fame, in raifing the point : on account therefore of the equal volocities of the points a and G, the action of E at the point A will be the fame as if it were exerted at G in the direction of the tangent CG. But again, fuppofe the equal weight F to be restored, and the point & will then be acted on by two equal and oppofite forces, which, destroying each other's effect, will not produce motion; confequently the lever will continue at rest.

T It is likewife evident, that if the radii a B and B G be not in a right line, the equal forces will nevertheless be in equilibrio, if they be applied in the directions of the tangents: thus, if B G be bent to the position в K, and the force & be there applied in the direction KH, the equilibrium will remain as before.

U If two contrary forces be applied to a lever at unequal distances from the fulcrum, they will equiponderate when the forces are to each other in the reciprocal proportion of their diftances. For, v Let A c (fig. 12) reprefent a lever, whose radius AB is three times as long as BC. At A is fufpended the weight E of one pound, and at c is fufpended the weight of three pounds. Then, I fay, these weights will equiponderate. With the radius BA describe the arc A K, interfecting C F at K. Draw the line BK, which may represent another arm to the lever; and it will be evidently of no confequence whether the thread cr be faftened at c or K; conceive it therefore to be fastened at K, and to act on the arm BK. Let AG reprefent the force of £; and K F, being three times as long, will represent

1

represent the force of F. This force KF may be refolved (23, T) into two others, K in the direc tion of BK, and K н in the direction of the tangent, and their quantities are determined by drawing the lines F H and F I, parallel respectively to KI and K H, Now, I has no effect in moving the arm в K. It is the force к H alone that tends to produce motion towards H. The triangles. B C K and K н F are fimilar, therefore, BK: BC: KF: KH. as 3 to 1, whence the force KH is of к F, as is likewife AG by the condition. Confequently HK and AG are equal, and being applied at the end of equal arms A B, BK, will be in equilibrio (56, т), which was to be proved; and the conclufion will be the fame, when the weights are to each other in any other ratio, provided the arms of the lever A B and Bc be reciprocally in the fame proportion,

But BK: BC,

By the resolution of force it appears, that if two w contrary forces be applied to a trait lever at dif tances from the fulcrum in the reciprocal proportion of their quantities, and in directions always parafiel to each other, the lever will remain at rest in any pofition,

For, let the forces, A E, CF (fig. 13) be refolved: x AE into G E parallel, and GA perpendicular to AC; and CF into HF parallel, and CH perpendicular to Ac; and the forces, which tend to produce motion, will in all pofitions be to each other in the ratio of the forces applied; i. e. AE: CF: AG CH, the triangles AGE and cнr being fimilar,

Many

B

Many curious and ufeful effects may be produced by levers, whofe arms are bent into an angle; but the limits of this work do not permit us to enlarge upon them.

It is evident, that all which has been faid concerning the lever is equally true, when the contrary forces are applied on the fame fide of the fulcrum.

On the lever A B, (fig. 14) if the weight of one pound be applied at A, and the weight F of three pounds at c, fo that their distances A B and C B, from the fulcrum B, may be as three to one, they will equiponderates which is proved by applying the reafoning at fig, 12 to the present figure,

Since, of the three forces which act on a lever, the two which are applied at the extremes are always in a contrary direction to that which is applied in the space between them; this laft force will fuftain the effects of the other two; or, in other words, if the fulcrum be placed between the weights, it will be acted upon by, or will fuftain their fum: but if the weights are on the fame fide of the fulcrum, it will be acted upon by their difference.

C On the principle of the lever are made, scales for weighing different quantities of various kinds of substances; the steelyard, which answers the fame purpose by a fingle weight, removed to different distances from the fulcrum on a graduated arm, according as the body to be weighed is more or lefs in quantity; and the bent lever balance, which, by the revolution of a fixed weight, increafing in power as it afcends in the arc of a circle,

indicates

indicates the weight of the counterpoife,

ABC

(fig. 15) is a bent lever, fupported on its axis or D fulcrum B in the pillar J H. At A is fufpended the scale E, and at c is affixed a weight: draw the hori zontal line. KG through the fulcrum, on which from A and C, let fall, the perpendiculars AK and CD; then if в K and BD are reciprocally, in proportion to the weights at A and C, they will be in equilibrio, but if not, the weight c will move along the arc FG, till that ratio is obtained. It is eafy to graduate the plate FG fo as to express the weight in E by the pofition of c.

The beam of the common balance is ufually a bent E lever, with equal arms. Its property of coming to reft in an horizontal pofition, when the extremes are equally loaded, is a confequence of its being bent, or, which is the fame thing, of its fulcrum being above the line, joining the two points on which the scales. are fufpended. For it is evident, (fig. 16) that there is but one pofition in which the lengths of the arms AB, BC, referred to the horizontal line DE, can be equal, and that is when the points A and C are on the fame level.

Balances that move with very little friction on the F fulcrum, and are exactly equibrachial, are highly. valued. But this laft property is of lefs importance than is commonly imagined. For, if two balances be equally fenfible, and one of them not equibrachial, it is certain, that if the ftandard weight be placed in one of the scales of this laft, and counterpoifed, and the ftandard weight be afterwards re

moved, any other body fubftituted in its place will have exactly the fame mass, if it be in equilibrio with the counterpoife. In fact, this is the best method of weighing, when great accuracy is required. Or if the weights be always put into one fcale and the quantities into the other, thefe laft will be proportionally true among each other, which is quite fufficient in all philofophical experiments. G On this principle alfo depends the motions of animals, the overturning or lifting great weights by means of iron levers called crows, the action of nutcrackers, pincers, and many other inftruments of the fame nature.

CHA P. III.

OF THE AXIS AND WHEEL, AND OF THE PULLEY, OR TACKLE.

H THE axis and wheel may be confidered as a

lever, one of the forces being applied at the cir-' cumference of the axis, and the other at the circumference of the wheel, the central line of the axis being as it were the fulcrum. Fig. 17 is a perspective view of the inftrument, and fig. 18 is a fection of the fame at right angles to the axis. Then, if AB, the femidiameter of the axis, be to BC, the femidiameter of the wheel, reciprocally as the power в is to the power F, the first of which is applied in the direction of the tangent of the axis, and the other in the direction of the tangent of the wheel, they will be in equilibrio

(56, v).