39. FARTHER a lever may be hung upon an axis, and then the two arms of the lever need not be continuous, but fixed to different parts of this axis; as in fig. 33, where the axis A B is supported by its two extremities A and B. To this axis one arm of the lever is fixed at the point C, the other at the point D. Now here, if a weight be hung at E, the extremity of that arm, which is fixed to the axis at the point C; and another weight be hung at F, the extremity of the arm, which is fixed on the axis at D; then these weights will equiponderate, when the weight at E bears the same proportion to the weight at F, as the arm D F bears to C E. 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.