This section is from the book "The Engineer's And Mechanic's Encyclopaedia", by Luke Hebert. Also available from Amazon: Engineer's And Mechanic's Encyclopaedia.

A modification of the lever, by means of which a weight may be raised to a considerable height. A slight attention to the nature of the lever will show that the extent of its influence in space is very small, depending upon the length of that arm to which the weight is attached; and as this arm becomes shorter in proportion to the increase of power obtained, so the height to which a body may be raised, speedily attains its limit. In the wheel and axle, no limit of this kind exists. Let a b, in the annexed cut, represent the diameter of the wheel, and c d that of the axle; then, if a power p be connected by means of a rope to the wheel, and a weight w to the axle, these two, when in equilibrio, will be to each other as c d to a b. That is, the power is to the weight as the diameter cf the axle to the diameter of the wheel; or, since the diameter of a circle is double its radius; as, the radius of the axle to the radius of the wheel. If a line f h g be drawn, connecting the parallel cords, and a perpendicular e h be let fall on it, it will be divided in the same ratio as the diameters or radii of the wheel and axle; and hence its relation to the lever becomes manifest.

It will be immediately seen that the power is to the weight, as/A to h g; that is, as the radius of the axle to the radius of the wheel. The velocity with which the power and weight will move, is, as in the other simple machines, inversely as the power gained. If the diameter of the wheel be 20 inches, and that of the axle 4 inches, the power obtained will be 20/4 = 5 times; or a power of one pound will balance a weight of five pounds; but the velocity with which the weight moves, is five times less than that of the power. The windlass by which water is drawn from wells, and the capstan used to raise the anchor on ship-board, are illustrations of the utility of this simple machine; but the most extensive employment of the wheel and axle is in combination, in which, under the name of wheel and pinion, it enters largely into the construction of the most complicated machinery. In the arrangement of a number of wheels and pinions for the purpose of gaining power, or velocity, each pinion is connected with the following wheel, and the power or weight is attached to the last pinion.

Thus, in the foregoing representation a b and c are three wheels; def, three axles or pinions, as it may be; the power p puts a into motion, the axle of which turns b, whose axle again influences c, on the axle of which th resistance is applied. The proportion between p and w in this and similar cases, will be found by multiplying together the diameters of the axles, and the diameters of the wheels. If the diameters of the wheels be 14, 9 and 7, and the axles be 3, 3 and 2, the power obtained will be

14 X 9 X 7 / 3 X 3 X 2 - 49, and as a consequence, the velocity of p must be 49 times greater than that of w. When wheels and pinions act upon each other as in watches and other machines, a number of teeth are cut in the circumference of each, in nearly the same proportion as the radii of the wheel and pinion. Sometimes, especially in heavy machinery, they are connected by bands, as in the annexed cut; but the calculated power is still the same at whatever angle they may be placed to each other, since the bands always act on that part of the wheel which is perpendicular to their own direction. In calculating the power of this machine, allowance must be made for the friction on the pivots, the weight and stiffness of the rope, and for the increased magnitude which a large rope gives to the wheel or axle.

Continue to: