PCLLEY, Doctrine of the. 1. If the equal weights w and P hang by the cord a a upon the pulley A, whose block, b, is fixed to the beam n 1, they will counterpoise each other, just in the same manner as if the cord were cut in the middle, and its two ends hung upon the hooks fixed in the pulley, at A and A, equally distant from its centre.

Hence, a single pulley, if the lines of direction of the power and the weight be taugents to the periphery, neither assists nor impedes the power, but only changes its direction.

The use of the pulley, therefore, is when the vertical direction of a power is to be changed into a horizontal one ; or an ascending direction into a descending one ; and on the contrary.

This is found a good provision for the safety of the work men employed in drawing with the pulley.

The change of direction by the means of a pulley has this farther advantage ; that if any power can exert more force in one direction than in another, we are here able to employ it in its greatest force. For instance, a horse cannot draw in a vertical direction, but draws with all its advantage in a horizontal one. By changing the vertical draught, therefore, into a horizontal one, a horse becomes qualified to raise a weight.

But the grand use of the pulley is, where several of them are combined ; thus forming what Vitruvius, and others after him, call polygpasta ; the advantages of which are, that the machine takes up but little room, is easily removed, and raises a very great weight with a moderate 2. If a weight, w hangs at the lower end of the moveable block, 27, of the pulley n, and the cord, G F, goes under the pulley, it is plain that the half, G, of the cord, bears one half of the weight, w, and the half, F, the other ; for they bear the whole between them. Therefore, whatever holds the upper end of either rope, sustains one-half of the weight ; and if the cord at F be drawn up so as to raise the pulley n to c, the cord will then be extended to its whole length, except that part which goes under the pulley ; and consequently, the power that draws the cord will have moved twice as fin• as the pulley n, with its weight w, rises ; on which account, a power whose intensity is equal to one half of the weight, will be able to support it, because, if the power moves (hy means of a small addition) its velocity will be double the velocity of the weight ; as may be seen by putting the cord over the fixed pulley c (which only changes the direction of the power, without giving any advantage to it) and hanging on the weight P, which is equal only to one-half of the weight w ; in which case there will be an equilibrium, and a little addition to P will cause it to descend, and raise w through a space equal to one-half of that through which P descends. hence, the advantage gained will be always equal

to twice the number of pulleys in the moveable or undermost block. So that, when the upper or fixed block, u, contains two pulleys, which only turn on their axes, and the lower or moveable block, u, contains two pulleys, which not only turn upon their axes, hut also rise with the block and weight, the advantage gained by this is as four to the working power. Thus, if one end of the rope, x Mt o o, be fixed to a hook at I, and the rope passes over the pulleys x and R, and under the pulleys a and P, and has a weight, T, of one pound, hung to its other end, at T ; this weight will balance and support a weight, w, of four pounds, hanging by a hook at the moveable block u, allowing the said block as a part of the weight. And if as much more power be added, as is sufficient to overcome the friction of the pulleys, the power will descend with four times as much velocity as the weight rises, and, consequently, through four times as much space. The two pulleys in the fixed block, x, and the two in the moveable block, r, are in the same case with those last mentioned ; and those in the lower block give the same advantage to the power.

It is necessary to observe, that if the lower pulleys do not rise all together in one block with the weight, as in the eases just recited, but act upon each other, and the weight is only fastened to the lowest of them, the force of the power is very much increased, each pulley doubling it. Thus, a power, whose intensity is equal to 8 lb. applied at a, will, by means of the lower pulley, A, sustain 10 lb. ; a power equal to 4 lb. at b, will, by means of a lower pulley, a, sustain the power of 8 lb. acting at a; a third power, equal to 3 lb. at c, will, by means of the pulley c, sustain the power of 41b. at b; a fourth power of 1 lb. at d, will, by means of the pulley n, sustain the power of lb. at e ; and this is not altered by having its rope carried over the upper pulley, or roller, E.

In the former cases, the f ?ree of the power is augmented in an arithmetical proportion of the number of ropes or pulleys; but in this, in a geometrical proportion.

3. If a power move a weight by means of several pulleys, the space passed over by the power will be to the space passed over by the weight, as the weight, to the power.

I fence, the smaller the force that sustains a weight by means of pulleys is, the slower is the weight raised ; so that what is saved in force, is spent in time.