two equations, namely, AB. b = BC. W, and BC. b AB. w;

the product of the two is AB hence then

√ww,

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

61. 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 and forward, to different distances, on the longer arm of the lever; and it is thus constructed :

I

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 1 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 I, and slide 1 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 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 1 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 bis 5 pounds; but if 1 be 2 pounds, then bis 10 pounds; and so on.

OF THE WHEEL AND AXLE.

62. PROP. 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,

P: W:: CA: CB.

b

D

IP

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 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 PW CA: CB.

B

W

63. 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.

64. 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,

66. And the same for all cranes, capstans, windlasses, and

1

such like; the power being to the weight, always as the ra 'dius 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 refer. red the gimblet and augur for boring holes.

67. 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, we must add half the thickness of the chord, when there is but one fold.

68. The wheel-and-axle has a great advantage over the simple lever, in point of convenience. For a weight can be raised but a little way by the lever; whereas, by the continu. al turning of the wheel and roller, the weight may be raised to any height, or from any depth.

69. By increasing the number of wheels, too, the power may be multiplied 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 Work, 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 a, 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

And

weight w; then Pw: CB. DE. FG: AC. BD. EF. 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: 103 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.

OF THE PULLEY.

70. 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.

71. PROP. If a power sustain a weight by means of a fixed pulley: the power and weight are equal.

For through the centre c of the pulley draw the horizontal diameter AB then will AB represent a lever of the first kind, its prop being the fixed centre c; from which the points A and B, where the power and weight act, being equally distant, the power P is consequently equal to the weight w.

AC

B

P

W

72. Corol. Hence, if the pulley be put in motion, the power P will descend as fast as the weight w ascends. So that the power is not increased by the use of the fixed pulley, even though the rope go over several of them. It is, however, of great service in the raising of weights, both by changing the direction of the force, for the convenience of acting, and by enabling a person to raise a weight to any height without moving from his place, and also by permitting a great many persons at once to exert their force on the rope at P, which they could not do to the weight itself; as is evident in raising the hammer or weight of a pile-driver, as well as on many other occasions. 73. PROP. If a power sustain a weight by means of one

moveable pulley; the power is but half the weight, if the portions of the sustaining cord are parallel to each other.

For, here AB may be con

sidered as a lever of the
second kind, the power act-
ing at A, the weight at c, P
and the prop or fixed point
at B; and because P: w::
CB AB, and CB = AB,
therefore P= w, or w
= 2P.

74. Corol. 1. Hence it is evident, that, when the pulley is put in motion, the velocity of the power will be

B

double the velocity of the weight, as the point r moves twice as fast as the point c and weight w rises. It is also evident, that the fixed pulley F makes no difference in the power r, but is only used to change the direction of it, from upwards to downwards.

75. Corol. 2. Hence we may estimate the effect of a combination of any number of fixed and moveable pulleys; by which we shall find that every cord going over a moveable pulley always adds 2 to the power; since each moveable pulley's rope bears an equal share of the weight: while each rope that is fixed to a pulley, only increases the power by unity.

Note. If the portions of the sustaining cords between the pulleys are not parallel, the forces will be reduced upon the principle of art. 31.