A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

[32.] The lever was first named above. This is a bar made use of to sustain and move great weights. The bar is applied in one part to some strong support; as the bar A B (in fig. 23, 24.) is applied at the point C to the support D. In some other part of the bar, as E, is applied the weight to be sustained or moved; and in a third place, as F, is applied another weight or equivalent force, which is to sustain or move the weight at E. Now here, if, when the level should be put in motion, and turned upon the point C, the velocity, wherewith the point F would move, bears the same proportion to the velocity, wherewith the point E would move, as the weight at E bears to the weight or force at F; then the lever thus charged will have no propensity to move either way. If the weight or other force at F be not so great as to bear this proportion, the weight at E will not be sustained; but if the force at F be greater than this, the weight at E will be surmounted. This is evident from what has been said above[60], when the forces at E and F are placed (as in fig. 23.) on different sides of the support D. It will appear also equally manifest in the other case, by continuing the bar B C in fig. 24. on the other side of the support D, till C G be equal to C F, and by hanging at G a weight equivalent to the power at F; for then, if the power at F were removed, the two weights at G and E would counterpoize each other, as in the former case: and it is evident, that the point F will be lifted up by the weight at G with the same degree of force, as by the other power applied to F; since, if the weight at E were removed, a weight hung at F equal to that at G would balance the lever, the distances C G and C F being equal.

33. If the two weights, or other powers, applied to the lever do not counterbalance each other; a third power may be applied in any place proposed of the lever, which shall hold the whole in a just counterpoize. Suppose (in fig. 25.) the two powers at E and F did not equiponderate, and it were required to apply a third power to the point G, that might be sufficient to balance the lever. Find what power in F would just counterbalance the power in E; then if the difference between this power and that, which is actually applied at F, bear the same proportion to the third power to be applied at G, as the distance C G bears to C F; the lever will be counterpoized by the help of this third power, if it be so applied as to act the same way with the power in F, when that power is too small to counterbalance the power in E; but otherwise the power in G must be so applied, as to act against the power in F. In like manner, if a lever were charged with three, or any greater number of weights or other powers, which did not counterpoize each other, another power might be applied in any place proposed, which should bring the whole to a just balance. And what is here said concerning a plurality of powers, may be equally applied to all the following cases.

34. If the lever should consist of two arms making an angle at the point C (as in fig. 26.) yet if the forces are applied perpendicularly to each arm, the same proportion will hold between the forces applied, and the distances of the center, whereon the lever rests, from the points to which they are applied. That is, the weight at E will be to the force in F in the same proportion, as C F bears to C E.

35. But whenever the forces applied to the lever act obliquely to the arm, to which they are applied (as in fig. 27.) then the strength of the forces is to be estimated by lines let fall from the center of the lever to the directions, wherein the forces act. To balance the levers in fig. 27, the weight or other force at F will bear the same proportion to the weight at E, as the distance C E bears to C G the perpendicular let fall from the point C upon the line, which denotes the direction wherein the force applied to F acts: for here, if the lever be put into motion, the power applied to F will begin to move in the direction of the line F G; and therefore its first motion will be the same, as the motion of the point G.

36. When two weights hang upon a lever, and the point, by which the lever is supported, is placed in the middle between the two weights, that the arms of the lever are both of equal length; then this lever is particularly called a balance; and equal weights equiponderate as in common scales. When the point of support is not equally distant from both weights, it constitutes that instrument for weighing, which is called a steelyard. Though both in common scales, and the steelyard, the point, on which the beam is hung, is not usually placed just in the same straight line with the points, that hold the weights, but rather a little above (as in fig. 28.) where the lines drawn from the point C, whereon the beam is suspended, to the points E and F, on which the weights are hung, do not make absolutely one continued line. If the three points E, C, and F were in one straight line, those weights, which equiponderated, when the beam hung horizontally, would also equiponderate in any other situation.

But we see in these instruments, when they are charged with weights, which equiponderate with the beam hanging horizontally; that, if the beam be inclined either way, the weight most elevated surmounts the other, and descends, causing the beam to swing, till by degrees it recovers its horizontal position. This effect arises from the forementioned structure: for by this structure these instruments are levers composed of two arms, which make an angle at the point of support (as in fig. 29, 30.) the first of which represents the case of the common balance, the second the case of the steelyard. In the first, where C E and C F are equal, equal weights hung at E and F will equiponderate, when the points E and F are in an horizontal situation. Suppose the lines E G and F H to be perpendicular to the horizon, then they will denote the directions, wherein the forces applied to E and F act. Therefore the proportion between the weights at E and F, which shall equiponderate, are to be judged of by perpendiculars, as C I, C K, let fall from C upon E G and F H: so that the weights being equal, the lines C I, C K, must be equal also, when the weights equiponderate. But I believe my readers will easily see, that since C E and C F are equal, the lines C I and C K will be equal, when the points E and F are horizontally situated.

37. If this lever be set into any other position (as in fig. 31.) then the weight, which is raised highest, will outweigh the other. Here, if the point F be raised higher than E, the perpendicular C K will be longer than C I: and therefore the weights would equiponderate, if the weight at F were less than the weight at E. But the weight at F is equal to that at E; therefore is greater, than is necessary to counterbalance the weight at E, and consequently will outweigh it, and draw the beam of the lever down.

38. In like manner in the case of the steelyard (fig. 32.) if the weights at E and F are so proportioned, as to equiponderate, when the points E and F are horizontally situated; then in any other situation of this lever the weight, which is raised highest, will preponderate. That is, if in the horizontal situation of the points E and F the weight at F bears the same proportion to the weight at E, as C I bears to C K; then, if the point F be raised higher than E (as in fig. 32.) the weight at F shall bear a greater proportion to the weight at E, than C I bears to C K.

39. Farther a lever may be hung upon an axis, and then the two arms of the lever need not be continuous, but fixed to different parts of this axis; as in fig. 33, where the axis A B is supported by its two extremities A and B. To this axis one arm of the lever is fixed at the point C, the other at the point D. Now here, if a weight be hung at E, the extremity of that arm, which is fixed to the axis at the point C; and another weight be hung at F, the extremity of the arm, which is fixed on the axis at D; then these weights will equiponderate, when the weight at E bears the same proportion to the weight at F, as the arm D F bears to C E.