INC 
“ method as calycina,” consisting of plants 
whose flowers want either the calyx or 
petals. 
INCORPORATION, power of. To the 
erection of any corporation the King’s con- 
sent is necessary, either impliedly or ex- 
pressly given : the King’s implied consent 
is to be found in corporations which exist 
by force of the common law, to which our 
former kings are supposed to have given 
their concurrence; of this sort are all bishops, 
parsons, vicars, churchwardens, and some 
others, who by common law have ever been 
held to have been corporations by virtue of 
their office. Another method of implied 
consent is with regard to all corporations by 
prescription ; such as the city of London, 
and many others, which have existed as 
corporations for time immemorial ; for 
though the members thereof can show no 
legal charter of incorporation, yet in cases 
of such high antiquity the law' presumes 
there once was one, and that by variety of 
accidents, which a length of time may pro- 
duce, the charter is lost or destroyed. The 
methods by which the King’s consent is ex- 
pressly given are either by act of parliament 
or charter ; but the immediate creative act is 
usually performed by the King alone, in 
virtue of his royal prerogative. See further 
Joint Stock. 
INCREMENT, is the small increase of 
a variable quantity. Sir Isaac Newton calls 
these increases “ moments,” and observes, 
that they are proportional to the velocity 
or rate of increase of the flowing or variable 
quantities, in an indefinitely small time. 
The notation of increment is different by 
different authors. The method of incre- 
ments, is a branch of analytics, in which a 
calculus is founded on the properties of 
successive values of variable quantities, and 
their differences, or increments. It is nearly 
allied to the doctrine of fluxions, and, in 
truth, arises out of it. Of the latter the 
great Newton was the inventor ; of the for- 
mer w'e have different treatises by Dr. Tay- 
lor, Mr. Emerson, and others. We shall 
give Mr. Emerson’s observations on the 
distinction betw'een the method of incre- 
ments and fluxions. “ From the method of 
increments,” he says, “ the principal foun- 
dation of the method of fluxions may be 
easily derived; for, as in the method of in- 
crements, the increment may be of any 
magnitude, so in the method of fluxions, it 
must be supposed infinitely small ; whence 
all preceding and successive values of the 
variable quantity will be equal, from which 
' v INC 
equality the rules for performing the princi- 
pal operations of fluxions are immediately 
deduced. Tiiat I may give the reader,” con- 
tinues he, “ a more perfect idea of the nature 
of this method : suppose the abscissa of a 
curve be divided into any number of equal 
parts, each part of Which is called the in- 
crement of the abscissa, and imagine so 
many parallelograms to be erected thereon, 
either circumscribing the curvilineal figure, 
or inscribed in it; then the finding the sum 
of all these parallelograms is the business of 
the method of increments. But if the parts 
of the abscissa be taken infinitely small, 
then these parallelograms degenerate into 
the curve ; and then it is the business of the 
method of fluxions to find the sum of all, or- 
the area of the curve. So that the method 
of increments finds the sum of any number 
of finite quantities ; and the method of 
fluxions the sum of any infinite number of 
infinitely small ones : and this is the essential 
difference between these two methods." 
Again : “ There is such a near relation be- 
tween the method of fluxions and that of 
increments, that many of the rules for the 
one, with little variation, serve also for the 
other. And here, as in the method of fluxions, 
some questions may be solved, and the in- 
tegrals found, in finite terms ; whilst in 
others We are forced to have recourse to in- 
finite series for a solution. And the like 
difficulties will occur in the method of in- 
crements, as usually happen in fluxions. 
For whilst some fluxionary quantities have 
no fluents, but what are expressed by se- 
ries, so some increments have no integrals 
but what infinite series afford ; which will 
often, as in fluxions, diverge and become 
useless.” By means of the method of in- 
crements, many curious and useful problems 
are easily resolved, which scarcely admit of 
a solution in any other way. As, suppose 
several series of quantities be given, whose 
terms are all formed according to some 
certain law which is given ; the method of 
increments will find out a general series, 
which comprehends all particular cases, and 
from which all of that kind may be found. 
The method of increments is also of great 
use in finding any term of a series pro- 
posed : for the law being given by which 
the terms are formed, by means of this ge- 
neral law the method of increments will 
help us to this term, either expressed in 
finite quantities, or by an infinite series. 
Another use of the method of increments is 
to find the sum of series, which it will often 
do in finite terms. And when the sum of a 
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