DYNAMICS. 
more horizontal than the latter, or upper 
part, would permit the force to act more 
powerfully on the body D. On the other 
hand, if the line of ascent were convex, as 
at A E B, tlie first part of the ascent being 
steepest, would be slowest ; and the latter, 
in consequence of its regular approach to 
the horizontal, would be proportionately 
rapid. Thus we see, that C is an uniform 
force on the plane A B ; an accelerating 
force on the convex ascent A E B ; and a 
retarding force on the concave, ascent 
A D B : we therefore deduce, that accele- 
rating and retarding motions may be de- 
scribed ' by arcs of which the axes are ea- 
sily ascertained. 
There is a kind of fluctuating, or alter- 
nate force to be found in the action of 
forcing pumps. In these the compression 
of the fluid, and of the intermediate air, 
demands a greater force, in proportion as 
the piston descends : and vice versa, as it 
ascends ; the gradual increase of compres- 
sion, furnished by the return of the fluid 
into the lower part of the cylinder, causes 
a gradual diminution of resistance to the 
upward motion, and consequently acts as 
an accelerating force. 
W'lien bodies at rest fall from heights, the 
times employed are, respectively, as the 
square roots of the cubes of the heights 
from which those bodies fall. We may, 
perhaps, form a ready estimate of this cir- 
cumstance when we recollect that gravity 
gives to all falling bodies an accelerating 
force. In the latitude of London, a heavy 
body falls nearly feet in the first se- 
cond ; which velocity is not only doubled 
in the next second, so as to amount to 
feet, but quadrupled by means of the addi- 
tional force gained by the continued ac- 
tion of gravity ; the third second of time 
will give 96i for its increase, and the fourth 
second will give 128|. In this we suppose 
the bodies not to be impelled downwards 
by any force, (exclusive of gravity) but to 
be in a state of rest, and allowed to descend 
simply by their own weight ; for instance, 
by cutting the line that suspends a weight. 
As bodies gain velocity in falling, so they 
lose velocity when projected upwards. If 
a body be impelled upwards, with the same 
force it had acquired in falling, its velo- 
city upwards would gradually decrease in 
the exact ratio that it increased in descend- 
ing, and with such a power it would reach 
to that height whence it had fallen, but no 
further. Hence, by ascertaining either the 
time of ascent, or of descent, the height 
will be discovered. It is worthy of notice 
that the acquired velocities are as follow : 
2, 4, 6, 8, 10, &c. upon each preceding se- 
cond respectively ; the spaces for each time 
being 1, 3, 6, 7, &c. respectively, and their 
constant differences 2. These laws of 
acceleration were ascertained, by Galileo, 
to prevail equally in the motion of bodies 
along inclined plane^, which may indeed, 
be fully proved by observing the progress 
of a ball as it descends a hill : but this will 
not hold good unless the body be at per- 
fect liberty ; for in cases of interruption, 
each gradation must be considered as the 
incipient motion; vt'ere it otherwise, no 
carriage could descend a long declivity, 
without the certainty of being dashed to 
pieces, nor ascend a hill at the same pace 
throughout. 
The force which accelerates, or retards, 
the motion of a body upon an inclined 
plane, is to the force of gravity, as the 
height of the plane to its length; or as the 
sine of the planes elevation to the radius ; 
and if the diameter of a circle be perpen- 
dicular to the horizon, and chords be drawn 
from either extremity, the time of descent 
down all the chords will be equal ; and each 
will be equal to the time of free descent 
through the vertical diameter. In the cir- 
cle, fig. 7, the line A B is a vertical dia- 
meter, the chords AC, AD, A E, and 
E B, are all of different lengths, and of 
different inclinations from tlie horizon. It 
is evident that, in every instance, the short- 
est chords have the least deviation from 
the horizontal ; consequently they must re- 
tard the progress of a body passing down 
them, much more than those which ap- 
proach to the vertical ; the latter, obvious- 
ly, give a more free passage to the body; 
and as ^hey become more vertical, approach 
nearer to the free descent at A B. 
When a body descends along any num- 
ber of contiguous planes, it will ultimately 
acquire the same velocity as would have 
been acquired by falling perpendicularly 
through the height of the tvhole number of 
planes. The times of descents along simi- 
lar arcs, similarly situated, are as the square 
roots of those arcs ; or as the square 
roots of the radii of their respective cir- 
cles. And if a body, being at rest, is suf- 
fered to fall down a curved surface, whicii 
is perfectly smooth, the velocity acquired 
will be equal to that which would result 
from falling the rame perpendicular height. 
It will require an incipient force to urge 
tlie body back, to the height whence it fell, 
