40. This is the case, if both the arms are perpendicular to the axis, and lie (as the geometers express themselves) in the same plane; or, in other words, if the arms are so fixed perpendicularly upon the axis, that, when one of them lies horizontally, the other shall also be horizontal. If either arm stand not perpendicular to the axis; then, in determining the proportion between the weights, instead of the length of that arm, you must use the perpendicular let fall upon the axis from the extremity of that arm. If the arms are not so fixed as to become horizontal, at the same time; the method of assigning the proportion between the weights is analogous to that made use of above in levers, which make an angle at the point, whereon they are supported.
[41.] From this case of the lever hung on an axis, it is easy to make a transition to another mechanical power, the wheel and axis.
42. This instrument is a wheel fixed on a roller, the roller being supported at each extremity so as to turn round freely with the wheel, in the manner represented in fig. 34, where A B is the wheel, C D the roller, and E F its two supports. Now suppose a weight G hung by a cord wound round the roller, and another weight H hung by a cord wound about the wheel the contrary way: that these weights may support each other, the weight H must bear the same proportion to the weight G, as the thickness of the roller bears to the diameter of the wheel.
43. Suppose the line k l to be drawn through the middle of the roller; and from the place of the roller, where the cord, on which the weight G hangs, begins to leave the roller, as at m, let the line m n be drawn perpendicularly to k l; and from the point, where the cord holding the weight H begins to leave the wheel, as at o, let the line o p be drawn perpendicular to k l. This being done, the two lines o p and m n represent two arms of a lever fixed on the axis k l; consequently the weight H will bear to the weight G the same proportion, as m n bears to o p. But m n bears the same proportion to o p, as the thickness of the roller bears to the diameter of the wheel; for m n is half the thickness of the roller, and o p half the diameter of the wheel.
44. If the wheel be put into motion, and turned once round, that the cord, on which the weight G hangs, be wound once more round the axis; then at the same time the cord, whereon the weight H hangs, will be wound off from the wheel one circuit. Therefore the velocity of the weight G will bear the same proportion to the velocity of the weight H, as the circumference of the roller to the circumference of the wheel. But the circumference of the roller bears the same proportion to the circumference of the wheel, as the thickness of the roller bears to the diameter of the wheel, consequently the velocity of the weight G bears to the velocity of the weight H the same proportion, as the thickness of the roller bears to the diameter of the wheel, which is the proportion that the weight H bears to the weight G. Therefore as before in the lever, so here also the general rule laid down above is verified, that the weights equiponderate, when their velocities would be reciprocally proportional to their respective weights.
45. In like manner, if on the same axis two wheels of different sizes are fixed (as in fig. 35.) and a weight hung on each; the weights will equiponderate, if the weight hung on the greater wheel bear the same proportion to the weight hung on the lesser, as the diameter of the lesser wheel bears to the diameter of the greater.
46. It is usual to join many wheels together in the same frame, which by the means of certain teeth, formed in the circumference of each wheel, shall communicate motion to each other. A machine of this nature is represented in fig. 36. Here A B C is a winch, upon which is fixed a small wheel D indented with teeth, which move in the like teeth of a larger wheel E F fixed on the axis G H. Let this axis carry another wheel I, which shall move in like manner a greater wheel K L fixed on the axis M N. Let this axis carry another small wheel O, which after the same manner shall turn about a larger wheel P Q fixed on the roller R S, on which a cord shall be wound, that holds a weight, as T. Now the proportion required between the weight T and a power applied to the winch at A sufficient to support the weight, will most easily be estimated, by computing the proportion, which the velocity of the point A would bear to the velocity of the weight. If the winch be turned round, the point A will describe a circle as A V. Suppose the wheel E F to have ten times the number of teeth, as the wheel D; then the winch must turn round ten times to carry the wheel E F once round. If wheel K L has also ten times the number of teeth, as I, the wheel I must turn round ten times to carry the wheel K L once round; and consequently the winch A B C must turn round an hundred times to turn the wheel K L once round. Lastly, if the wheel P Q has ten times the number of teeth, as the wheel O, the winch must turn about one thousand times in order to turn the wheel P Q, or the roller R S once round. Therefore here the point A must have gone over the circle A V a thousand times, in order to lift the weight T through a space equal to the circumference of the roller R S: whence it follows, that the power applied at A will balance the weight T, if it bear the same proportion to it, as the circumference of the roller to one thousand times the circle A V; or the same proportion as half the thickness of the roller bears to one thousand times A B.
[47.] I shall now explain the effect of the pulley. Let a weight hang by a pulley, as in fig. 37. Here it is evident, that the power A, by which the weight B is supported, must be equal to the weight; for the cord C D is equally strained between them; and if the weight B move, the power A must move with equal velocity. The pulley E has no other effect, than to permit the power A to act in another direction, than it must have done, if it had been directly applied to support the weight without the intervention of any such instrument.