166. A Fourth kind is sometimes added, called the Bended Lever. As a hammer drawing a nail. W C P 167. In all these instruments the power may be represented by a weight, which is its most natural measure, acting downward: but having its direction changed, when necessary, by means of a fixed pulley. PROPOSITION XXΧΙ. 168. When the Weight and Power keep the Lever in Equilibrio, they are to each other Reciprocally as the Distances of their Lines of Direction from the Prop. That is, P: W :: CD: CE; where CD and CE are perpendicular to wo and AO, the Directions of the two Weights, or the Weight and Porver w and A. But, because of the parallels, the two triangles CDF, CEB are equiangular, therefore Hence, by equality, CD:CE:: CF: Cв. PW:: CD: CE. That is, each force is reciprocally proportional to the distance of its direction from the fulcrum. And it will be found that this demonstration will serve for all the other kinds of levers, by drawing the lines as directed. 169. Corol. 1. When the angle A is = the angle w, then is CD: CE :: CW:CA::P: W. Or when the two forces act perpendicularly on the lever, as two weights, &c; then, in case of an equilibrium, D coincides with w, and E with P; consequently then the above proportion becomes also p : w:: cw: CA, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces. 170. Corel. 170. Corol. 2. If any force e be applied to a lever at A; its effect on the lever, to turn it about the centre of motion 2, is as the length of the lever ca, and the sine of the angle of direction CAE. For the perp. CE is as CA X S. LA. 171. Corol. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power by the distance of its direction, is equal to the product of the weight by the distance of its direction. That is, PX CE = W X CD. 172. Corol. 4. If the lever, with the weight and power fixed to it, be made to move about the centre c; the momentum of the power will be equal to the momentum of the weight; and their velocities will be in reciprocal proportion to each other. For the weight and power will describe circles whose radii are the distances CD, CE; and since the circumferences or spaces described, are as the radii, and also as the velocities, therefore the velocities are as the radii CD, CE; and the momenta, which are as the masses and velocities, are as the masses and radii; that is, as ŕ X CE and w × CD, which are equal by cor. 3. 173. Corol. 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two. 174. Corol. 6. If several weights P, Q, R, s, act on a straight lever, and keep it in equilibrio; then the sum of the products on one side of the prop, will be equal to the sum on the other side, made by multiplying each weight by its distance; namely, PX AC + QX BC = R X DC + S X EC. For, the effect of each weight to turn the lever, is as the weight multiplied by its distance; and in the case of an equilibrium, the sums of the effects, or of the products on both sides, are equal. 175. Corol. 7. Because, whert two weights and R are in equilibrio, Q: R::CD: Cв; Ca C therefore, by composition, Q+ R:Q::BD:CD, and, Q+ R:R::BD:CB. VOL. II. N R Tha 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. 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. 177. 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 to pounds or ounces in the scale E: then we have these two equations, namely, AB. b = Bc.w. and BC. b = AB . τω; the product of the two is AB. BC hence then b= AB.BC. ww; b2 = ww, the mean proportional, which is the true weight of the body b. 178. 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 back ward : ward 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 I 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 I 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 I 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 I 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 I be 2 pounds, then b is 10 pounds; and so on. OF THE WHEEL AND AXLE. PROPOSITION XXXII. 179. 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 CB, where the Power acts. That is, PW::CA: CB. HERE the cord, by which the power pacts, goes about 1 the circumference of the wheel, while that of the weight w goes round its axle, or another smaller wheel, attach-ed 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: Cв. B b P M 180. 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 CB; 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. 181. Corol. 2. If the power do not act at right angles to the radius cb, but obliquely; draw CD perpendicular to the direction of the power; then, by the nature of the lever, 183. 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 acts; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and augur for boring holes. 1 184. 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 |