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.