The balance, an instrument of very extensive use in comparing the weights of bodies, is a lever of the first kind, whose arms are of equal length. The points from which the weights are suspen ded being equally distant from the cen tre of motion, will move with equal velo city ; consequently if equal weights be applied, their momenta will be equal, and the balance will remain in equilibrio. In order to have a balance as perfect as possi ble, it is necessary to attend to the follow ing circumstances : 1. The arms of the beam ought to be exactly equal, both as to weight and length. 2. The points from which the scales are suspended, should be in a right line, passing through the centre of gravity of the beam ; for by this, the weights will act directly against each other, and no part of either will be lost, on account of any oblique direction. 3. If the fulcrum be placed in the centre of gravity of the beam, and if the fulcrum and the points of suspension he in the same right line, the balance will have no tendency to one position more than ano ther, but will rest in any position it may be placed in, whether the scales be on or off, empty or loaded. If the centre of gravi ty of the beam, when level, be immedi ately above the fulcrum, it will oversct by the smallest action ; thatis, the end which is lowest will descend ; and it will do this with more swiftness, the higher the cen tre of gravity be, and the less the points of suspension be loaded. But if the cen tre of gravity of the beam ly below the fulcrum, the beam will not rest in any position but when level ; and if disturbed from that position and then left at liberty, it will vibrate, and at last come to rest on the level. In a balance, therefore, the fulcrum ought always to be placed a little above the centre of gravity. Its vibrations will be quicker, and its ho rizontal tendency stronger, the lower the centre of gravity, and the less the weight upon the points of suspension. 4. The friction of the beam upon the axis ought to be as little as possible ; because, should the friction be great, it will require a con siderable force to overcome it ; upon which account, though one weight should a little exceed the other, it will not pre ponderate, the excess not being sufficient to overcome the friction, and bear down the beam. 5. The pivots, which form the axis or fulcrum, should be in a straight line. and at right angles to the beam, fi The arms should be as long as possible, relatively to their thickness, and the pur poses for which they are intended, as the longer they are the more sensible is the balance. They should also be made as stiff and inflexible as possible ; for if the beam be too weak, it will bend, and be come untrue. 7. The rings, or the piece on which the axis bears, should be hard and well polished, parallel to each other, and of an oval form, that the axis may always keep its proper bearing; or remain always at the lowest point. 8. If the arms of a balance be unequal, the weights in equipoise will be unequal in the same proportion. The equality of the arms is of use, in scientific pursuits, chiefly in the making of weights by bisection. A ba lance with unequal arms will weigh as ac curately as another of the same work manship with equal arms, provided the standard weight itself be first counter poised, then taken out of the scale, and the thing to be weighed be put into the scale, and adjusted against the counter poise. Or, when proportional quantities only are considered, the bodies under ex amination may be weighed against the weights, taking care always to put the weights in the same scale ; for then, though the bodies may not be really equal to the weights, yet their proportions amongst each other will be the same as if they had been accurately so. 9. Very delicate balances are not only useful in nice experiments, but are likewise much more expeditious than others in common weighing. if a pair of scales, with a cer tain load, be barely sensible to one-tenth of a grain, it will require a considerable time to ascertain the weight to that de gree of accuracy, because the turn must be observed several times over, and is ve ry small. But if no greater accuracy were required, and scales were used, which would turn with one-hundredth of a grain, a tenth of a grain more or less would make so great a difference in the turn, that it would be seen immediately.
The statera, or Roman steel-yard, is a lever of the first kind, and is used for finding the weights of different bodies, by one single weight placed at different dis tances from the prop or centre of motion D, fig. 6. For, the shorter arm D G is of
such a weight as exactly to counterpoise the longer arm D X. if this arm be divid ed into as many equal parts as it will contain, each equal to G D, the single weight P (which we may suppose to be one pound) will serve for weighing any thing as heavy as itself, or as many times heavier as there are divisions in the arm DX, or any quantity between its own weight and that quantity. As for example, if P be one pound, and placed at the first division 1 in the arm D X, it will balance one pound in the scale at AV ; if it be re moved to the second division at 2, it will balance two pounds in the scale ; if to the third, three pounds ; and so on to the end of the arm 17 X. if any of these in tegral divisions be subdivided into as many equal parts as a pound contains ounces, and the weight P be placed at any of these subdivisions, so as to counterpoise what is in the scale, the pounds and odd ounces therein will by that means be as certainAl. In the Danish and Swedish steel-yard, the body to be weighed, and the constant weight, are fixed at the ex tremities of the steel-yard, but the point of suspension or centre of motion moves along the lever till the equilibrium takes place The centre of motion therefore shews the weight of the. body.
The wheel and axle, or axis in peritro chio, is a machine much used, and is made in a variety of forms. It. consists of a wheel with an axle fixed to it, so as to turn round with it ; the power being ap plied at the circumference of the wheel, the weight to be raised is fastened to a rope which coils round the axle.
A 13 (fig. 7.) is a wheel, and C Dan ax le fixed to it, and which moves round with it. If the rope which goes round the wheel be pulled, and the wheel turned once round, it is evident that as much rope will be drawn off as the circumference of the wheel ; but while the wheel turns once round, the axle turns once round; and consequently the rope by which the weight is suspended will wind once round the axis, and the weight will be raised through a space equal to the circumfer ence of the axis. The velocity of the therefore, will be to that of the weight, as the circumference of the wheel to that of the axis. In order, therefore, that the power and the weight may be ill equilibrio, the power must be to the weight as the circumference of the wheel to that of the axis. Circles being to . each other as their respective diameters, the. power is to the weight, as the diame ter also of the axis to that of the wheel Thus, suppose the diameter of the wheel to be eight inches, and the diameter of the axis to be one inch then one ounce acting as the power P, will balance eight ounces as a weight W ; and a small addi tional force will cause the tvlieel to turn with its axis, and raise the weight ; and for every inch which the weight rises the power will increase eight inches.
The wheel and axis may be considered as a kind of perpetual lever, (fig. 8.) of which the fulcrum is the centre of the axis, and the long and short arms the di ameter of the wheel and the diameter of the axis. From this it is evident, that the longer the wheel, and the smaller the axis, the stronger is the power of this machine ; but then the weight must rise slower in proportion. A capstan is a cy linder of wood, with holes in it, into which are put bars, or levers, to turn it round ; these ate like the spokes of a wheel without the rim. Sometimes the axis is turned by a winch fastened to it, which, in this respect, serves for a wheel, and is more powerful, in pro portion to the largeness of the circle it describes, compared with the diameter of the axle. When the parts of the axis differ in thickness, and weights are sus pended at the different parts, they may be sustained by one and the same power applied to the circumference of the wheel, provided the product arising from the multiplication of the power into the diameter of the wheel, be equal to the sum of the products arising from the multiplication of the several weights into the diameters of those parts of the axis from which they are suspended. In con sidering the theory of the wheel and axle, we have supposed the rope that goes round the axis to have no sensible thickness ; but as in practice this cannot be the case, if it is a thick rope, or if there be several folds of it round the axis, you must measure to the middle of the outside rope to obtain the diameter of the axis, for the distance of the weight from the centre is Increased by the coiling up of the rope.