the circumference of the wheel, while
that of the weight w goes round its
axle, or another smaller wheel, attach-
ed to the larger, and having the same
axis or centre c. So that BA is a lever
moveable about the point c, the power
P acting always at the distance BC, and
the weight w at the distance GA; there-
fore P w CA; CB.
P:

B

180. Corol. 1. If the wheel be put in motion; then, the spaces moved being as the circumferences, or as the radii, the velocity of w will be to the velocity of P, as CA to CB; that is, the weight is moved as much slower, as it is heavier than the power; so that what is gained in power, is lost in time. And this is the universal property of all machines and engines.

181. Corol. 2. If the power do not act at right angles to the radius CB, but obliquely; draw CD perpendicular to the direction of the power; then, by the nature of the lever,

182. To this power be-
long all turning or wheel
machines, of different radii.
Thus, in the roller turning E
on the axis or spindle cE,
by the handle CBD; the
power applied at в is to
the weight w on the roller
as the radius of the roller
is to the radius cв of the
handle.

183. And the same for all cranes, capstans, windlasses, and such like; the power being to the weight, always as the radius or lever at which the weight acts, to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and auger for boring holes.

184. But all this, however, is on supposition that the ropes or cords, sustaining the weights, are of no sensible thickness. For, if the thickness be considerable, or if there be several folds of them, over one another, on the roller or barrel; then we must measure to the middle of the outermost rope,

for

the

1

the radius of the roller; or, to the radius of the roller we
must add half the thickness of the chord, when there is but
one fold.

185. The wheel-and-axle has a great advantage over the
simple lever, in point of convenience. For a weight can be
raised but a little way by the lever; whereas, by the continual
turning of the wheel and roller, the weight may be raised to
any height, or from any depth.

186. By increasing the number of wheels too, the power may be multiplied to any extent, making always the less wheels to turn greater ones, as far as we please; and this is commonly called Tooth and Pinion Work, the teeth of one circumference working in the rounds or pinions of another, to turn the wheel. And then, in case of an equilibrium, the power is to the weight, as the continual product of the radii of all the axles, to that of all the wheels. So, if the power

ין ་

turn the wheel e, and this turn the small wheel or axle R,
and this turn the wheel s, and this turn the axle r, and this
turn the wheel v, and this turn the axle x, which raises the
DE
DE. FG: AC. BD. EF.
weight w; then P: W:: CB
And in
the same proportion is the velocity of w slower than that of P.
Thus, if each wheel be to its axle, as 10 to 1; then P: W::
13 : 103 or as 1 to 1000. So that a power of one pound will
balance a weight of 1000 pounds; but then, when put in mo-
tion, the power will move 1000 times faster than the weight.

OF THE PULLEY.

187. A PULLEY is a small wheel, commonly made of wood or brass, which turns about an iron axis passing through the centre, and fixed in a block, by means of a cord passed round its circumference, which serves to draw up any weight. The pulley is either single, or combined together, to increase the power. It is also either fixed or moveable, according as it is fixed to one place, or moves up and down with the weight and power.

PROPOSITION XXXIII.

188. If a Power sustain a Weight by means of a Fixed Pulley;
the Power and Weight are Equal.

FOR, through the centre c of the pulley
draw the horizontal diameter AB: then will
AB represent a lever of the first kind, its
prop being the fixed centre c; from which
the points A and B, where the power and
weight act, being equally distant, the pow-
er P is consequently equal to the weight

W.

189. Corol. Hence, if the pulley be put
in motion, the power p will descend as fast
as the weight w ascends. So that the
power is not increased by the use of the
fixed pulley, even though the rope go over
several of them. It is, however, of great service in the
raising of weights, beth by changing the direction of the force,
for the convenience of acting, and by eňabling a person to
raise a weight to any height without moving from his place,
and also by permitting a great many persons at once to exert
their force on the rope at r, which they could not do to the
weight itself; as is evident in raising the hammer or weight
of a pile-driver, as well as on many other occasions.

PROPOSITION XXXIV.

190. If a Power sustain a Weight by means of One Moveable
Pulley; the Power is but Half the Weight.

FOR, here AB may be considered as a lever of the second

kind,

1

kind, the power acting at a,
the weight at c, and the prop
or fixed point at в ; and be-
cause P w :: CB AB, and P
CBAB, therefore r=1w,
or w=2p.

191. Corol. 1. Hence it is evident, that when the pulley is put in motion, the velocity of the power will be double the velocity of the weight, as the point P

B

moves twice as fast as the point c and weight w rises. It is also evident, that the fixed pulley r makes no difference in the power P, but is only used to change the direction of it, from upwards to downwards.

192. Corol. 2. Hence we may estimate the effect of a combination of any number of fixed and moveable pulleys; by which we shall find that every cord going over a moveable pulley always adds 2 to the powers; since each moveable pulley's rope bears an equal share of the weight; while each rope that is fixed to a pulley, only increases the power by unity.

OF THE INCLINED PLANE.

193. THE INCLINED PLANE, is a plane inclined to the horizon, or making an angle with it. It is often reckoned one of the simple mechanic powers; and the double inclined plane makes the wedge. It is employed to advantage in raising heavy bodies in certain situations, diminishing the powers that ustain them by laying them on the inclined planes.

PROPOSITION

4

PROPOSITION XXXV.

194. The Power gained by the Inclined Plane, is in Proportion as the Length of the Plane is to its Height. That is, when a Weight w is sustained on an Inclined Plane; Bc, by a Power P acting in the Direction Dw, parallel to the Plane; then the Weight w, is in proportion to the Power P, as the Length of the Plane is to its Height; that is, w: P:: BC: AB.

For, draw AE perp. to the plane BC, or to Dw. Then we are to consider that the body w is sustained by three forces, viz. 1st, its own weight or the

by

force of gravity, acting perp. to Ac, or parallel to BA; 2d,
the power P, acting in the direction wD, parallel to BC, or BE;
and 3dly, by the re-action of the plane, perp. to its face, or
parallel to the line EA. But when a body is kept in equili-
brio by the action "of three forces, it has been proved, that
the intensities of these forces are proportional to the sides
of the triangle ABE made by lines drawn in the directions of
their actions; therefore those forces are to one another as
the three lines

the weight of the body w is as the line
the power r is as the line

AB, BE, AE; that is,

AB,

BE,

and the pressure on the plane as the line AE.

But the two triangles ABE, ABC are équiangular, and have therefore their like sides proportional; that is,

the three lines

AB, BE, AE,

are to each other respectively as the three BC, ab, ac,
or also as the three

BC, AE, CE,

which therefore are as the three forces w, p, p,
where Ρ denotes the pressure on the plane. That is, w: P : :
BC AB, or the weight is to the power, as the length of the
plane is to its height.

See more on the Inclined Plane, at p. 144, &c.

195. Scholium. The Inclined plane comes into use in some situations in which the other mechanical powers cannot be conveniently applied, or in combination with them. As, in sliding heavy weights either up or down a plank or other plane laid sloping or letting large casks down into a cellar, or drawing them out of it. Also, in removing earth from a lower situation to a higher by means of wheel-barrows, or otherwise, as in making fortifications, &c.; inclined planes, made of boards, laid aslope, serve for the barrows to run upon.

Of