fit the consequence of the want of an affinity 
'tween these two bodies, but of some other 
ause. 
We may present a body to another with 
fhich it is capable of combining in two dif- 
xent states, either insulated or already com- 
ned with some other body. Thus we may 
[resent lime to nitric acid, either in the state 
pure lime, or combined with carbonic 
id, and consequently in the state of a car- 
inat. 
[ In the first case the affinity is opposed by 
he cohesion, and combination does not take 
Pace unless that force can be overcome, 
ience the reason that combination jseldom 
Ucceeds well, unless some one of the bodies 
o be combined is iiuid, or is assisted bv heat, 
ifhich has the property of diminishing the 
orce of cohesion. Indeed there are not 
hinting numerous instances of solid bodies 
ombimng together without the assistance of 
teat ; but they are always bodies which have 
he property of becoming liquid in the act of 
ombination. Thus common salt and snow, 
nuriat of lime and snow, &c. combine ra- 
•idly when mixed, and are converted into 
tquids. 
It is to the force of cohesion that the diffi- 
ulty of dissolving the diamond, the sapphire, 
nd many other natural bodies, is to be as- 
ribed, though composed of ingredients very 
eadily acted on by solvents when their co- 
psive force is sufficiently diminished. If 
.tire alumina is formed into a paste, and 
bated sufficiently, it becomes so hard that 
0 acid can act upon it ; yet its nature is not 
1 the least changed. By proper trituration 
may be again rendered soluble ; and when 
Precipitated from this new solution, it has 
ecovered all its original properties. The 
fleet of the fire then was merely to increase 
he cohesion, by separating all the water, and 
Mowing the particles to approach nearer 
kch other. 
I Even when the cohesive force of the par- 
ades to be combined is not very great, it 
by be still sufficient to prevent combination 
l|om taking place, provided the other body 
an only approach it in a very small mass, 
lence the reason that carbonic "acid gas, and 
ther elastic fluids, have scarcely any action 
n the greater number of bodies, though 
hey combine with them readily when the 
nee of cohesion presents no opposition. 
|hus the oxygen of the atmosphere does not 
ombine with sulphur in its natural stafe, 
hough it unites with it readily when the sul- 
jhur is combined with hydrogen and potash, 
odies which diminish its cohesion very con- 
jjderably, or when it is converted into its in- 
Igrant particles by the action of heat. 
!’ In the second case, or when the body pre- 
sided to be combined with another is already 
i combination with some other body, it does 
ot altogether leave the old body and coni- 
ine with the new, but it is divided between 
iem in proportion to the mass and the affi- 
|ty of these bodies. Thus, when lime, al- 
?ady in combination with phosphoric acid, 
presented to sulphuric acid, it does not al- 
jgether leave the phosphoric to combine 
hh the sulphuric acid ; but it divides itself 
etween these two acids, part combining 
Ith the one and part with the other, accord- 
jg to the respective quantities of each of 
iese acids, and the strength of their affinity 
•x the lime. Chemists had formerly, hi a 
COMBINATIONS. 
great measure, overlooked the modification 
produced on the action of heterogeneous 
bodies on each other by the different propor- 
tions ot each; supposing that in all cases a 
substance A, which has a stronger affinity for 
C than B lias, is capable of taking C alto- 
gether from 13, provided it be added in suf- 
ficient quantity, how great soever the propor- 
tion of 13 exceeds that of A. Several excep- 
tions, indeed, have been pointed out to this 
general law. Thus Cavendish ascertained, 
that lime-water is incapable of depriving air 
completely of carbonic acid. But Berthollet 
has demonstrated, that this pretended law 
never holds ; that no substance whatever is 
capable of depriving another of the whole of 
a third, with which it is combined, except 
in particular circumstances, however strong 
its affinity for that third may be, and in how 
great a proportion soever it be added. Thus 
no proportion of lime whatever is capable of 
depriving the carbonat of potash of the whole 
of its acid. Neither does sulphuric acid de- 
compose phosphat of lime completely, nor 
ammonia sulphat of alumina, nor potash sul- 
phat of magnesia. 
In short, it may be considered as a general 
law in chemistry, that the smaller the pro- 
portion of a body in combination with a 
given quantity of another body is, with the 
greater energy is it retained ; so that at last 
the force of its affinity becomes stronger than 
any direct force that can be applied to sepa- 
rate it. Hence the impossibility of depriving 
sulphuric acid and several other bodies com- 
pletely of water. 
Berthollet has shown also, that every 
body, how weak soever its affinity for an- 
other may be, is capable of abstracting part 
of that other from a third, how strong soever 
the affinity of that third is, provided it be 
applied in sufficient quantity. Thus potash 
is capable of abstracting part of the acid from 
sulphat of barytes, from oxat of lime, phos- 
phat of lime, and carbonat of lime. Soda 
and lime abstract part of the acid from sul- 
phat of potash, and nitric acid abstracts part 
of the base from oxalat of lime. Hence it 
follows that substances are capable of decom- 
posing each other reciprocally, provided 
they be added respectively in the proper 
quantity. Indeed this was known formerly 
to be the case, though it had not been con- 
sidered as a general law till Berthollet drew 
the attention of chemists to it. Sulphuric 
acid decomposes nitrat of potash altogether 
by the assistance of heat. The nitric acid is 
driven off, and there remains behind sulphat 
of potash with an excess of acid. On the 
other hand, if nitric acid is poured into 
sulphat of potash in sufficient quantity, it 
takes a part ot the base from the sulphuric 
acid. In the same manner phosphoric acid 
decomposes inuriat of lead, and muriatic acid 
on the other hand decomposes phosphat of 
lead. 
COMBINATIONS, in law are persons as- 
sembled together unlawfully, with an intent to 
do unlawful acts ; and these offences are pu- 
nishable before such acts are carried into effect, 
in order to prevent the consequence of com- 
binations and conspiracies. 9- Hep. 57-. 
Combinations denote the alternations or 
variations of any number of quantifies, let- 
ters, sounds, or the like, in ail possible ways. 
lather Mersenne gives the combinations 
of all the notes and sounds in music,, as far as 
3 97 
64 ; the sum of which amounts to a number 
expressed by 90 places of figures. And the 
number of possible combinations of the 24 
letters of the alphabet, taken first two by 
two, then three bv three, and so on, accord- 
ing to Prestet’s caclulation, amounts to 
139 1 724288887252999425 12849340.2200. 
Father Truchet, in Mem. de PAcad. shews 
thaf two square pieces, each divided diago- 
nally into two colours, may be arranged and 
combined 64 different ways, so as to form so 
many different kinds of chequer-work : a 
thing that may be of use to masons, pa- 
viours, &c. 
Doctrine of Combinations. 
I. Having given any number of thing*, with the 
number in each combination ; to find the number of com- 
binations. 
1. When only two are combined together. - 
One thing admits of no combination. 
Two, a and b, admit of one only, viz. ab. 
1 hree, a, b, c, admit of three, viz. ab, ac, be. 
Four admit of six, viz. ab, ac, ad, be, bd, cd. 
Five admit of ten, viz. ab, ac, ad, ac , be, bd, bey 
cd, ce, de. 
Whence it appears that the numbers of com- 
■ binations, of two-and-two only, proceed accord- 
j ing to the triangular numbers 1, 3, 6, 10, 15, 21, 
See. which are produced by the continual addi- 
| tion of the ordinal series 0, 1, 2, 3, 4, 5, See. And 
if n be the number of things, then the general 
| formula for expressing the sum of all their com- 
binations by twos, will be 1 
I .00 
2.1 
2~ 
1 .2 
I hus, if n — 2 ; this becomes 
T , „ . . 3.2 
If « = 3 ; it is 
2 
ir „ • • 4.3 
II n ~ 4 : it is 
2 
6, Ac. 
2. When three are combined together ; then 
Three things admit of one order, abc. 
Four admit of 4 ; viz. abc, abd , acd, bed. 
Five admit of 10 ; viz. abc, abd, abc, acd , ace , ade, 
bed, bee, bde, cde. And so on according to the first 
pyramidal numbers 1, 4, 10, 20, &c. which are 
formed by the continual addition of the former, 
or triangular numbers 1, 3, 6, 10, &c. And the 
general formula for any number/; of combina- 
• , , , . n . n — 1 . n 2 
tions, taken by threes, is 
So. 
3 . 2 
. if n ~ 3 ; it is — — . 
1 . 2 
If « ~ 4 ; it is - 
1 . 2 . 3 . 
if _ - • 5.4.3 
II n~5 \ it is — y = 10, &c. 
Proceeding thus, it is found that a general* 
formula for any number n of. things, combined 
by m at each time, is s — — 1 •” ” ~ 3 
1 • 2. 3; 4.&c 
continued to m factors, or terms, or till the last 
factor in the denominator be m . 
So, m 6 things, combined by 4s, the number 
6 . 5 . 4 . 3 
= 15. 
of combinations is 
1 . 2 . 3 . 4 
3 - adding all these series together, their 
sum will be the whole, number of possible com- 
binations of n things combined both by twos, by 
thiees, by lours, &c. And as the said series are 
evidently the co-efficients of the power « of a 
binomial; wanting only the first two, 1 and«; 
therefore the. said sum, or whole number of all 
such combinations, will be 
1 + F> 1 - n 1, or 2» — i7 — 1. Thus if the 
number of things be 5; then 2' — 5 — 1 — 
32 — 6 = 26,. 
