227. =No Gain of Power in this Lever.=--I will now illustrate the truth that there is no gain or saving of power in this lever, though at first thought it would seem that there is. Let _a b_, Fig. 159, represent a lever, and _e_ its fulcrum. Let the arm _a e_ be twice as long as _e b_. A pound, therefore, suspended from _a_ will balance two pounds at _b_. If, now, when the weights are suspended, the long arm be raised so that the lever shall be in the position represented by the line _c d_, and then let go, the one pound at _c_, balancing the two pounds at _d_, will bring the lever again to the position _a b_. It will be perceived that the end of the long arm of the lever moves through the space _a c_, which is larger than _b d_, through which the end of the short arm moves, in the same time. The one-pound weight, in fact, falls two feet in raising the two-pound weight one foot, and it moves twice as far as a one-pound weight suspended at _i_ would. If a one-pound weight could raise a two-pound weight without thus moving through twice as much space we might then say that there is an actual gain of power in the lever. But it evidently makes no difference whether one pound moves through two feet or two pounds through one foot; the force is the same in both cases. For the momentum or force of a moving body is in proportion to its weight and velocity, § 201; and therefore the pound weight, moving through two feet, has as much momentum as the two-pound weight moving through one foot in the same time. The small weight does the same amount of work that the larger one would by moving twice as far in the same time as the larger, just as a boy, who carries a load half as large as a man, will do as much work as the man if he carry it twice as fast. [Illustration: Fig. 160.] 228. =The See-Saw.=--We see the same thing illustrated in the see-saw, Fig. 160. The man, being much heavier than the boy, is nearer the prop, and as they move up and down the boy passes over a much larger space than the man, describing an arc in a much larger circle. 229. =Archimedes's Lever.=--Archimedes said that if he could have a lever long enough and a prop strong enough he could move the world by his own weight. But he did not think how far he would have to move to do this, from the vast difference between his weight and the weight of the earth. "He would have required," says Dr. Arnot, "to move with the velocity of a cannon-ball for millions of years to alter the position of the earth by a small part of an inch."