ME C H A N I C S. 
63 3 
Cafe 6 . In the fyflcm of pulleys defiribed at Prep.XV. each 
fucceeding pulley moves twice as fart as the preceding ; 
therefore, W’s velocity : C’s velocity :: 1 : 2 
C’s velocity : B’s velocity :: 1 : 2 
B’s velocity : P’s velocity :: 1 : 2 &c. 
comp. W’s velocity : P’s velocity :: 1 : 2 X - X 3 
X See. :: P : W. 
Cafe 7. In the fyjiem of pulley;, Prop. XVI. if the weight 
he railed 1 inch, the pulley B will defeend 1 inch, and 
tiie pulley A will defeend 2 + 1 inches; in the fame man¬ 
ner, the next pulley will defeend :X J + i + > inches, or 
4 + 2 + 1 inches; &c. therefore P’s velocity : W’s velo¬ 
city :: 1+2 + 4+&C. : 1 :: W : P. 
Cafe 8. Let a body be uniformly raifed along the inclined 
plane B A (fig. 51.) from B to P, by a power ailing pa¬ 
rallel to P V ; upon BP deferibe a femicircle E O P, cut¬ 
ting B C in M ; produce V P to O, join B O, P M, M O. 
Then, fincetbe angles BOP, BMP, in the femicircle, are 
right angles, OP and M P are fpaces uniformly deferibed 
in the fame time, by the power and weight in their re- 
fpeflive directions. Prop. XXII. Alfo, becaufe POM 
/PBM-/PCV, and /OPMc=zPVC, the tri- 
angles P O M, P V C, are fimilar, and OP : MP :: VC : 
PV; or. The power’s velocity : the weight’s velocity :: 
the weight : the power, in the cafe of an equilibrium. 
Cafe 9. In the ifoceles wedge, xC is the only effective 
part of the refiftance DC, fig. 52. Draw VO perpendicu¬ 
lar to C D produced 5 then, if the wedge be moved uni¬ 
formly from C to V, CO is the fpace uniformly deferibed 
by the refilling force, (Prop. XXI.) hence, The power’s 
velocity : the velocity of the relifting force :: CV : CO 
:: Cx :: CF :: the refiftance : the power. 
Cafe 10. In the ferew, whilft the power uniformly de- 
feribes the circumference of a circle, the weight is uni¬ 
formly raifed through the diftance between two contiguous 
threads; therefore, P’s velocity : W’s velocity :: the cir¬ 
cumference of the circle ; the diftance between two 
threads W : P. 
Cafe 11. In any combination of the mechanical powers, let 
P : W, W : R, R : S, See. be the ratios of the power 
and Weight in each cafe, when there is an equilibrium; 
then, P’s velocity : W’s velocity :: W : P 
W’s velocity : R’s velocity :: R : W 
R’s velocity : S’s velocity :: S : R See. 
comp. P’s velocity ; S’s velocity S : P. 
Other Methods of accumulating Power, zJhick do not properly 
come under the Denomination of any of the Mechanical Powers 
already deferibed. 
From the account already given of the fix mechanical 
powers, it is evident, that they can do no more than ac¬ 
cumulate, or, if we may ufe the expreflion, comprefs, any 
degree of velocity into a fmall fpace. The velocity thus 
compreffed, becomes what we call power, and is capable 
of again imprefling the original degree of velocity upon 
a body of an equal or nearly-equal lize to the firlt which 
originally impreffed it; but in every cafe the abfolute 
quantity of motion, or of power, remains the fame, with¬ 
out a poflibility of augmentation or diminution by levers, 
ferews, pulleys, or wedges. It follows therefore, that if 
by any method we can preferve for a certain time a fmall 
quantity of motion, that fmall quantity will, at the end 
of the time fpecified, amount to an aftonifhing power, 
which we could fcarcely at firft have imagined to proceed 
from lo fmall a caufe. Thus, though a man cannot raif'e 
a ton weight from the ground at once, he is eafily capa¬ 
ble of railing 100 pounds at once from the ground, and 
this for a confiderable number of times in fucceflion. It 
is plain, therefore, that in a very fiiort time a man could 
in this manner raif'e the ton weight, if it were divided into 
20 parts, as effeflually as by a lever or other machine.; 
though the fatigue confequent upon Hooping down and 
railing up his body fo often would no doubt make the 
toil much greater. Even by means of a lever, however, 
before 3 man could raile a ton weight one foot from the 
ground, with the trouble of exerting a force equal to io@ 
pounds, he mult have a lever 20 or 21 feet in length, and 
exert a conftant force of 100 pounds, while he goes up 
through a fpace of 20 feet, or pulls down a rope through 
that Ipace. The lever, therefore, only accumulates the 
power exerted in pulling or carrying the weight of 100 
pounds through 20 feet, and difeharges it all upon the 
fpace of one foot; whence it is plain, that any other things 
w hich could do this would raif'e the ton weight as effec¬ 
tually as the lever. 
One method of accumulating a great power is by fuf- 
pending a very heavy body by a chain or ftrong rope of 
confiderable length. This body may be put in motion 
by a very fmall degree of power more than is requifite for 
bending the rope, and will acquire a vibrating motion 
like a pendulum ; by continuing the impulfe as the body 
returns, it will continually acquire greater and greater 
force, the arc through which it moves becoming con¬ 
tinually larger, until at laft it might be made to overcome 
almoft any obltacle : and upon this principle the battering- 
rams of old were conftrufled. Neverthelefs, the power 
of one flroke of this engine never could exceed the accu¬ 
mulated power of the impulfes given to it in order to 
produce that ftroke, or even quite equal it, becaufe the 
ftiffiiefs of the rope, and the refiftance of the air, muft al¬ 
ways take fomething from it. 
Another method of accumulating force is by means of 
a very heavy wheel or cylinder, moveable about an axis. 
A fmall force will be f'ufficient to put this wheel in mo¬ 
tion ; and, if long continued, will accumulate in fuch a 
manner as to produce fuch effebts in railing weights and 
overcoming refinances, as could not by any means be ac- 
compliflied by the application of the original moving 
force. On this fubjeft Mr. Atwood has demonflrated that 
a force of 20 pounds applied for 37 feconds to the cir¬ 
cumference of a cylinder of 10 feet radius, and weighing 
4713 pounds, would, at the diftance of one foot from the 
centre, give an impulfe to a mufket-ball equivalent to 
what it receives from a full charge of gunpowder. The 
fame effebt would be produced in fix minutes and ten fe¬ 
conds by a man turning the cylinder with a winch one 
foot long, in which he ccnftantly exerted a force of 20 
pounds. In this cafe, however, as well as the former, 
there is not any abfolute accumulation of power ; for the 
cylinder has no principle of motion in itfelf, and cannot 
have more than it receives. 
This accumulation of motion, however, in heavy w heels, 
is of great fervice in the conftrublion of machines for va¬ 
rious purpofes, rendering them greatly more powerful and 
eafy to be worked by animals, as well as more regular and 
fteady when fet in motion by water, or any inanimate 
power. Hence the ufe of flies, ballad-wheels, &c. which are 
commonly fuppofed to increafe the power of a machine, 
though in reality they take fomething from it, and abt 
upon a quite different principle. In all machines in which 
flies are ufed, a confiderably greater force muft at firft be 
applied than is neceflary to move the machine without 
it, or the fly muft have been fet in motion f'ome time 
before it is applied to the machine. It is this fuperfluous 
power which is collected by the fly, and ferves as a kind of 
refervoir from which the machine may be fupplied when 
the animal flackens his efforts. This, we muft obferve, 
will always be the cafe with animals; for none are able 
to exert a great power with abfolute conftancy ; forne in¬ 
tervals of reft, even though almoft .imperceptible, are re- 
quilite, othenvife the creature’s ftrength would in a fiiort 
time be entirely exhaufted. When he begins to move ike 
machine, he is vigorous, and exerts a great power, in 
confequence of which he overcomes not only the refift¬ 
ance of the machine itfelf, but communicates a confi¬ 
derable degree of power to the fly. The machine, when 
moving, yields for a time to a i'malLer impulfe ; during 
which time the fly itfelf abis as a moving power, and the 
animal recovers the ftrength he had loft. By degrees, 
however, the motion of the machine decreafes, and the 
animal 
