■hedral crystals, it passes into an octahedron ' 
and, on the other hand, an octahedral crys- 
tal put into a solution that would afford cu- 
bic crystals becomes itself a cube. Now, 
how difficult a matter it is to proportion the 
•different ingredients with absolute exactness, 
'must appear evident to all. 
2d, The secondary forms are sometimes 
owing to the solvent in which the crystals 
are formed. Thus if common salt is dissolved 
in water, and then crystallized, it assumes the 
form of cubes ; but when crystallized in urine, 
it assumes the form not of cubes, but of re- 
gular octahedrons. On the other hand, mu- 
| riat of ammonia, when crystallized in water, 
assumes the octahedral form, but in urine it 
crystallizes in cubes. .. 
3d, But even when the solvent is the same, 
and the proportion of ingredients, as far as can 
be ascertained, exactly the same, still there 
are a variety of secondary forms which are 
usually found to make their appearance. 
These secondary forms have been happily 
explained by the theory of crystallization for 
which we are indebted to the sagacity of Mr. 
Hauy; a theory which, for its ingenuity, 
clearness, and importance, must ever rank 
high, and which must be, considered as one of 
the greatest acquisitions which mineralogy 
: and even chemistry have hitherto attained. 
According to this theory, the additional mat- 
ter which envelopes the primitive nucleus con- 
j sists of thin slices or lay ers of particles laid 
one above another upon the faces of that nu- 
I cleus ; and each layer decreasing in size, in 
I consequence of the abstraction of one or more 
rows of integrant particles from its edges or 
| angles. 
Let us suppose that ABFG (fig. 6). is a 
cube composed of 729 small cubes : each of 
i its sides will consist of 81 squares, being the 
I external sides of as many cubic particles, 
which together constitute the cube. Upon 
ABCD, one of the sides of this cube, let us 
apply a square lamina, composed of cubes 
| -equal to those of which the primitive crystal 
! consists. It will of course be composed of 
49 cubes, 7 on each side; so that its lower 
base o nfg (lig. 7.) will fall exactly on the 
square marked with the same letters in lig. 
[ 6 . 
1 Above this lamina let us apply a second, 
l ?np u (fig. 10), composed of 25 ; it will be 
situated exactly above the square marked 
with the same letters (fig. 6). Upon this 
second let us apply a third lamina, v x // z 
i (fig. 8.), consisting only of 9 cubes; so that 
I its base shall rest upon the letters v x y z 
i (fig. 0). Lastly, on the middle square r let 
i us place the small cube r (fig. 5.), which will 
represent the last lamina. 
It is evident that by this process a qua- 
! drangular pyramid has been formed upon the 
; face ABCD (fig. 6), the base of which is 
this face, and the vertex the cube r ( fig. 5). 
| (By continuing the same operation upon the 
other five sides of the cube, as many similar 
pyramids will be formed ; which will enve- 
lope the cube on every side. 
It is evident, however, that the sides of these 
pyramids will not form continued planes, but 
that, owing to the gradual diminution of the 
laminae of the cubes which compose them, 
these sides will resemble the steps of a stair- 
case. We can suppose however (what must 
■certainly be the case), that the cubes of which 
the nucleus is formed are exceedingly small, 
5 
CRYSTALLIZATION. 
almost imperceptible ; that therefore a vast 
number of laminae are required to form the 
pyramids, and consequently that the channels 
which they perform are imperceptible. Now 
DCBE (lig. 9-) being the pyramid resting 
upon the face ABCD (lig. (j.), and CBOG 
(lig. 9.) the pyramid applied to the next 
face BCG li (fig. 6.), if we consider that 
every thing is uniform from E to O (lig. 9-), 
in the manner in which the edges of the 
laminae- of superposition (as the abbe 
Hauy calls the laminae which compose the 
pyramids) project beyond each other, it will 
readily be conceived that the face CEB of 
the first pyramid ought to be exactly in the 
same plane with the face COB of the adjacent 
pyramid, and that therefore the two faces to- 
gether will form one rhomb ECOB. But all 
the sides of the six pyramids amount to 24 
triangles similar to CEB ; consequently they 
will form 12 rhombs, and the figures of the 
whole crystal will be a dodecahedron. 
Thus we see that a body which has the 
cube for the primitive form of its crystals, 
may have a dodecahedron for its secondary 
form. The formation of secondary crystals, 
by the superposition of lamina? gradually de- 
creasing in size, was first pointed out by 
Bergman. But Hauy has carried the subject 
much farther: he has not only ascertained all 
the different ways by which these decrements 
of the lamina; may take place, but pointed 
out the method of calculating all the possible 
varieties of secondary forms which can result 
from a given primitive form ; and conse- 
quently of ascertaining whether or not any 
given crystal can be the secondary form of a 
given species. 
The decrements of the lamina; which cover 
the primitive nucleus in secondary crystals 
are of four kinds : 
1. Decrements on the edges, that is, on 
the edges of the slices which correspond with 
the edges of the primitive nucleus. 
2. Decrements on the angles ; that is to 
say, parallel to the diagonals of the faces of 
the primitive nucleus. 
3. Intermediate decrements ; that is to 
say, parallel to lines situated obliquely be- 
tween the diagonals and edges of the primi- 
tive nucleus. 
4. Mixed decrements. In these the su- 
perincumbent slices, instead of having only 
the thickness of one integrant particle, have 
the thickness of two or more integrant par- 
ticles ; and the decrement, whether parallel 
to the edges or angles, consists hot of the abs- 
traction of one row of particles, but of two 
or more. Hauy denotes these decrements 
by fractions, in which the numerator indi- 
cates the number of rows of particles which 
constitutes the decrement, and the denomi- 
nator represents the thickness of the lamina:. 
Thus 1 denotes lamina; of the thickness of 
3 
three integrant particles, decreasing by two 
rows of particles. 
An example of the first law of decrement, 
or of decrement on the edges, has been given 
above in the conversion of the cubic nucleus 
to a rbomboidal dodecahedron. In that exam- 
ple the decrement consisted of one row of par- 
ticles, and it took place on all the edges. But 
these decrements may be more rapid : in- 
stead of one, they may consist of two, three, 
four, or more rows ; and instead of taking ) 
place on all the edges, they may be confined i 
to one -or two of them, while no decrement j 
463 
at all takes place on the others. Each of 
these different modifications must produce a 
different secondary crystal. Besides this, 
the lamina; may cease to be added before 
they have reached their smallest possible 
size ; the consequence of which must be a 
different secondary lorm. Thus, in the ex- 
ample given above, if the superposition of 
laminae had ceased before the pyramids were 
completed, the crystal would have consisted 
of 18 faces ; 6 squares parallel f o the faces of 
the primitive nucleus, and 12 hexahedrons 
parallel to the faces of the secondary dode- 
cahedron. This is the figure of the barat of 
lime and magnesia found atLuneburg. 
The second law in which the decrement is 
on the angles, or parallel to the diagonals of 
the faces of the primitive nucleus, will be un- 
derstood from the following example : Let it 
be proposed to construct around the cube 
ABGFXiig. 11), considered as a nucleus, a 
secondary solid, in which the lamina; of su- 
perposition shall dec rease on all sides by sin- 
gle rows of cubes, but in a direction parallel 
to the diagonals. Let ABCD (fig. 12), the 
superior base of the nucleus, be divided into 
81 squares, representing the faces of the small 
cubes of which it is composed*-- f ig- 13 re- 
presents the superior surface of the first la- 
mina of superposition ; which must be placed 
above ABCD (lig. 12) in such a manner, that 
the points abed (fig. 13) answer to the 
points abed (fig. 12). By this disposition 
the squares A a, lib, C c, Y)d (fig. 12), which 
compose the four outermost rows of squares 
parallel to the diagonals AC, BC, remain un- 
covered. It is evident also, that the borders 
QV, ON, IL, GF (fig. 13), project by 
one range beyond the borders AB, AD, Cl), 
BC (fig. 13); which is necessary, that the 
nucleus may he enveloped towards these 
edges : for if this were not the case, re-en- 
tering angles would be formed towards the 
parts AB, BC, CD, I)A, of the crystal ; 
which angles appear to be excluded by the 
laws which determine the formation of sim- 
ple crystals, or, which comes to the same 
thing, no such angles are ever observed in 
any crystal. The solid must increase, then, 
in those parts to which the decrement does 
not extend. But as this decrement is alone 
sufficient to determine the form of the secon- 
dary crystal, we may set aside all the other 
variations which intervene only in a subsidia- 
ry manner ; except when it is wished, as in 
the present case, to construct artificially a so- 
lid representation of a crystal, and to exhibit 
all the details which relate to its structure. 
The superior face of the second lamina will 
be AGLK (lig. 14). It must be placed so 
that the points a, b, c, d, correspond to 
the points a, b, c, d (fig. 13); which will 
leave uncovered a second row of cubes at 
each angle, parallel to the diagonals AC and 
BD. The solid still increases towards the 
sides. Tire large faces of the laminae of su- 
perposition, which in fig, 13 were octagons, 
in tig. 14 arrive at the form of a square ; and 
when they pass that term they decrease oi» 
all sides; serthat the next lamina has for its 
superior face the square KMLS (fig. 15) 
less by one range in every' direction than the 
preceding lamina (lig. 14). This square must 
be placed so that the points e, f, g, h (lie. 
15), correspond to the points e,f,g, h (fig. 
14). Figures 16, 17,18, and 19, represent 
the four lamina; which ought to rise succes- 
