418 CRYSTALLOGRAPHY. 
fituated parallel to the fuperior bafe A B C D of the 
nucleus, and which will be confounded with that bafe 
when the fections have made the faces of the octahedron 
to difappear entirely. This is the ftrudture of the octa¬ 
hedral fulphuret of lead and of muriat of foda. Decre¬ 
ments which take place in this manner have been called 
by Mr. Hauy decrements on'the angles. 
There are certain cryltals in which the decrements on 
the angles do not take place in lines parallel to the dia¬ 
gonals, but parallel to lines fituated between the diago¬ 
nals and the edges. This is the cafe when the fubtrac- 
tions are made by ranges of double, triple, &c. molec'ulae. 
F'g- 33> exhibits an inftance of the fubtradtions in ques¬ 
tion f and it is feen that tire moleculae which compofe 
the range reprefented by that figure are afforted in fuch 
a manner as if of two there were formed only one ; fo 
that we need only to conceive the cryftal compofed of 
parallelopipedons having their bafes equal to the fmall 
rectangles abed , edfg, hgil, See. to reduce this cafe un¬ 
der that of the common decrements on the angles. To 
this particular kind of decrement Mr. Hauy has given 
the name of intermediate. 
In other cryftals the decrements, either on the edges 
or on the angles, vary according to laws, the proportion 
of which cannot be expreffed but by the fraction | or 
It may happen, for example, that each lamina exceeds 
the following by two ranges parallel to the edges, and 
that it may at the fame time have an altitude triple that 
ot a fimple molecule. Fig. 34, reprefents a vertical geo¬ 
metrical fedtion of one of the kinds of pyramids which 
would refult from this decrement; the etfedt of which 
may be readily conceived, by confidering that A B is a 
horizontal line taken on the upper bafe of the nucleus, 
bazr the fedtion of the firft lamina of fuperpofition, 
gfen that of the fecond, See. Thefe decrements Mr. 
Hauy Jias called mixed. Both of thefe laft fpecies of de¬ 
crements occur but rarely ; Mr. Hauy found them only 
in certain metallic fubftances. 
All the meVamorphbfes to which cryftals are fubjedt- 
cd depend, according to Mr. Hauy, on the laws of 
ftricture juft explained, and others of the like kind. 
Sometimes the decrements take place at the fame time 
on all the edges ; as in the dodecahedron having rhom- 
bufes for its planes, as before mentioned ; or on all the 
angles, as in the odtahedron originating from a cube. 
Sometimes they take place only on certain edges or 
certain angles. Sometimes there is.an uniformity be¬ 
tween them ; fo that it is one fingle law by one, two, 
three, ranges, &c. which adts 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 ; 
and this happens above all when the nucleus has not a 
fymmatrical form ; for example, when it is a purallelo- 
pipedon, the faces of which differ by their refpedtive 
inclinations, or by the meafure of their angles. In 
certain -cafes the decrements on the edges concur 
with the decrements on the angles to produce the fame 
cryftailir.e form. It happens alfo fo me times that the 
fame edge, or the fame angle, is fubjedted to fever .1 
laws of decrement that fucceed each other. Tn (Fort, 
there are cafes where the fepOndary cryftal has faces pa¬ 
rallel to thole-of the primitive form, and which com¬ 
bine with t he. faces produced by the decrements to mo¬ 
dify the figure of the cryftal. The cryftals ariling from 
a fingle/ law of-decrement are called by Mr. Hauy Jim- 
piejicondary.forms .; thole which arife from 1'cveral 'fimul- 
taneous laws of decrement, he calls compound fecondary 
forms. 
“‘If arn.idft this diverfity of laws (he obferves), fome- 
times inf ttla'ted,• femetimes united by combinations more 
or lels complex, the number of the ranges fubtracbed 
were-itfelf extremely variable ; for example, were thefe 
decrements by twelve, twenty, thirty, or forty ranges,, 
or more, as might abfoltuely be pqjlible, the multitude 
of tiu forms which might exift in each, kind of mineral 
would be immenfe, and exceed what could be imagined. 
But the power which effedts the fubtradtions feems to 
have a very limited adtion. The fubtradtions, for the 
mod part, take place by one or two ranges of molecules. 
T have found none which exceeded four ranges, except 
in a variety of calcareous fpar, forming part of the col- 
ledtion of C. Gillet Laumont, the ftructure of which de¬ 
pends on a decrement by fix ranges ; fo that if there exift 
laws which exceed the decrements by four ranges, there 
is reafon to believe that they rarely take place in nature. 
Yet, notwithftanding thefe narrow limits by which the 
laws of cryftallization are circumfcribed, I have found, 
by confining myfelf to two of the limpleft laws, that is 
to fay, thofe which produce fubtradtions by one or two 
ranges, that calcareous fpar is fufceptible of two thou- 
fand and forty-four different forms: a number which 
exceeds more than fifty times that of the forms already 
known ; and if we admit into the combination decre¬ 
ments by three and four ranges, calculation will give 
8,388,604 pofiible forms in regard to the fame fubftance. 
This number may be ftill very much augmented in con- 
fequence of decrements either mixed or intermediary. 
“ The ftria? remarked on the furface of a multitude 
of cryftals afford a new proof in favour of theory, as 
they always have diredtions parallel to tire projedting 
edges of the laminae of fuperpofition, which mutually 
go beyond each other, unlefs they arife from fome par¬ 
ticular want of regularity. Not that tire inequalities 
refulting from the decrements muft be always fenfible, . 
fuppoling the form of the cryftals had always that degree 
of finiftring of which it is fufceptible ; for, onpaccount of 
the extreme nrinutenefs of the molecules, the. fur-face 
would appear of a beautiful poliflr, and the ftrire would . 
elude our fenfes. There are therefore fecondary cry¬ 
ftals where they are not at all obferved, while they 
are very vifible in other cryftals of the fame nature 
and form. In the latter cafe, the adtion of the. caufes 
which produce cryftallizatibn not having fully enjoyed 
all the conditions neceffary for’perfedting that fo de¬ 
licate operation of nature, there have been ftarts and in-, 
terruptions in their progrefs, fo that, the law of conti¬ 
nuity not having been exadtly obferved, there have re¬ 
mained on tire furface of tire, cryftal vacancies.apparent 
to our eyes. The fmall deviations are attended with 
this advantage, that they point.out the direction accord-, 
ing to which the ftria: are arranged inlines on the perfect 
forms where they e’fcape our organs, and thus contribute 
to unfold to us the real meclranifm of the ftrudture. 
“ The fmall vacuities which the edges of the laminae 
of fuperpofition leave on the furface of even theYuoft per¬ 
fect fecondary cryftals, by their, re-entering and falient 
angles, thus afford a fatisfadtory folution of the diffi¬ 
culty a little before mentioned ; which is, that, the frag¬ 
ments obtained by divifion, the external fides. of which 
form part of the faces of the fecondary cryftal, are not 
like thofe drawn from the interior part. For this di¬ 
verfity, which is only apparent, arifes from the fides. 
in queftion being compofed ofa multitude of fmall planes, 
really inclined to one another, but which, on account 
of their fmallnefs, pref'ent the appearance of one plane : 
fo that if the divifion could reach its utmoft bounds, 
all thefe fragments would be refolved into molecules 
fimilar to each other, and to thofe fituated towards the 
centre. 
“ The fecundity of the laws on which the variations, 
of cryftalline forms depend, is not confined to the pro-, 
ducing of a multitude of very different forms with the 
fame molecules.- It often happens alfo, that molecules 
of different figures arrange themielves in fuch a manner: 
as gives rife to like polyhedra in different kinds of mi¬ 
nerals. Thus the dodecahedron with rhombufes for 
its planes, which we obtained by -.combir.ing cubic mole¬ 
cules, exifts in the granite .with a ftrudture Compofed of 
fmall tetrahedra, having ifofceles triangular faces ; and 
I have found it in fparry Suor (Jluat of limeJ, where 
there 
