G32 MECH 
Prop. XIV. In a fyfiem where the fame fring pajfes round 
any number of pulleys, and the parts of it between the pulleys are 
parallel, P : W :: i : the number of f rings at the lower 
block. —Since the parallel parts, or firings at the lower block, 
are in the diretlion in which the weight afts, they ex¬ 
actly fupport the whole weight; alfo, the tenfion in every 
point of thefe firings is the fame, otherwife the fyftem 
would not be at reft, and confequently each of them fuf- 
tains an equal weight; whence it follows that, if there be 
rt firings, each fuftains an nth part of the weight; there¬ 
fore, in fig. 33. P fuftains an nth part of the weight, or 
P : W :: 1 :: 1 : n. 
n 
Cor. If two fyftemsof this kind be combined, in which 
there are m and n firings, refpedtively, at the lower blocks, 
as at fig. 34. P : W :: 1 : mn. 
Prop. XV. In a fyfem where each pulley hangs by a feparate 
fring, and the firings are parallel, P : W :: 1 : that power 
of 2. whofe index is the number of moveable pulleys .—In this 
fyftem, fig. 35. a firing paffes over the fixed pulley A, and 
under the moveable pulley B, and is fixed at E; another 
firing is fixed at B, paffes under the moveable pulley C, 
and is fixed at F; &c. in fuch a manner that the firings 
are parallel. Then, by Prop. XII. when there is an equi¬ 
librium, 
P : the weight at B :: 1 : 2 
The weight at B : the weight at C :: 1 : 2 
The weight at C : the weight at D :: 1 : 2 See. 
Comp. P: VI :: 1 : 2 X 2 X 2 X See. continued to as 
many faftors as there are movable pulleys ; that is, when 
there are n fuch pulleys, P : W :: 1 : 2 n. 
Cor. 1. The power and weight are wholly fuftained at 
A, E, F, G, See. which points l'uftain refpedtively, 2 P, P, 
s P, 4 P, See. 
Cor: z. When the firings are not parallel, P : W :: 
rad. : 2 cos. of the angle which the firing makes with the 
direction in which the weight aids, in each cafe. 
Prop. XVI. In a Jyfiem of n pulleys each hanging by a fe¬ 
parate firing, where the firings are attached to the weight as is 
reprefented in fig. 36. P : W :: 1 : 2"—1.—A firing, 
fixed to the weight at F, paffes over the pulley C, and is 
again fixed to the pulley B ; another firing, fixed at E, 
paffes over the pulley B, and is fixed to the pulley A; See. 
in fuch a manner that the firings are parallel. Then, if P 
be the power, the weight fuftained by the firing DA is P ; 
alfo the prefiure downwards upon A, or the weight which 
the firing A B fuftains, is 2P ; therefore the firing E B 
fuftains 2P; Sec. and the whole weight fuftained is P -f- 
2 P -}-4P -f- Sec. Hence, P ; W :: 1 : 1 -J- 2 -j- 4 + Sec. 
to n terms :: 1 : 2" — 1. 
Cor. 1. Both the power and the weight are fuftained 
at H. 
Cor. 2. When the firings are not parallel, the power in 
each cafe, is to the correfponding prefiure upon the centre 
of the pulley :: rad. : 2 cof. of the angle made by the 
firing with the direction in which the weight adds. Alfo, 
by the refolution of forces, the power in each cafe, or pref- 
fure upon the former pulley, is to the weight it fuftains 
2: rad. : cof. of the angle made by the firing with the 
direction in which the weight adds. 
Of the Inclined Plane. 
An inclined plane is like one-half of a wedge which 
has been cut in two equal parts lengthwife. A weight 
railed, or a refiftance moved, by an inclined plane, moves 
only through a fpace equal to the height of that machine, 
in the time that a power drives it through a fpace equal 
to its whole length. All edged tools, which are cham¬ 
fered (or ground down only on one fide to the edge) are 
inclined planes, as far as the chamfer goes from the edge. 
Prop. XVII. In the inclined plane, the power and weight 
balance each other, when the power is to the weight as the fine of 
the inclination of the plane is to the fine of the angle which the 
line oj the direlhon of the power makes with the perpendicular to 
the plane.—Let a weight be fupported on the inclined plane 
1NICS. 
C. A, fig. 37. by a power afHiig lit any given dire&ion 
P D. Let the whole force, whereby the weight would 
defcend perpendicularly, be reprefented by P B : and, re¬ 
viving PB into two forces, one of which, B D, is per¬ 
pendicular to the plane C A, and the other, P D, is in the 
direction of the power; the force B D is deftroyed by the 
re-affion of the plane, and the force PD will be fuftained 
by an equal power, acting in the direction P D. There¬ 
fore, when there is an equilibrium, the power is to the 
weight as PB to PD ; that is, as the fine of the angle 
P B D, or its equal CAB, to the fine of the angle P D B. 
When P D is in the direction of the plane, this ratio be¬ 
comes that of C D to C B,. or of the height of the plane 
C B, to C A its length. When the direction of the power 
P D is parallel to the bafe of the plane, the ratio of the 
power to the weight becomes that of E D to E B 5 or of 
C B, the height of the plane, to B A, the bafe. When 
the direction of the power coincides with the perpendicu¬ 
lar B D, the ratio of the power to the weight becomes that 
of the fine of a finite angle, to the fine of an angle inde¬ 
finitely diminilhed. From which it appears, that no finite 
power is fufficient to fupport a weight upon an inclined 
plane, if that power adts in a diredtion perpendicular to 
the plane. 
The inclined plane forms a very ufeful part of ma¬ 
chinery in the railing of great weights. It is fuppofed 
that in all the edifices of remote antiquity, where great 
mafles of ftone were employed, as in the pyramids of 
Egypt, and the druidical temples of this country, thefe 
vaft blocks were elevated on inclined planes of earth, or 
of fcaffolding, with the aftiftance alfo of levers and rollers. 
Inclined planes are frequently ufed for drawing boats out 
of one canal into another; and fometimes the local cir- 
cumftances are fuch, that this may be done with great 
convenience, merely to allow a loaded boat to defcend, 
and to turn the axis which raifes an empty one. See the 
article Canal, vol. iii. p. 792 & feq. 
Inclined planes are ufed in many places for railing and 
lowering coal-waggons, fo that one is brought up by the 
force of that which defeends. Inclined planes are alfo 
univerfally employed for facilitating the afeent of heights, 
by men or by animals; they may be either uniform, as 
roads, or the general inclination of the furface may be 
fuperfeded by the formation of feparate fteps or flairs. The 
inclination of the furface may be governed by the propor¬ 
tion of the ftrength of the animal to its weight, the force 
required to fupport any weight on a plane being to the 
whole weight as the height of the plane to its length; and, 
if the plane be a little lefs inclined than the exact equili¬ 
brium would require, the animal will be able to acquire 
a fufficient velocity at firlt to carry it eafily up the afeent 
witli a motion nearly equable. The ftrength of a labourer 
may be advantageoufly employed in afeending a given 
height by a flight of fteps, and placing himfelf on a ftage 
which may raile a weight by its defeent; but it appears 
that the force of other animals is lefs calculated for exer¬ 
tions of this kind. 
Of the Screw. 
The ferew is a mechanical power chiefly ufed in pref- 
fing or fqueezing bodies clofe, and fometimes in railing 
weights. It is a very ftrong machine, though it cannot 
be accounted a Ample one, as no ferevv can be made ule 
of without a lever or winch to aflift in turning it. Tha 
ferew is chiefly diftinguilhed by its fpiral thread, of which 
a tolerable conception may be obtained by cutting a piece 
of paper into the form of an inclined plane, and then 
wrapping it round a cylinder, as in Plate III. fig. 38. 
t . The ferew may be confidered as compofed of the lever and 
the inclined plane ; as will be evident from a more minute 
account of the manner in which it may b& conceived to be 
generated. If an ifofceles triangle BFG turn about the 
axe AZ, fig. 39. there will be generated by that revolution 
two conic frultums united by their greater ends. Con¬ 
ceive now, that, befides the motion of rotation, this tri¬ 
angle has alfo a motion of tranflation in the direction of 
3 the 
