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 momenta and radii; that is, as P x CE and W × CD, which are equal by corol 3.

Corollary 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. Corollary 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

B

C

D

E

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.

therefore, by composition, Q + K : Q :: BD : CD,

and, Q+R: R :: BD : CB.

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.

29. SCHOLIUM. Upon the foregoing principles depends the nature of scales and beams, for weighing all sorts of commodities. For, if the weights be equal, then will the distances be equal also, which gives the construction of the common scales, which ought to have these properties :

E

1st, The points of suspension of the scales and the centre of motion of the beam, ABC, must be in a straight line; 2d, The arms AB, BC must be of an equal length: 3d, That the centre of gravity be in the centre of motion B: 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 equilibro when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal; but the de

ceit will be shown by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.

30. 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 w pounds or ounces in the scale E; then we have these two equations;

b = √Ww,

hence, then

and

the mean proportional, which is the true weight of the body b.

31. 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 backwards and forwards 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, from 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

H

E

B

make a notch in the beam, marking it with a cypher 0. Then hang on at A a weight W equal to I, 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 I, and sliding I back to balance it, and there mark it with 2. Do the same at 3, 4, 5, &c., by making W equal to 3, 4, 5, &c. times I; 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 backwards and forwards till it just balance the body b, which suppose to be at the number 5; then is equal to 5 times the weight of I. So, if I be 1 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

PROP. VI.

32. In the Wheel and Axle; the Weight and Power will be m 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 Wright hangs, to the Radius of the Wheel CB, where the Power acis. That is, P: W:: CA: CB.

HERE the cord, by which the power P acts, goes about the circumference of the wheel, while that of the weight W goes round its axle, or another smaller wheel, attached to the larger, and having the same 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 : CB.

B

W

Corollary. 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.

Corollary. 2. If the power do not act at right angles to the radius C5, but obliquely; draw CD perp. to the direction of the power; then, by the nature of the lever, P: W:: CA: CD.

SCHOLIUM.

33. To this power belong all turning or wheel machines, of different radii. Thus, in the roller turning on the axis or spindle CE, by the handle CBD; the power applied at B is to the weight W on the roller, as the radius of the roller is to the radius CB of the handle.

34. 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.

35. 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 radius of the roller; or, to the radius of the roller and half the thickness of the cord, when there is but one fold.

36. The wheel-and-axle has a great advantage over the simple lever, in point of convenience. For a weight can be raised only a little way by the lever. But, by the continual turning of the wheel and roller, the weight may be raised to any height, or from any depth.

37. By increasing the number of wheels too, the power may be mul

tiplied to any extent, making always the less wheels to turn greater ones, as far as we please; and this is commonly called Tooth and Pinion, the teeth of one circumference working in the rounds or Pinions

of another, to turn

the wheel. And then, in case of an equilibrium, the power is to the weight, as the continual product of the radii of all the axles, to that of all the wheels. So, if the power P turn the wheel Q, and this turn the small wheel or axle R, and this turn the wheel S, and this turn the axle T, and this turn the wheel V, and this turn the axle X, which raises the weight W; then

P: W:: CB. DE. FG: AC. BD. EF.

And in the same proportion is the velocity of W slower than that of P. Thus, if each wheel be to its axle, as 10 to 1; then P: W:: 13: 10s or as 1 to 1000. So that a power of one pound will balance a weight of 1000 pounds; but then when put in motion, the power will move 1000 times faster than the weight.

38. If ropes are used for the action of the power and weight, we must consider the forces applied to the axes of the ropes. Hence if R, r denote the radii of the wheel and axle, and T, t half the thickness of the ropes, we have P: Wr+t: R+1.

OF THE PULLEY.

39. A PULLEY is a small wheel, commonly made of wood or brass, which turns about an iron axis passing through the centre, and fixed in a block, by means of a cord passed round its circumference, which serves to draw up any weight. The pulley is either single, or combined together, to increase the power. It is also either fixed or moveable, according as it is fixed to one place or moves up and down with the weight and power.

When a power sustains a weight by means of a fixed pulley, the power and weight are obviously equal; for if through the centre of the pulley a horizontal line be drawn, it will represent a lever of the first kind, whose prop or fulcrum is the fixed centre; hence the points where the power and weight act, are equally distant from the centre, and therefore the power must be equal to the weight. No mechanical advantage, however, is gained by the fixed pulley, though it is still of great utility in the raising of weights, both by changing the direction of the force, and also by enabling several persons to exert their united forces.

PROP. VII.

40 In the single moveable pulley, and the strings parallel, the power's to the weight as 1 : 2; but if the strings produced make an angle = 24; then P: W:1: 2 cos .

Through the centre of the pulley draw the vertical p line WAC, and take AC to represent the weight W, where A is the point of intersection of the strings produced. Draw CB parallel to AH; then since the string is equally stretched throughout, we have AB BC and angle BAC = 4; whence

P: W::AB: AC:: sin : sin 24 :: 1:2 cos p; and when the strings are parallel P:W: :1:2, for $=0. Cor. 1. If w= weight of the moveable block, then 2PW+w.

B

Cor. 2. In the system where there are two blocks of pulleys, the one fixed and the other moveable, and the same rope passing round all the pulleys, then we have simply a combination of the preceding case; and therefore

nPW + w

where n = number of strings at the moveable block and w, its weight. If the strings are not parallel, the cosine of the angle made with the vertical in each case must be introduced, as above.

Cor. 3. In the system where each pulley hangs by a separate string, we have merely a repetition of the single moveable pulley; and the strings being parallel, we get

2"PW+w1 + 2w2 + 2ow, +

....

where n is the number of moveable pulleys, and w1, wa, w,, the weights of the pulleys including the blocks respectively.

41. In the system of pulleys, where each string is attached to the weight, and the strings parallel, we have P: W::1:2"-1, where n is the number of pulleys.

Let w1, w2, w、, &c. be the weights of the pulleys, and let the strings passing over the pulleys w1, w, w, &c. be attached to the weight at the points Pa Pa P, &c.; then we have