C RTSTA L L 
fcnt the appearance of a Ample ridge. _ By a neceflavy 
conlequence the pyramid will have for faces two trape¬ 
ziums, fitch as D M N C, refulting from the firft decre¬ 
ment ; and two ifofceles triangles, fuch as CN B, which 
will be produced by the fecond decrement. Here the 
face which correfponds to A B C D, fig. 7, has twenty- 
five fquares on each fide, as may be feen in fig'. 14, and 
the ftrufture of the pyramid in queftion may be imitated 
artificially, by obferving the order and number of the ' 
cubes reprefented in the fame figure. 
Let us" fuppofe, moreover, that, in regard to the la¬ 
minae of fuperpofition which arife on the face BCGH, 
fig. 7, the decrements follow the fame laws, but by crofs 
directions ; fo that the more rapid of the two may take 
place in proceeding from B C or G H towards the fum- 
mit of the pyramid, and the flower in proceeding from 
C G or B H towards the fame fummit. The pyramid 
refulting front thefe decrements will be placed in a di- 
redlion oppofite to that refting on A B C D, and will have 
the pofition exhibited in fig. 17, where it is feen that 
the edge KL, which terminates the pyramid, inftead of 
being parallel to C D like the edge M N, fig. 14 and 15, 
is, on the contrary, parallel to B C. We fliall then con¬ 
ceive what ought to be done in order that the pyramid 
which will reft on DC G F, fig. 7, may be turned as re¬ 
prefented in fig. 16, and may have its terminating edge 
PR parallel to C G, fig. 7. We need not notice the py¬ 
ramids that will reft on the three other faces of the cube, 
becaufe it is evident that each of thefe pyramids ought 
to-ftund like that which arifes on the oppofite face. 
The fides of all the fix pyramids thus formed amount 
to twelve trapeziums and twelve triangles. Every tri¬ 
angle is evidently contiguous and in the fame plane with 
a trapezium of the neareft pyramid ; confequently the 
fecondary cryftal thus formed confifts of twelve fides, 
each of which is a pentagon. Several other examples 
have been given by Mr. Hauy; but thefe are fufficient 
to fliew in what manner the various fecondary forms of 
cryftals are conftrudted, according to the theory of that 
ingenious philofopher. 
In his refearches on this fubjeft, Mr. Hauy perceived, 
that fome cryftals a (Turned fecondary forms which could 
not be accounted for by any decrement whatever along 
the edges. Thus, for inftance, fome bodies, the primary 
form of which is cubic, are fometimes found cryftallized 
in regular odtagons. Mr. Hauy explains the formation 
of thefe fecondary cryftals, by fuppofing that the decre¬ 
ment took place parallel, not to the edges, but to the 
diagonals of the faces of the primary cubes. In order to 
comprehend this, let us fuppofe A B C D, fig. 18, to be 
the fur face of a lamina compofed of fmall cubes, the 
bafes of which are reprefented by the little fquares in 
the figure. It is evident, that the cubes a, b, c, d, e,f, 
g , h, i, are in the direction of the diagonal of the fquare 
A B C D ; that the row of cubes q, v, k, u, x,y, z, is pa¬ 
rallel to the diagonal; as alfo the row n, t, l, m,p, o,r,s\ 
and that the whole figure might be divided into rows of 
fquares, each of which would be parallel either to the 
diagonal AC or D B. Now we may conceive that the 
laminae of fuperpofition, inftead of decreafing by rows of 
cubes parallel to the edges A B, AD, decreafe by rows 
parallel to the diagonals. 
Let it be propoted to conftruft around the cube A B 
G F, fig, 19, confidered as a nucleus, a fecondary foiid, 
in which the laminae of fuperpofition fhall decreafe on 
all fides by Angle rows of cubes but in a direction paral¬ 
lel to the diagonals. Let A B C D, fig. 20, the fuperior 
bafe of tiie nucleus, be divided into eighty-one fquares, 
reprefenting the faces of the fmall cubes of which it is 
compofed. Fig. 21, reprefents the fuperior furface of 
the firft lamina of fuperpofition; which muft be placed 
above A B C D, fig. 20, in fuch a manner that the points 
b‘, c', d', fig, 2i, anfwcr to the points a, b, c, d, fig. 20. 
Bv this difpofition the fquares A a, B b, C c, D d, fig. 20, 
which compofe the four outermoft rows of fquares pa- 
Vgl. V- No. 283. 
OGRAPIIY.. 41? 
rallel to the diagonals AC, B D, remain uncovered. It 
is evident alfo, that the borders Q^V, O N, I L, G F, 
fig. 21, project by one range beyond the borders A B, 
AD, C D, B C, fig. 20, which is neceflary, that the 
nucleus may be enveloped towards thefe edges : for, it 
this were not the cafe, re-entering angles would be form¬ 
ed towards the parts A B, B C, C D, D A, ot the cryftal ; 
which angles appear to be excluded by the laws which 
determine the formation of fimple cryftals ; or, which 
comes to the fame thing, no fuch angles are ever cb- 
ferved in any cryftal. The foiid muft increafe, then, 
in thole parts to which the decrement does noc extend. 
But as this decrement is a'one fufficient to determine 
the form of the fecondary cryftal, we may fet afide all 
the other variations which intervene only in a fubfidiary 
manner, except when it is wifhed, as in the prefent cafe, 
to conftrudt artificially a foiid reprefentation of a cryftal, 
and to exhibit all the details which relate to its ftrufture. 
The fuperior face of the fecond lamina will be A/G' 
L' K', fig. 22. It muft be placed fo that the points a", l", 
c",d“, correfpond to the points a', b',c',d', fig. 21, which 
will leave uncovered a fecond row of cubes at each angle, 
parallel to the diagonals A C and B D. "1 lie foiid ftill 
increafes towards the Tides. The large faces of the la¬ 
minae of fuperpofition, which in fig. 21, were octagons, 
in fig. 22, arrive at that of a fquare; and when they pafs 
that term they decreafe on all fides; fo that the next 
lamina lias for its fuperior face the fquare B' M' L' S', 
fig. 23, lefs by one range in every direction than the pre¬ 
ceding lamina, fig. 22. This fquare muft be placed lo 
that the points e',f, g', h!, fig. 23, correfpond to the 
points e,f,g,h, fig. 22. Figures 24, 25, 26, and 27, re- 
prefent the four laminae which ought to rife fuccefiively 
above the preceding; the manner of placing them being 
pointed out by correfponding letters, as w r as done with 
refpedt to the three firft laminae. The laft lamina z, 
fig. 28, is a Tingle cube, which ought to be placed upon 
the fquare z, fig. 27. 
The laminae of fuperpofition, thus applied upon the 
fide A B C D, fig. 20, evidently produce four faces, which 
correfpond to the points A, B, C, D, and form a pyra¬ 
mid. Thefe faces, having been formed by laminae, which 
began by increafing, and afterwards decreafed, muft be 
quadrilaterals of the figure reprefented in fig. 29, in 
which the inferior angle C is the fame point with the 
angle C of the nucleus, fig. 19 and 20; and the diagonal 
L Q^reprefents L' G' of the lamina A' G' L' K.', fig. 22, 
And as the number of laminae compofing the triangle 
LCLC, fig. 29, in the Cryftallography Plate II. is much 
fmaller than that of the laminae forming the triangle 
ZLQ^it is evident that the latter triangle will have a 
much greater height than the former. 
The furface, then, of the fecondary cryftal thus pro¬ 
duced, muft evidently confift of twenty-four quadrilate¬ 
rals, (for pyramids are raifed on the other five Tides ot 
the primary cube exactly in the fame manner,) difpofed 
three and three around each foiid angle of the nucleus. 
But, in confequence-of the decrement by one range, the 
three quadrilaterals which belong to each foiid angle, as 
C, fig. 18, will be in the fame plane, and will form an 
equilateral triangle Z I N, fig. 30. The twenty-four qua¬ 
drilaterals, then, will produce eight equilateral triangles; 
one of which is reprefented at fig. 31, in fuch a manner 
as to fliew, on a fimple view, the afibrtment of the cubes 
that concur to form it; and the fecondary foiid will be 
a regular odtahedron. Fig. 32, thews this oftahedron in 
which the cubic nucleus is inclofed, fo that each of its 
foiid angles C, D, F, G, &c. correfponds to the center 
of one of the triangles I Z N, I P N, PI S, S I Z, &c. of 
the oftahedron. It may be readily feen, that to extract 
this nucleus, it would be necetrary to divide the octal)e. 
dron in its eight foiid angles, by febtions parallel to the 
oppofite edges. For example, the feftion made in the 
angle Z ought to be parallel to the edges I S, I N, T N, 
T S, and hence will refult a fquare which will itfelf be 
5 O fituatccl 
