended between L and O, and then the square E. This increment corresponds with that which would take place if the dodecahedron began by being a cube proportioned to its bulk, and which increased afterwards with the addition of continually decreasing laminae.

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

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

Such is the ingenious theory of the structure of crystals, which the author observes, is in this similar to other theories, that it sets out from a principal fact, on which it makes all facts of the same kind to depend, and which are only as it were corollaries. This fact is the decrement of the lamina 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 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.