4 20 
CRYSTAL! 
and 37, of the rhomboid ; from which it follows, that 
thefe two divifions will detach each a regular tetrahe¬ 
dron b, a , g, e, or d, s,f p, fig. 37, fo that the rhomboid 
will be found converted into a regular octahedron ej, 
fig. 38, which will be the real nucleus of the cube; fince 
it is produced by divifions finiilarly made, in regard to 
the eight folid angles of the cube. If we fuppofe the fame 
cube to be divifible, throughout its whole extent, by 
tedious anologous to the preceding, it is clear that each 
of the (mail rhomboids of which it is the alfemblage, 
will be found, in like manner, fubdivided into an octa T 
hicdron, with two regular tetrahedra applied on tire two 
oppofite faces of the octahedron. 
By taking the odahedron for nucleus, we may con- 
ftrud around this nucleus a cube by regular fubtradions 
of fmall complete rhomboids. For example, if we fup¬ 
pofe decrements by a (ingle range of th.ele rhomboids, 
having b for their point of departure, and made in a di¬ 
rection parallel to the inferior edges gf eg, dc, df, ot 
the four triangles, which unite to form the folid angle b, 
there will refult four faces, which will_be found on a 
level, and, like the octahedron with fix folid angles, (i- 
rnilar decrements around the other five angles will pro¬ 
duce twent) faces, which, taken four arid four, will be 
equally on a level, which will make, in the whole, fix 
diftind faces, (ituated as thofe ot the cube, fig. 33 ; fo 
that the refult will be precifely the fame as in the cafe 
of the rhomboid confidercd as nucleus. 
In whatever manner we proceed to fubdivide either 
the cube, the rhombus, or the octahedron, we (hall al¬ 
ways have folids of two forms, that is to fay, odahedra 
and tetrahedra, without ever being able to reduce the 
refult of the divition to unity. But the moleculse of a 
eryttal being neceflarily fimilar, it appeared to be pro¬ 
bable that the ftrudure was as it were interlperfed with 
a multitude of fmall vacuities, occupied either by the 
water of cryftallization, or by fome other fubtlance ; fo 
that, if it were poflible to carry the divifion to its limits, 
one of the two kinds of folids in queftion would difap- 
pear, and the whole cryftal would be found compofed 
only of moleculse of the other form. 
This idea is the more admiffible, as each odahedron 
being enveloped by eight tetrahedra, and each tetrahe¬ 
dron being equally enveloped by four oCtahedra, which¬ 
ever of the forms we imagine to be fupprelled, the folids 
that remain will join exactly by their edges; fo that, in 
this refpeCt, there will be continuity and uniformity 
throughout the whole extent of the mafs. The manner 
in which each octahedron is enveloped by eight tetrahe¬ 
dra may be readily conceived, if we take care that in di¬ 
viding the cube, fig. 35, only by the fix fedions given 
by the rhombus, we may depart at pleature from any 
two, a,s-, 0, h ; c, n ; i, r; of the eight folid angles, pro¬ 
vided that thefe two angles be oppofite to each other. 
But, if we depart from the angles a, s t the rhomboid win 
have the pofition thewn fig. 37. On the other hand, if 
we depart from the folid angles 0, k , thefe angles will 
become the fummits of a new rhomboid, fig. 39, com¬ 
pofed of the fame octahedron as that of fig. 38, with two 
new tetrahedra applied on the faces bdf, tgp, fig. 39, 
which were unoccupied on the rhomboid of fig. 37. 
Fig. 40 and 41 reprefent, one, the cafe in which the two 
tetrahedra repofe on the faces die , fgp , of the odahe- 
dron; the other, that in which they would reft on the 
faces bfg, dep. It is thence feen, that whatever may be 
the two folid angles of the cube alfumed for the points 
of departure, we fhall always have the fame odahcdion, 
with two tetrahedra contiguous by their fummits to the 
two folid angles in queftion; and as there are eight of 
thefe folid angles, the central odahedron will be circum- 
fenbed by eight tetrahedra, which will reft on its faces. 
The fame efted will take place if we continue the divi¬ 
fion always parallel to the firft fedions. Each face of 
the odaliedron, then, however fmall we may tnppofe 
that odahedron to be, adheres to a face of the tetrahe- 
OGRAPII Y. 
dron, and reciprocally. Each tetrahedron then is enve¬ 
loped by four odahedra. 
Tire ftrudure here explained is that of fparry fiuor. 
By dividing a cube of this fubtlance we may, at pleafure, 
extrad rhomboids, having the angles formed by their 
planes equal to 120° or regular odahedra, or tetrahedra, 
equally regular. There are a fmall number of other 
fubftances, fiich as rock cryftal, carbonate of lead (fparry 
lead), &c. which being mechanically divided beyond the 
term at which we Ihould have a rhomboid or parallelo- 
pipedon, givealfo parts of various different forms aflorted 
together in a manner even more complex than in fparry 
fiuor. Ihefe mixed ftrudures necefiarily occafion un¬ 
certainly refpeding the real figure of the integral mole, 
cola. 1 which belong to the fubftances in queftion. It has, 
however, been obferved, that the tetrahedron is always 
one of thofe folids which concur to the formation of 
fmall rhomboids or parallelopipedons that would be 
drawn front the cryftal by a firft divition. On the other 
hand, there are fubftances, which, being divided in all 
pofiiblc diredions, refolve themfelves only into tetrahe¬ 
dra. Of this number are garnet, blend, and tourmaline. 
In ftiort, feveral minerals are divifible into right tri¬ 
angular prifms. Such as the apatite, the primitive ’form 
of which is a regular right hexahedral prifm, divifible 
parallel to its bates and its planes, from which necefta- 
rily refult right prifms with three planes, as may be fees 
by infpeding fig. 43, which reprefents one of the bafes 
of the hexahedral prifm divided into frnali equilateral 
triangles, which are the bafes of fo many moleculse, and 
which, being taken two and two, form quadrilateral 
prifms with rhombufes for their bafes. By adopting 
then the tetrahedron in fuch doubtful cafes, we thould 
reduce, in general, all the forms of integral moleculse to 
three, remarkable by their fimplicity, viz. the parallelo- 
pipedon, the (impleft of all the foiids which have faces pa¬ 
rallel two and two ; the triangular prifm, the fimpleft of 
all prifms; and the tetrahedron, which is the fimpleft of 
pyramids. This fimplicity may furnifh a reafon for the pre¬ 
ference given to the tetrahedron in fparry fiuor, and the 
other fubftances here fpoken of. But the eftential objed 
is, that the different forms to which the mixed ftructures 
in queftion condud, are aflorted in fuch a manner, that 
their aflemblage is equivalent to afum of fmall parallelo¬ 
pipedons, as we have feen to be the cafe in regard to 
fparry fiuor; and that the laminae of fuperpofition, ap¬ 
plied on the nucleus, decreafe by fubtradions of one or 
more ranges of thefe parallelopipedons; fo that the bafts 
of the theory exifts independently of the choice which 
might be made of any of the forms obtained by the me¬ 
chanical divifion. 
By the help of this refult, the decrements to which 
cryftals are fubjed, whatever be their primitive forms, 
are found brought back to thofe which take place in 
fubftances where this form, as well as that of the mole- 
culae, are indivifible parallelopipedons; and theory lias 
the advantage of being able to generalize its objed, by 
conneding with one fad that multitude of fads which 
by their diverfity feem to be little fufceptible of con¬ 
curring in a common point. What has been here faid 
will be better illuftrated by a few examples of the man¬ 
ner in which we may reduce to the theory of the paral- 
lelopipedon that of the forms different from that folid. 
Crystals, the Molecules of which are Tetrahedra, 
with Isosceles Triangular Faces, 
Garnet, i. Primitive garnet, fig. 43, Grenat a douze 
faces, Daubenton Tab. Miner, edir. 1792, p. 5. Grenat do - 
decaedre a plans rhombes , De 1 ’Ifle Cryjlallographie , tom, ii. 
p. 322, var. 1 .—Geometrical Character, Refpedive inclina¬ 
tion of any two of the faces of the dodecahedron 120°. 
Angles of the rhombus CLGH, C or G = 109° 28' 16": 
L or H sz 78° 31' 44". 
Though garnets of the primitive form be in general 
vitreous on the fradures, there are perceived on them, 
however, 
