or 



192 H. Helmholtx on the Intei^action of Natural Forces, 



How, then, can we measure tliis expenditurej and compare it 

 in the case of different machines? 



I must here conduct you a portion of the way — as short a 

 portion as possible — over the uninviting field of mathematico- 

 mechanical ideas, in order to bring you to a point of view from 

 which a more rewarding prospect will open. And though the 

 example which I shall here choose, namelyj that of a water-mill 

 with iron hammer, appears to be tolerably romantic, still, ala?, 

 I must leave the dark forest valley, the spark-emitting anvil, and 

 the black Cyclops wholly out of sight, and beg a moment's at- 

 tention to the less poetic side of the question, namely, the ma- 

 chinery. This is driven by a water-wheel, which in its turn is 

 set in motion by the falling water. The axle of the water-wheel 

 bas at certain places small projections, thumbs, which, durin^ 

 the rotation, lift the heavy hammer and permit it to fall again. 

 The falling hammer belabors the mass of metal, which is intro- 

 duced beneath it ^ The work therefore done by the machine 

 consists, m this case, in the lifting of the hammer, to do which 

 the gravity of the latter must be overcome. The expenditure 

 of force will in the first ^place, other circumstances -being equal, 

 be proportional to the weight of the hammer; it will, for exam- 

 ple, be^double when the weight of the hammer is doubled. But 

 the action of the hammer depends not upon its weight alone, but 

 also upon the height from which it falls. If it falls through two 

 feet, it will produce a greater effect than if it falls through only 

 one foot. It is, however, clear that if the machine, with a cer- 

 tain expenditure of force, lifts the hammer a foot in> height, the 

 same amount of force must be expended to raise it a second foot 

 in height. The work is therefore not only doubled when the 

 weight of the hammer is increased twofold, but also when the 

 space through which it falls is doubled. From this it is easy to 

 see that the work must be measured by the product of the weight 

 into the space through which it ascends. And in this way, in- 

 deed, do we measure in mechanics. The unit of work is a foot- 

 pound, that is, a pound weight raised to the height of one foot. 



While the work in this case consists in the raising of the 



m 



heavy hammer-head, the driving force which sets the Tatter in 



motion is generated by falling water. It is not necessary that ^ 



the water should fall vertically, it can also flow in a moderately 

 inclined bed; but it must always, where it has water-mills to 

 set in motion, move from a higher to a lower position. Experi- 

 ment and theory coincide in teaching, that when a hammer of a 

 hundred weight is to be raised one foot, to accomplish this at 

 least a hundred weight of water must fall through the space of 

 one foot; or what is equivalent to this, two hundred weight 

 must fall half a foot, or four hundred weight a quarter of a foot, 

 &c* In short, if we multiply the weight of the falling water by 



