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pyramids, they are as the spherical surface 
intercepted by them is to the surface of 
the sphere, and therefore cones are as the 
versed sines of half their angles to the 
diameter of the circle : these three sorts of 
infinite quantity are analogous to a line, 
surface, and solid ; and, after the same man- 
ner, cannot he compared, or have no pro- 
portion the one to the other. 
INFINITESIMALS, among mathema- 
ticians, are defined to be infinitely small 
quantities. In the method of infinitesimals, 
the element, by which any quantity in- 
’creases or decreases, is supposed to be in- 
finitely small, and is generally expressed by 
two or more terms, some of which are infi- 
nitely less than the rest, which being neg- 
lected as of no importance, the remaining 
terms form what is' called the difference of 
the proposed quantity. The terms that are 
neglected in this manner, as infinitely less 
than the other terms of the element, are 
the very same which arise in consequence 
of the acceleration, or retardation, of the 
generating motion, during the infinitely 
small time in which the element is generat- 
ed ; so that the remaining terms express 
the elements that would have been pro- 
duced in that time, if the generating motion 
had continued uniform : therefore those 
differences are accurately in the same ratio 
to each other as the generating motions or 
fluxions. And hence, though in this method 
infinitesimal parts of the elements are neg- 
lected, the conclusions are accurately true 
without even an infinitely small error, and 
agree precisely with those that are deduced 
by the method by fluxions. 
In order to render the application of this 
method easy, some analogous principles 
are admitted, as that the infinitely small 
elements of a curve are right lines, or that 
a curve is a polygon of an infinite number 
of sides, which being produced, give the 
tangents of the curve , and by their inclina- 
tion to each other measure the curvature. 
This is as if we should suppose, when the 
base flows uniformly, the ordinate flows 
with a motion which is uniform for every 
infinitely small part of time, and increases 
or decreases by infinitely small differences 
at the end of every such time. 
But however convenient this principle 
may be, it must be applied with caution 
and art on various occasions. It is usual 
therefore, in many cases, to resolve the 
element of the curve into two or more in- 
finitely small right lines; and sometimes,' 
it is necessary, if we would avoid error, to 
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resolve it into an infinite number of such 
right lines, which are infinitesimals of the 
second order. In general, it is a postulatum 
in this method, that we may descend to 
the infinitesimals of any order whatever, as 
we find it necessary ; by which means, any 
error that might arise in the application of 
it may be discovered and corrected by a 
proper use of this method itself. See Mac- 
laurin’s Fluxions. 
INFLAMMATION. See Medicine 
and Surgery. 
Inflammation, in chemistry, is com- 
bustion attended with flame ; under the 
article Combustion, we have referred to 
the spontaneous inflammation of certain 
bodies, in peculiar circumstances, and like- 
wise to the combustion of living individuals 
in the human species. We shall in this 
place mention some of the causes of sponta- 
neous inflammation. The heat produced 
by friction ; the slacking of lime when in 
contact with combustible matter, the fer- 
mentation of hay, dunghills, &c. are well 
known. Many vegetable substances, high- 
ly dryed and heaped together will heat, 
scorch, and at last burst in a flame.’ A 
mixture of linseed, or rape oil, with almost 
any dry vegetable fibre, as hemp, cotton, 
matting, &c. and still more if united to cer- 
tain carbonaceous matters, will in time, if 
in a warm place, burst out into a flame. 
To this circumstance many alarming and 
destructive fires are to be imputed, which 
at the time w r ere supposed to have been 
occasioned by wilful crime. In 1781 a 
large magazine of hemp was destroyed in 
this way, at Cronstadt : and in the sum- 
mer of 1794 an accident of tins sort hap- 
pened at Gainsborough, with a bale of 
yarn accidentally soaked in rape oil, which 
after remaining in the warehouse for several 
days began to smoke, and finally to burst 
out in a most violent flame. A similar 
accident happened at Bombay. A bottle 
of linseed oil had been thrown down in the 
night, the oil had penetrated into a chest 
of coarse cotton cloth, and in the morning 
the cloth was found reduced nearly to a 
cinder, and the wood of the chest com- 
pletely charred in the inside. An experi- 
ment was immediately made to ascertain 
the true cause : a piece of the same cloth 
was dipped in the same sort of oil, and shut 
up in a box, and in three hours it was found 
scorching hot, and on opening the box it 
burst into a flame. Hence the spontaneous 
combustion of wool, or woollen yarn, which 
has sometimes happened when large quail- 
