C U B 
C 1* 
Y 
CUB 
sively above the preceding y (he manner of 
placing them being pointed out by corre- 
sponding letters, as was done with respect to 
the first three laminae. The last lamina z 
(lig. 20) is a single cube, which ought to be 
placed upon the square z (fig. 19). 
I he laminae of superposition, thus applied 
tipon die side ABCD (fig. 12), evidently pro- 
duce four faces, which correspond to the 
points A, B, C, D, and form a pyramid. 
'1 hese faces, having been formed by laminae 
which began by increasing, and afterwards 
decreased, must be quadrilaterals of the fi- 
gure represented in fig. 21 ; in which the in- 
ferior angle C is the same point with the an- 
gle C of the nucleus (figs. 11 and 12), and 
the^d+agonal LQ represents LG of the lami- 
jeti. AGLK (fig. 14). And as the number 
of .laminae composing the triangle LQC (fig. 
21), is much smaller than that of the laminae 
forming the triangle ZLQ, it is evident that 
4 he latter triangle will have a much greater 
height than the former 
! he surface, then, of the secondary crys- 
tal, thus produced, must evidently consist of 
24 quadrilaterals, for pyramids are raised on 
the other 5 sides of the primary cube exactly 
m the same manner, disposed three and three 
around each solid angle of the nucleus. But 
in consequence of the decrement by one 
range, the three quadrilaterals which belong 
to each solid angle, as C (fig 11.), will be in 
the same plane, and will form an equilateral 
triangle ZIN (fig. 22.). The 24 quadrilaterals 
then will produce 8 equilateral triangles ; and 
consequently the secondary crystal will be a 
regular octahedron. This is the structure 
of the octahedral sulphuret of lead, and of 
iuuriat of soda. 
'i he third iaw is occasioned by the abs- 
traction ot double, triple, &c. particles. 
Fig. 23. exhibits an instance of the subtrac- 
tions in question : and t he moleculae which 
compose the range represented by that figure 
are assorted in -such a manner as if of two 
-there were formed only one ; so that we need 
only to conceive the crystal composed of pa- 
rallelopipedons having their liases equal to 
the small rectangles a b c :d, e dfg h i l, 
&c. to reduce this case under that of the com- 
mon decrements on the angles. 
This particular decrement, as well as the 
fourth law, which requires no farther expla- 
nation, is uncommon. Indeed Hauy has 
met with mixed decrements only in some me- 
tallic crystals. 
These different laws of decrement account 
for all the different forms of secondary crys- 
tals. But in onjer to see the vast number of 
secondary forms which may result from them, 
it is necessary to attend to the different mo- 
difications which result from their acting se- 
parately or together. These modifications 
may be reduced to seven : 
1. The decrements take place sometimes 
on all the edges, or all the angles, at once. 
2. Sometimes only on certain edges, or 
certain angles. 
3. Sometimes they are uniform, and consist 
©f one, two, or more rows. 
4. Sometimes they vary from one edge to 
the other, or from one angle to another. 
5. Sometimes decrements on the edges and 
angles take place at the same time. 
6. Sometimes the same edge or angle 
is subjected successively to different laws of 
decrement. 
7. Sometimes the secondary crystal has 
faces parallel to those of the primitive nucleus, 
from the superposition of lamina; not going 
beyond a certain extent. 
Hence Mr. Hauy has divided secondary 
forms into two kinds, namely, simple and 
compound. Simple secondary crystals are 
those which result froqi a single law’ of decre- 
ment, and which entirely conceal the pri- 
mitive nucleus. Compound secondary crys- 
tals are those which result from several laws 
of decrement at once: or from a single law 
which has not reached its limit, and which of 
course has left in the secondary crystal faces 
parallel to those of the primitive nucleus. 
“ If amidst this diversity of laws (observes 
Mr. Iiauy) sometimes united by combinations 
more or less complex, the number of ranges 
subtracted was itself extremely variable; lor 
example, were these decrements by 12,20, 
30, oi*40 ranges, or more, as might absolutely 
be possible ; the multitude of the forms which 
might exist in each kind of mineral would be 
immense, and exceed what could be imagined. 
But the power which effects the subtractions 
seems to have a very limited action. These 
subtractions, for the most part, take p'ace ! v 
one or two ranges of molecules. I have found 
none which exceed four ranges, except in a 
variety of calcareous spar forming part of the 
collection ot C. Gillet Laumont, the structure 
of which depends on a decrement by six 
ranges ; so that if there exist laws which ex- 
ceed the decrements by tour ranges, there 
is reason to believe they rarely take place in 
nature. Yet notwithstanding these narrow 
limits by which the laws of crystallization are 
circumscribed, I have found,, by confining my- 
selftotwo of the simplestlaws,viz. those which 
produce subtractions by one or two ranges, 
that calcareous spar is susceptible of 2044 dif- 
i ferent forms, a number which exceeds more 
than 50 times that of the forms already 
known: and if we admit into the combination 
decrements by three or four ranges, calcula- 
tion will give 8,388,604 possible forms in re 
gard to the same . ubstance. This number 
may still be very much augmented incon- 
sequence of decrements either mixed or in- 
termediary. 
In the crystallization of metals, of salts, 
and other substances, soluble in caloric alone, 
as well as those soluble in water by the aid oi 
caloric, the latter substance is given out in 
abundance (as in the freezing, or, more pr 
perly, crystallization of water) : hence t m 
temperature of large masses of hot saline 
solutions diminishes at last very slowly during 
the increase of the crystals which have begun 
to form in them. 
CRYSTALLINE-HUMOUR. See Op- 
tics. 
CUBfEA, a genus of the decandria mo- 
nogynia class and order. The calyx is tur- 
binate, five-parted, unequal, permanent; 
petals five, unequal y filaments vil'io^e, three 
shorter; germ pedicebed ; legume villose, 
six or seven-seeded. There are two species. 
CUBE, a regular or solid body, consisting of 
six equal sides or faces, which are squares. A 
die is a small cube. 
It is also called a hexahedron, because of its six 
sides, and is the second of the five Platonic or 
regular bodies. 
The cube is supposed to be generated by the 
motion of a square plane, along a line equal and 
perpendicular to one of its sides. 
To describe a rete, or nel, for forming a tube, or 
•with iv Itch it m«y. le covered . — Describe six squares 
38 figure 27, upon card, paper, pasteboard, or 
the like, of the size of the faces of the proposed 
cube ; and cut it half through bv the lines AB* 
CD, EF,. AC, BD ; then fold up the several 
squares till their edges meet, and so form the 
cube, or a covering over one, as in the figure 
annexed. Gee PI. Miscel. fig. 27. 
T° determine the surface and solidity of a cube. , 
Multiply one side by itself, which will give one 
square or face : then this multiplied by 6, the 
number of faces, will give the whole surface. 
Also multiply one side twice by itself, that is, 
cube it, and that will be the solid content. 
Cubes, or Cubic Numbers, are formed by 
multiplying any numbers twice by themselves. 
So the cubes of 1, 2, 3, 4, 5 , 6 , &c. 
_ are 1, 8, 27, 64, 125, 216, &c. 
The third differences of the cubes of the na- 
tural numbers al e all equal to each other, being 
the constant number 6. For, let m\ n \ p\ be 
any three adjacent cubes in the natural scries as 
above, that is, whose roots m, «, p , have the 
common difference 1 ; then, because 
« =*» 4* 1, theref. n ’ = m' -f- 3 m 1 -f- 3m -4- 1, 
p = n 1- 1, ther°f. p ! — -f- 3n 2 -|- -(- 1 ; 
so that the difference between the 1st and 2d, 
and between the 2d and Sd cubes, are 
n ‘ vi ‘ J zz 3m 2 — 3 m -j— 1, 
\ f — »* =: 3/B -j- 3« -f- 1, 
and the difference of these differences, is 
^ • 1,1 ~~ 4" ® • n ** = 3 . n 4~ rn -j- 1 — 
6 . m 4- L the 2d difference. 
In like manner the next 2d difference is 6 . n -f- 1 : 
henc e fhe diff erence oi these two 2d differences 
is 6 . n — m — 6, which is therefore the constant 
3d difference of all the series of cubes. And 
hence that series of cube.; will be formed by ad- 
dition only, viz. adding always the od difference 
6 to find the column or series of 2d differences, 
then inese added always for the 1st differences, 
anu lastly, these always added for the cube 1 ? 
themselves, as below : 
Cubes* 
0 
1 
8 
27 
64 
125 
216 
343 
j’ ^ the 1st differences;. 
3d Difs. 
2d Difs. 
1st Difs. 
6 
6 
1 
6 
12 
7 
6 
18 
19 
G 
24 
37 
6 
30 
61 
6 
36 
91 
6 
42 
127 
6 
48 
169 
Peletarins, among various speculations con- 
cerning square and cub’c numbers, shews that 
the continual sums of the cubic numbers, whose ; 
roots are 1, 2, 3, &c. form the series of situates 
whose roots are 1, 3, 6, 10, 15, 21, &c. 
Thus, 1 — l — ig 
1 4 " 8 — 9 — 3 2 , 
1 4 - 8 4- 27 = 36 ~ 6 2 , 
1 4p 8 4* 27 4* 64 = 100 — log 8-zc. 
Or, in general, T -f- 2 2 T 3 3 -f- 4' & c . to rd 1 
— 14-24-34-4 n \ 2 — i n . „ -j_ j. 
It is also a curious property, that any number, 
and the cube of it, being divided by 6, leave 
the same remainder ; the series of remainders 
being 0, 1 , 2, 3, 4, 5, continually repeated. Or 
th:U the differences between the numbers and 
their., cubes, divided by 6, leave always 0 re- 
maining; and the quotients, with their succes—i 
six e differences, form the several orders of* 
figurate numbers. Thus, 
Num. 
Cubes. 
Difs. 
Quot. 
1 Dif. 
2 Dif. 
1 
1 
0 
0 
0 
0 
2 
8 
6 
1 
1 
1 
3 
27 
24 
4 
3 
2 
4 
64 
60 
10 
6 
3 
5 
125 
120 
20 
10 
4 
6 
216 
210 
35 
15 
5 . 
7 
343 
336 
56 
21 . 
6 
