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For, the effect of each weight to turn the level, is as the weight multiplied into its distance; and in the case of an equi. librium, the sums of the effects, or of the products on both sides, are equal. The same would also follow from art. 26.
therefore, by composition, q + R : Q :: BD : CD, and, q + R : R :: BD : cB. That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.
59. On the foregoing principles depends the nature of scales and beams, for weighing all sorts of goods. For, if the weights be equal, then will the distances be equal also, which gives the construction of the common scales, which ought to have these properties :
1st. That the points of suspension of the scales and the centre of motion of the beam, A, B, c, should be in a straight line : 2d, That the arms AB, Bc, be of an equal length : 3d, That the centre of gravity be in the centre of motion B, or a little below it : 4th, That they be in equilibrio when empty : 5th, That there be as little friction as possible at the centre B. A defect in any of these properties makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibrio when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal ; but the deceit will be detected by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.
60. To find the true weight of any body by such a false balance :-First weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required. For, if any body b weigh w pounds or ounces in the scale D, and only w pounds or ounces in the scale E : then we have these Wol. II. 22
61. The Roman Statera, or Steelyard, is also a lever, but of unequal brachia or arms, so contrived, that one weight only may serve to weigh a great many, by sliding it backward and forward, to different distances, on the longer arm of the lever ; and it is thus constructed :
Let AB be the steelyard, and c its centre of motion, whence the divisions must commence if the two arms just balance each other : if not, slide the constant moveable weight 1 along from B towards, c, till it just balance the other end without a weight, and there make a notch in the beam, marking it with a cipher 0. Then hang on at A a weight w equal to 1, and slide 1 back towards B till they balance each other; there notch the beam, and mark it with 1. Then make the weight w double of 1, and sliding 1 back to balance it, there mark it with 2. Do the same at 3, 4, 5, &c. by making w equal to 3, 4, 5, &c. times 1 ; and the beam is finished. Then, to find the weight of any body b by the steelyard : take off the weight w, and hang on the body b at A ; then slide the weight 1 backward and forward till it just balance the body b, which suppose to be at the number 5; then is b equal to 5 times the weight of 1. So, if I be one pound, then b is 5 pounds ; but if 1 be 2 pounds, then b is 10 pounds; and so on.
OF THE WHEEL AND AXLE.
62. Prop. In the wheel-and-axle; the weight and power will be in equilibrio, when the power P is to the weight w reciprocally as the radii of the circles where they act; that is, as the radius of the axle cA, where the weight hangs, to the radius of the wheel ch, where the power acts. That is, P : W : : CA. : CB.
Here the cord, by which the power P acts, goes about the circumference of the wheel, while that of the weight w goes round its axle, or another smaller wheel, attached 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 cA; therefore P : w :: CA : CB.
63. 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 ch; 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.
64. Corol. 2. If the power do not act at right angles to the radius ch, but obliquely; draw cd perpendicular to the direction of the power; then, by the nature of the lever, P : W - : CA : CD.
65. To this mechanical power belong all turning or wheel machines, of different radii. Thus, in the roller turning on the axisor spindle cr, by the handle cBD ; the power applied at B is to the weight won the roller as the radius of the roller is to the radius cB of the handle.
66. 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 aets; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and atgur for boring holes.
67. 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 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.
68. 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.
69. 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 p turn the wheel Q, and this turn the small wheel or axle R,
and this turn the wheel s, and this turn the axle T, and this
turn the wheel v ; and this turn the axle x, which raises the
: weight w; then f : w :: cm. DE . FG : Ac. Bd . EF. 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 l ; then. p : w :: 1* : 10° or as 1 to 1000. So that a power of one pound will balance a weight of 1000 pounds ; but then, when put in motion, the power will move 1000 times faster than the weight.
OF THE PULLEY.
70. 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.
71, PRop. 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 power P is consequently equal to the weight w. 72. 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, both by changing the direction of the force, for the convenience of acting, and by enabling 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 P, 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.
73. Prop. If a power sustain a weight by means of one