MOTION. 
22 g 
resolution of motion, and is of tiie utmost im- 
portance in mechanics. 
Suppose a body A (Plate Miscel. fig. 1 63) 
to be acted upon by another body in the di- 
rection AB, while at the same time it is im- 
pelled by another in the direction AC, then 
it will move in the direction AD ; and if the 
lines AB, AC, are made of lengths propor- 
tionate to the forces, and the lines CD, DB, 
drawn parallel to them, so as to complete the 
parallelogram ABDC, then the line which the 
body A will describe, will be the diagonal 
AD; and the length of this line will represent 
the force with which the body will move. It 
is evident, that if a body is impelled by equal 
forces acting -at right angles to each other, 
that it will move in the diagonal of a square; 
but whatever may be the direction, or degree 
of force by which the two powers act, the 
above method will always give the direction 
and force of the moving body. 
It follows from this, that if we know the 
effect which the joint action of two powers 
has upon a body, and the force and direc- 
tion of one of them, it is easy to find that of 
the other. For, suppose AD to 1 ).? the di- 
rection and force with which the body moves, 
and AB to be one of the impelling forces, 
then, by completing the parallelogram, the 
other power AC is found. 
Instances in nature of motion produced by 
several powers acting at the same time, are 
innumerable. A ship impelled by the wind 
and tide is one well known. A paper kite, 
acted upon by the wind and the string, is an- 
other. 
Motion is said to be accelerated, if its ve- 
locity continually increases ; to be uniformly 
accelerated, if its velocity increases equally 
in equal times. 
Motion is said to be retarded, if its velo- 
city continually decreases: and to be uni- 
formly retarded, if its velocity decreases 
equally in equal times. 
If you suppose a body to be put in motion 
by a single impulse, and. moving uniformly, 
to receive a new impulse in the same direc- 
tion, its velocity will be augmented, and it 
will goon with the augmented velocity. 
If at each instant of its motion it receives a 
new impulse, the velocity will be continually 
increasing; and if this impulse is. always 
equal, the velocity will be uniformly accele- 
rated. 
The regularly increasing velocity with 
which a body fails to the earth, is an instance 
of accelerated motion, which is caused by 
the constant action of gravity. To illustrate 
this, let us suppose the time of descent of a 
falling body to be divided into a number of 
very small equal parts; the impression of 
gravity, in the first small instant, would make 
the body descend with a proportionate and 
uniform velocity ; but in the second instant, 
the body receiving a new impulse from gra- 
vity, in addition to the first, would move with 
twice the velocity as before; in the third in 
stant, it would have three times the velocity, 
and so on. 
To illustrate the doctrine of accelerated 
motion, let us suppose that, in the triangle 
ABC (fig. Miscel. 1 64), A B expresses the time 
which a body takes to fall, and BC the velo- 
city acquired at the end of tile fall. Let AB 
be divided info a number of equal parts, in- 
definite!) small, and from each of these divi- 
sions suppose lines, as DE, drawn parallel to 
BC ; it is evident from what has been said, 
that those lines wii 1 express the velocities of 
the falling body in the several respective 
points of time, each being greater than the 
other, by a certain quantity of increase, which 
follows from the nature of the triangle. Now, 
the spaces described in tiie same time, are in 
proportion to the velocities; and the sum of 
the spaces described in all the small portions 
of time, is equal to the space described from 
the beginning of the fall. But the sum of all 
the lines parallel to BC, taken indefinitely 
near to each other, constitutes the area of the 
triangle. Therefore the space described by 
a failing body, in the time expressed by AB, 
with an uniformly accelerated velocity, of 
which the last degree is expressed by BC, 
will be represented by the area of the tri- 
angle ABC. 
Let us now suppose that gravity ceased to 
act, and that the body moved during another 
portion of time, BF, equal to AB, with the 
acquired velocity represented by BC. As 
the sp.ice moved over is found by multiply- 
ing the velocity by the time, the rectangle 
CF will represent the space moved over 
in this second portion of time, which is twice 
the triangle ABC, and consequently twice 
the space is moved over with the accelerating 
velocity in the same time. 
But if we suppose gravity still to act, be- 
sides the space CF, which it would have 
moved overby its acquired velocity, we must 
add the triangle CGH, for the effect of the 
constant action of gravity; therefore, in this 
second portion of time, the body moves over 
three times the space as in the first. In like 
manner, it mav be easily seen by the figure, 
that in the next portion it would move over 
five times the space; in the next seven times, 
and so on, in arithmetical progression. And 
as the velocities of falling bodies are in pro- 
portion to the spaces run over, it follows, 
that the velocities in each instant increase, as 
the numbers 1, 3, 5, 7, 9, &c. 
It follows from this, that the space run 
over is as the square of the time; that is, in 
twice the time, a body will fall with four times 
the velocity; in thrice the time, with nine 
times the velocity, &c. for, in the first time, 
there was but one space run over; the square 
of 1 is 1 : at the end of the second time there 
are four spaces run over, one in the first, and 
three in the second; the square of 2 is 4; at 
the end of the third time there are nine spaces 
run over ; the square of 3 is 9 : and so on. 
This may be seen in the figure. 
It is found bv experiment, that a body fall- 
ing from a height, moves at the rate of 16 — L 
feel in the first second; and, as has been 
shewn above, acquires a velocity of twice 
that, or 32-t feet in a second. At the end of 
the next second, it will have fallen 64^ feet, 
the space being as the square of the time; the 
square of 2 is 4, and 4 times 16 -L is 64 J. By 
the same rule you may find, that in the third 
second it will tall 144 feet; in the next 256 
feet, and so on. It is to be understood, how- 
ever, that by this velocity is meant what bo- 
dies would acquire, if they were to fall 
through a space where there was no air; for 
its resistance considerably diminishes their 
velocity in falling. 
It has been already shewn, that if two 
forces act uniformly upon a body, they will 
cause it to move in a straight line; but if one 
of the forces is not uniform, but either acce- 
lerating or retarding, the moving hotly will 
describe a curve line. 1 i a ball is projected 
from a cannon, it receives from it an impulse, 
which, if there was no resistance from the air, 
and if it was not acted upon by gravity, 
would cause it to move always in a straight 
line ; but as soon as it leaves the mouth ot the 
cannon, gravity acts upon it, and makes it 
change its direction. Jt then describes a 
curve, called a parabola. This is the founda- 
tion of the theory of projectiles, and the art 
of gunnery ; but it is not now considered to 
be of so much importance as it formerly was, 
as it is found that the resistance of the air, 
and other causes, have so much effect upon 
projected bodies, that they describe curves 
very different from what they ought to do 
according to this theory; and therefore it is 
much less applicable to practice than other- 
wise it would be. 
The force with which a body moves, or 
which it would exert upciranother bod) op- 
posed to it, is always in proportion to its ve- 
locity multiplied by its weight, or quantity of 
matter. This force is called the momentum 
of the body: for if two equal bodies move 
with different velocities, it is evident that their 
forces, or momenta, are as their velocities; 
and if two bodies move with the same velo- 
city, their momenta are as the quantities of 
matter; therefore, in all cases, their momenta 
must be as the products of their quantities of 
matter, and their velocities. This rule is the 
foundation of mechanics. 
In consequence of the vis inertia: of matter,, 
all motion produced by one force only act- 
ing upon a body, must be rectilinear; for it 
must receive some particular direction from 
the power that impressed it, and must retain 
that direction until it is changed by some- 
other power. Whenever, therefore, we see 
a body moving in a curvilinear direction, we 
may be certain that it is acted upon by two- 
forces at least. When one of the two forces 
ceases to act, the body will move again in a 
straight line. Thus a stone in a sling is mov- 
ed round by the hand, while it is pulled to- 
wards the centre of the circle, which it de- 
scribes, by the string: but when the string is- 
let go, the stone flies off in a tangent to the 
circle. 
Every body moved in a circle lias a ten- 
dency to fly off from its centre,, which en- 
deavour of receding is called the centrifugal 
force: and it is opposed to the centripetal 
force ; or that whiclv, by drawing bodies to- 
wards the centre, makes them revolve in a 
curve. These two forces arc called together 
central forces. 
The centre of gravity of a body is that, 
point about which all the parts of a body t]o 
in any situation exactly balance each other. ' 
Hence, if a body is suspended or sup- 
ported by this point, the body will rest in any 
position in which it is put. Also, whatever 
supports that point bears the weight of the 
whole body; and while it is supported, tl;e 
body cannot fall. We may therefore con- 
sider the whole weight of a body as centred 
in this point. 
The common centre of gravity of two or 
more bodies is the point about which they 
would equiponderate, or rest, in any position. 
If the centre of gravity of two bodies,. A and 
B, (Plate Miscel. fig. 165) is connected by tiie- 
