(251.) We have here omitted the consideration of the thickness of the rope. When this is considered, the force must be conceived as acting in the direction of the centre of the rope, and therefore the thickness of the rope which supports the power ought to be added to the diameter of the wheel, and the thickness of the rope which supports the weight to the diameter of the axle. It is the more necessary to attend to this circumstance, as the strength of the rope necessary to support the weight causes its thickness to bear a considerable proportion to the diameter of the axle; while the rope which sustains the power not requiring the same strength, and being applied to a larger circle, bears a very inconsiderable proportion to its diameter. (252.) In numerous forms of the wheel and axle, the weight or resistance is applied by a rope coiled upon the axle; but the manner in which the power is applied is very various, and not often by means of a rope. The circumference of a wheel sometimes carries projecting pins, as represented in _fig. 88._, to which the hand is applied to turn the machine. An instance of this occurs in the wheel used in the steerage of a vessel. In the common _windlass_, the power is applied by means of a _winch_, which is a rectangular lever, as represented in _fig. 89._ The arm B C of the winch represents the radius of the wheel, and the power is applied to C D at right angles to B C. In some cases no wheel is attached to the axle; but it is pierced with holes directed towards its centre, in which long levers are incessantly inserted, and a continuous action produced by several men working at the same time; so that while some are transferring the levers from hole to hole, others are working the windlass. The axle is sometimes placed in a vertical position, the wheel or levers being moved horizontally. The _capstan_ is an example of this: a vertical axis is fixed in the deck of the ship; the circumference is pierced with holes presented towards its centre. These holes receive long levers, as represented in _fig. 90._ The men who work the capstan walk continually round the axle, pressing forward the levers near their extremities. In some cases the wheel is turned by the weight of animals placed at its circumference, who move forward as fast as the wheel descends, so as to maintain their position continually at the extremity of the horizontal diameter. The _treadmill_, _fig. 91._, and certain _cranes_, such as _fig. 92._, are examples of this. In water-wheels, the power is the weight of water contained in buckets at the circumference, as in _fig. 93._, which is called an over-shot wheel: and sometimes by the impulse of water against float-boards at the circumference, as in the under-shot wheel, _fig. 94._ Both these principles act in the breast-wheel, _fig. 95._ In the paddle-wheel of a steam-boat, the power is the resistance which the water offers to the motion of the paddle-boards.