MECHANIC S. 
C 5 but, by tbe fuppofition, AX AC-f By BC 
DxDC = ExEC+FxFCj therefore r + s + t x 
X C p -^q X YC, and the weights r, s, t, placed at 
X, balance the weights p, q, placed at Y; alfo A, B, D, 
balance the former weights, and E, F, the latter; confe- 
quently, A, B, D, will balance E and F. This we lhall 
refer to in the inveftigation of the centre of gravity. 
Cor. i. If the weights do not act in parallel directions, 
inftead of the diftances we mult fubftitute the perpendicu¬ 
lars, drawn from the centre of motion, upon the directions. 
Cor. 2. In Cor. 7. Prop. VIII. the lever is fuppoled to 
be without weight, or the arms AC, C D, to balance each 
other. In the formation of the (teelyard, fig. 28. the 
longer arm C B is heavier than C A, and allowance mull 
be made for this excefs. Let the moveable weight, P, 
when placed at E, keep the lever at relt; then, when W 
and P are fufpended upon the lever, and the whole re¬ 
mains at reft, W fuftains P, and alfo a weight which 
would fupport P when placed at E ; therefore W x A C 
= PXDC + PXEC = PX DE; and, fince A C and P 
are invariable, W CC E D ; the graduation mult therefore 
begin from E; and, if P, when placed at F, fupport a 
weight of one pound at A, take F G, G D, &c. equal to 
each other, and to E F j and, when P is placed at G, it will 
fupport two pounds; when at D, it will fupport three 
pounds, &c. This is the common lteelyard. 
Of the Wheel and Axle. 
The wheel and axle, or axis in peritrockio, has a near ana¬ 
logy to the lever; the power is applied at the circumfer¬ 
ence of the wheel, and the weight is raifed by a rope that 
is gathered up (while the machine turns round) on the 
axis. The power may be conceived as applied at the ex¬ 
tremity of the arm of a lever equal to the radius of the 
wheel, and the weight as applied at the extremity of a lever 
equal to the radius of the axle; only thofe arms do not meet 
at one centre of motion, as in the lever, but in place of 
this centre we have an axis of motion, viz. the axis of the 
whole engine. But, as this can produce no difference, 
it follows that the power and weight are in equilibrio 
when they are to each other inverfely as the diftances of 
their diredtions from the axis, or axle, of the engine ; or 
when the power is to the weight as the radius of the roller 
is to the radius of the wheel, the power being fuppofed 
to adt in a perpendicular to this radius; but, if the power 
adt obliquely to this radius, fubftitute a perpendicular 
from the axis on the direction of the power, in the place 
of the radius. Thus, if A B D E, fig, 29. reprefent the 
cylindrical roller, H P N the wheel, L M the axis, or right 
line upon which the whole engine turns, Q the point of 
the furface of the roller where the weight W is applied, 
P the point where the power is applied, K Q the radius of 
the roller, C P the radius of the w heel ; then, if the power 
P sft with a diredtion perpendicular to C P, the power 
and weight will fuitain each other when P is to W as K Q 
to C P or C PI ; but, if the power adt in any other direc¬ 
tion PR, let C R be perpendicular from C, the centre of 
the wheel, on that direction ; then P and W will fuitain 
each other when P is to WasKQtoCR; becaufe, in 
this cafe, a power P has the fame effedt as if it was ap¬ 
plied at the point R of its direction, adting in a right line 
perpendicular to CR. See Axis, vol. ii. p. 585. 
Of the Pulley. 
A pulley is a fmall wheel, or rundle, having a channel 
round it, and turning on an axis, ferving, by means of a 
rope which Aides in its channel, for the railing of weights. 
The Latins call it trochlea ; and the leamen, when fitted 
with a rope, a tackle. An alfemblage of feveraf pulley's is 
called a lyltem of pulleys, or polyfpafion ; fome of which 
are in a block or cafe, which is fixed ; and others in a 
block which is moveable, and riles with the weight. The 
moveable w heel or rundle, is called the Jheave , or Jhiver ; 
the axis on which it turns, the gudgeon ; and the fixed 
piece of wood or iron, into which it is put, the block. 
631 
One of the ufes of the pulley ts when the vertical direc¬ 
tion of a power is to be changed into an horizontal one, 
or an afcending diredtion into a defcending one ; and the 
contrary. This is found a good provifion for the fafety 
of the workmen employed in drawing with the pulley. 
The change of diredlion by the means of a pulley has this 
farther advantage; that, if any power can exert more force 
in one diredtion than another, w'e 3re here able to employ 
it in its greateft force. Thus, e. g. a horfe cannot draw 
in a vertical diredtion, but draws with all its advantage in 
a horizontal one. By changing the vertical draught, 
therefore, into a horizontal one, a horfe becomes qualified 
to raife a weight. 
But the grand ufe of the pulley is, where feveral of them 
are combined ; thus forming what Vitruvius, and others 
after him, call polyfpajla ; the advantages of which are, 
that the machine takes up but little room, is eafily removed, 
and railesa very great weight with a moderate force. But 
pulleys have a great deal of fridtion, on the following ac¬ 
counts: 1. Becaufe the diameters of their axes bear a very' 
confiderable proportion to their own diameters. 2. Be- 
caufe in working they are apt to rub againft one another, 
or againft the tides of the block. 3. Becaufe of the ftiff- 
nefs of the rope that goes over and under them. 
Ttie pulley is faid to be fixed or moveable , according 
the centre of motion is fixed or moveable. 
Prop. XI. In the Jingle fixed pulley, there is an equilibrium, 
when the power and weight are equal. —Let a power and weight 
P V/, fig. 30. equal to each other, adt by means of a per¬ 
fectly flexible rope P D W, which pafles over the fixed 
pulley A D B ; then, whatever force is exerted at D in tire 
diredtion DAP, by the power, an equal force is exerted 
by the weight in the direction D B W 5 thefe forces will 
therefore keep each other in equilibrio. 
Cor. 1. Converfely, when there is an equilibrium, the 
power and weight are equal. 
Cor. 2. The propofition is true in whatever diredlion the 
power is applied; the only'alteration made, by changing 
its diredtion, is in the preflure upon the centre of motion. 
Prop. XII. In thefingle moveable pulley, whofe firings are pa¬ 
rallel, the power is to the weight as 1 to 2. In this and the 
following propofitions, the power and weight are fup¬ 
pofed to be in equilibrio. 
A firing fixed at E, fig. 31. pafles under the moveable 
pulley A, and over the fixed pulley B ; the weight is an¬ 
nexed to the centre of the pulley A, and the power is ap¬ 
plied at P. Then, fince the firings E A, BA, are in the 
diredtion in which the weight acts, they exactly fuitain 
it; and they are equally ftretched in every point, there¬ 
fore they fuftain it equally between them ; or each fuf¬ 
tains half the weight. Alfo, whatever weight A B fuf¬ 
tains, P fuftains; therefore, P : W :: j : 2. 
Prop. XIII. In general, in the Jingle moveable pulley, tha 
power is to the. weight, as radius to twice the cofine of the angle 
which either Jlring makes with the dir.edion in which the weighs 
ads. —Let A VV, fig. 32. be the diredtion in which the 
weight adts ; produce B D till it meets A W in C ; .from 
A draw A D at right angles to A C, meeting B-C in D ; 
then, ii C D be taken to reprefent the power at P, or ths 
power which adts in the direction D B, C A will reprs- 
fent that part of it which is efredtive in fultaining the 
weight, and AD will be counteradted by an equal and op- 
polite force, arifing from the tenlion of the firing. C E. 
Alfo the two firings are equally effedtive in fultaining tbe 
weight; therefore 2 AC will reprefent the whole weight 
fuftained ; confequently, P : W :: CD s 2 AC :: rad. 
: 2 cos. D C A. 
Cor. 1. If the figure be inverted, and E and B be con- 
fidered as a power and weight which fuftain each other 
upon the fixed pulley A, W is the preflure upon the cen¬ 
tre of motion ; confequently. The power : the preflure :: 
radius : 2Cos. DC A, 
Cor. 2. When the firings are parallel, the angle D C A 
vanilhes, and its cofine becomes the radius ; in this cafe, 
Ths power ; the preflure ;; 1 ; 2, 
Prop, 
