As the arcs A _a_, B _b_, and D _d_, described by the ends of the balance or points of suspension, are proportionable to their sines _e a_, _g b_, and _d f_, as also the radii or distances C A, C B, and C D; in the case of this common sort of balance, the arcs described by the weights, or their points of suspension, or the distances from the center, may be taken for velocities of the weights hanging at A, B, or D, and, therefore, the acting force of the weights will be reciprocally as their distances from the center. Scholium.--The distances from the center are taken here for the velocities of the bodies, only because they are proportionable to the lines _e a_, _b g_, and _f d_, which are the true velocities; for there are a great many cases wherein the velocities are neither proportionable to the distances from the center of motion of a machine, nor to the arcs described by the weights or their points of suspension. Therefore, it is not a general rule that weights act in proportion to their distances from the center of motion; but a corollary of the general rule that weights act in proportion to their velocities, which is only true in some cases. Therefore, we must not take this case as a principle, which most workmen do, and all those people who make attempts to find the perpetual motion, as I have more amply shewn in the Phil. Trans., No. 369. But to make this evident even in the balance, we need only take notice of the following experiment:--A C B E K D is a balance in the form of a parallelogram passing through a slit in the upright piece N O standing on the pedestal M, so as to be moveable upon the center pins C and K. To the upright pieces A D and B E of this balance are fixed at right angles the horizontal pieces F G and H I. That the equal weights P W must keep each other in æquilibrio, is evident; but it does not at first appear so plainly, that if W be removed to V, being suspended at 6, yet it shall still keep P in æquilibrio, though the experiment shews it. Nay, if W be successively moved to any of the points 1, 2, 3, E, 4, 5, or 6, the æquilibrium will be continued; or if, W hanging at any of those points, P be successively moved to D, or any of the points of suspension on the cross-piece F G, P will at any of those places make an æquilibrium with W. Now, when the weights are at P and V, if the least weight that is capable to overcome the friction at the points of suspension C and K be added to V, as u, the weight V will overpower, and that as much at V as if it was at W. From what we have said above, the reason of this experiment will be very plain.