The wheel and axle consists of a cylinder revolving upon an axis, and having a wheel of larger diameter immoveably affixed to it. The power is applied to the circumference of the wheel, the weight to that of the axle.

[Let a b, be a wheel, cd, fig. 6, its axle, and suppose the circumference of the wheel to be eight times as great as the circumference of the axle; then a power, P, equal to one pound, hanging by the cord I, which goes round the wheel, will balance a weight, W, of eight pounds, hinging by the rope K, which goes round the axle; and as the friction on the pivots, E F, or gudgeons of the axle is but small, a small addition to the power, will cause it to descend, and raise the weight; but the weight will rise with only an eighth part of the velocity wherewith the power descends, and, consequently, through no more than an eighth part of an equal space in the same time. If the wheel be pulled round by the handles, S S, the power will be increased in proportion to their length. G, is a ratchet-wheel on one end of the axle, with a catch II, to fall in its teeth.

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The law of equilibrium is, that " the power must be to the weight as the radius of the axle is to that of the wheel."

This instrument is, evidently, nothing but a modification of the lever; it may he aided as a continuously acting lever, in fact, it is sometimes called "the perpetual lever." In its mode of action, the common lever operates in an intermitting way, and, as it were, by small steps at a time. A mass which is forced up by a lever a short distance must be temporarily propped, and the lever readjusted before it can be brought into action again ; but the wheel and axle continues its operation constantly in the same direction.

The inconvenience of having a large wheel and very slender axle may he avoided, without lessening the mechanical advantage, by employing a machine called the " Chinese wheel and axle," which consume of two cylinders, one larger than the other, turning about the same axis. The weight is attached to a pulley, which plays on a long cord, which is coiled round both axles in contrary directions. When the winch is turned, one end of the cord uncoils from the smaller cylinder, and is wound round the larger; thus the weight is elevated at each turn, through a space equal to half of the difference between the circumference of the two cylinders. Therefore the advantage of this machine, with its pulley, is in the ratio of the diameter of the larger cylinder to half its excess above that of the lesser one. (Fig. 7.)

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That this is its mode of action may be understood from considering fig. 8, in which let c be the common centre of the axle, c b, and of the wheel c a, a the point of application of the power P, and b that of the weight, W. Draw the line a c b ; it evidently represents a lever of the first order, of which the fulcrum is c, and from the principles of the lever, it is easy to demonstrate the law of equilibrium of this machine, as just given. Further, it is immaterial in what direction the power be applied, as P' at the point a' for a c b still forms a bent lever, and the same principle still holds good. The effect of the wheel depends upon the superiority of the radius, or diameter of the wheel, to that of the axle. In fig. 9 we see that the weight, "W, corresponds to the counteracting force, P, in an inverse ratio to the arms of the lever; that is, inversely to the radii, a b, and d c, of the wheel. Let us suppose that the radius, a b, of the axle, is four times less than the radius, d c, of the wheel, we may equipoise a weight of eighty pounds by a force of twenty pounds. Sometimes the wheel is replaced by a winch, as in fig. 10; it is then called a "windlass," if the motion is vertical: but if it be horizontal, as in fig. 11, the machine is called a "capstan," which differs from the windlass in having its revolving axis placed vertically. The circumference is pierced with holes which receive long levers, called capstan-bars, by which it is worked by men, who walk round the capstan and make it revolve by pressing the ends of the levers forward.

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The tread-mill is another variety. In this case the weight of several people treading on the circumference of a long wheel causes it to revolve. The paddle-wheel of a steamboat acts on the same principle ; the water, which offers a resistance to the motion of the paddle-boards, is the power.

Wheels and axles are often made to act upon one another by the aid of cogs, as in clockwork and mill machinery. In these cases the cogs on the periphery of the wheel] take the name of teeth, those on the axle the name of leaves, and the axle itself is called a pinion.

The law of equilibrium of such machines may be easily demonstrated to be, that the power multiplied by the product of the number of teeth, in all the wheels, is equal to the weight multiplied by the product o] the number of leaves in all the pinions. A system of wheel and pinion work is represented at fig 12. It is scarcely necessary to ob serve, that in it as in all other cases the law of virtua velocities hold good - the powe multiplied by the velocity of the power is equal to the weight multiplied by the velocity of the weight

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1m the construction of such machinery, attention has to be paid to the form of the teeth, so that they may not scrape or jolt upon one another. Several of them should be in contact at once, to diminish the risk of fracture and the wear.

If the teeth of a wheel be in the direction of radii from its centre it is called a spur-wheel.

If the teeth are parallel to the axis of the wheel it is called a crown-wheel.

If the teeth are oblique to the axis of the wheel it is called a beveled wheel.

By combining these different forms of wheels suitably together, the resulting motion can be transferred to any required plane. Thus, by a pair of beveled-wheels motion round a vertical axis may be transferred to a horizontal one, or, indeed, one in any other direction.

When a pinion is made to work on a toothed-bar, it constitutes a rack. This contrivance is under the same law as the wheel and axle.