CRYSTALI 
however, laminae fituated parallel to the rhombufes 
which compol'e their furface. Let us fuppofe the dode¬ 
cahedron divided in the direction of its laminae, and, 
for the greater fnnplicity, let us make the feclions pafs 
- through the center. One of thefe feclions, viz. that 
which will be parallel to the two rhombufes DLFN, 
BHOR, will concur with a hexagon which would pafs 
through the points E, C, G, P, I, A, by making the tour 
of the cryflal. A fecond feiftion parallel to the two 
rhombufes G L F P, BEAR, will coincide with ano¬ 
ther hexagon (hewn by the points D, C, H, O, I, N. If 
the divifion be continued parallel to the other eight 
rhombufes, taken two and two, we (hall find that the 
planes of the fefiiions will be confounded with four new 
hexagons analogous to the preceding. But by refuming' 
all thefe hexagons, it is feen that their (ides correfpond, 
feme of them with the fmall diagonals of tire rhombufes 
of the dodecahedron, viz. thofe which would be drawn 
from C to G, from A to I, from C to B, &c. and others 
would correfoond with the different ridges EC, GP, 
PI, EA, &c. 
The planes then of the fefiiions, palling through the 
fides and through the fmall diagonals of the twelve rhom¬ 
bufes, will fubdivide the whole furface into twenty-four 
ifofceles triangles, which will be the halves of thefe 
rhombufes. And iince the planes of the fefiiions pafs 
alfo through the center of the cryflal, they will detach 
twenty-four pyramids with three faces, the bafes of 
which, if we ehoofe, will be the external triangles that 
make a part of the furface of the dodecahedron, and of 
which the fummits will be united in the center. More¬ 
over, if we take, for example, the fix tetrahedra which 
Irave for external faces the halves of the three rhombufes 
CEDL, CLGH, CEB H, thefe fix tetrahedra will form 
a rhomboid reprefer.ted by fig, 44; and in which the 
three inferior rhombufes DLGS, GHBS, DEBS, re- 
fult from three divifions which pafs, one through the 
hexagon D L G O R A, fig. 43 ; the fecond through the 
hexagon G H B A N F, and the third through the hexa¬ 
gon BE D F P O. Fig. 44 reprefents alfo the two tetra¬ 
hedra, the bafes of which make part of the rhombus 
C 1 , G H. One of thefe is marked by the letters L, C, 
G, S, and the other by the letters H, C, G, S. By ap¬ 
plying what has been laid to the other nine rhombufes, 
which are united three and three around the points F, 
A, H, fig. 44, we fhall have three new rhomboids; front 
which it follows, that the twenty-four tetrahedra, con- 
lidered fix and fix, form four rhomboids ; fo that the do¬ 
decahedron may be conceived as being itfelf immediately 
coinpofed of thefe four rhomboids, and in the lad analy- 
fis of twenty-four tetrahedra. The dodecahedron having 
-eight fclid angles, each formed by three planes, we 
might have confidered them as being the alfemblage of 
the four rhomboids, which would have for exterior fum¬ 
mits the four angles G, B, D, A; from which it refults 
that any one of the faces, fuch as C L G O, is common 
to two rhomboids, one of which would have its fummit 
in C, and the other in G, and which would thernfelves 
have a common part in the interior of the cryflal. 
It may be farther remarked, that a line G S, fig. 44, 
drawn from any one, G, fig. 43, of the folid angles com- 
ofed of three planes, as far as the center of the dodeca- 
edron, is, at the fame time, the axis of the rhomboid, 
which would have its fummit in G, and one of the edges 
of that which would have its fummit in C, fig. 43 and 44. 
The compofing rhomboids then have this property, that 
their axis is equal to the fide of the rhombus. With a 
little attention it will be eafily feen, that in each tetra¬ 
hedron, fuch as C L G S, fig. 44, all the faces are equal 
and fimilar ifofceles triangles. If we fhould continue the 
divifion ot the dodecahedron by fefiiions palling between 
thofe which we have fuppofed to be directed towards 
the center, and which fhould be parallel to them, we 
fhould obtain tetrahedra always fmaller, and arranged 
311 fuch a manner, that, taking them in groupes of fix, 
Vol. V. No. 284. 6 r ? 
< O G R A P H Y. 441 
they would form rhomboids of a bulk proportioned to 
their own. 
The tetrahedra, winch would be the term of the divi¬ 
fion, were it pofiible for us to reach, it, ought to be .con¬ 
fidered as the real mo! ecu (as of the garnet. But we fhall 
fee, that, in the paifage to the fecondary forms, the la¬ 
minae of fuperpofition, which envelop the nucleus, really 
decreafe by ranges of fmall rhomboids, each of which is 
the affemblage of fix of thefe tetrahedra. The fiiiphure 
of zinc or blend has the fame ftruclure as the garnet. 
2. Trapezoidal Garnet, fig. 45. Grenataz^faccs, 
Daub. Grenat a z\facettes, trcipezoidales, De l’lfle.— Geo¬ 
metrical Charadler. Refpefitive inclination of the trape¬ 
zoids united three and three around the fame folid angle 
D, C, G, &c. 146° 26' 33";. of the trapezoids united four 
and four around the fame folid angle u, x, r, &c. i3i°48' 
36". Angles of any one of the trapezoids m D u L, L~ 
78° 27' 46" •, D 2=: 117° 2' 8" ; m or u — 82° 15' 3". The 
value of the angle L is the fame as that of the acute angle 
of the nucleus of calcareous fpar. 
This variety refults from a feries of lamina: decreafing 
at the four edges, on all the faces of the primitive dode¬ 
cahedron. For the more fnnplicity, let us firft confider 
the effe'fil of this decrement in regard to the rhombus 
C L G H, fig. 43. We have juft feen that this rhombus 
was fuppofed to belong in common to two rhomboids, 
which fhould have for fummits, one, the point C, and 
the other, the point G. Let us fuppofe that the laminae 
applied on this rhombus decreafe’ towards their four 
edges by fubtrafilions of a fingle range of fmall rhom¬ 
boids, in fuch a manner that, in regard to the two edges 
C L, C H, circumflances are the fame as if the rhombus 
belonged to the rhomboid which has its fummit in C ; 
and that, in regard ter the other two edges G L, G H, 
the effefil is the fame as if the rhombus belonged to the 
rhomboid having its fummit in G. This difpofition is 
admiflible here in confequence of the particular ftruclure 
of the dodecahedron, which permits us to obtain fmall 
rhomboids, feme of which have their faces parallel to 
the faces of that with its fummit in C, and the reft to 
that having its fummit in G. 
The refults of the four decrements being thus per¬ 
fectly fimilar to each other, the laminae of fuperpofition, 
applied on the rhombus CLGH, and on each of the 
other rhombufes of the dodecahedron, will form as many 
right quadrangular pyramids, which will have for bafes 
thefe fame rhombufes. In fig. 46, may be feen the py¬ 
ramids which reft on the three rhombufes CLDE, C E 
BH, C G H B, fig. 43, and which have for fummits the 
oints m, e, s, fig. 46 ; but on account of the decrement 
y a fimple range, the adjacent triangular faces, fuch as 
E m C, EsC of the two pyramids that belong to the 
rhombufes CLDE, CE BU, are on a level, and form a 
quadrilateral EjjCj. But we had twelve pyramids, and 
confequently forty-eight triangles. Dividing by two, we 
fhall then have twenty-four quadrilaterals, which will 
compofe the furface of the fecondary cryflal. But, be- 
caufe the rhomboidal bafes 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 fides m E, E s, 
of the quadrilateral will be longer than tile fides C m, Cs. 
Moreover, we fhall evidently have m E equal to Es, and 
C m equal to Cs. The quadrilaterals will then be trape¬ 
zoids, which w$ll have their fides equal two and two. 
There feems to be no cryftalline form where the ftriae, 
when they exift, point out, in a more fenfible manner 
than in this, the mechanifm of the firufiture. We may 
here fee the feries of decreafing rhombufes which' form 
each of the pyramids CLDE®, CEBHs, See. fig. 46, 
and fometimes the furrows are fo deep that they produce 
a kind of flair, the fteps of which have a more particular 
polifh and brilliancy than thofe of their facets, which are 
parallel to the faces C E D L, C FI B E, &c. of the nucleus. 
If the decrements flop abruptly at a certain term, fo 
that the pyramids are not terminated, the twenty-four 
5 P trapezoids 
