492 Prof. Helmholtz ow the Interaction of Natural Forces. 



laboiirer is valued more highly than a weak one), partly, how- 

 ever, by the skill which is brought into action. A machine, on 

 the contrary, which executes work skilfully, can always be 

 multiplied to any extent ; hence its skill has not the high value 

 of human skill in domains where the latter cannot be supplied 

 by machines. Thus the idea of the quantity of work in the case 

 of machines has been limited to the consideration of the expendi- 

 ture of force; this was the more important, as indeed most 

 machines are constructed for the express purpose of exceeding, 

 by the magnitude of their effects, the powers of men and animals. 

 Hence, in a mechanical sense, the idea of work is become iden- 

 tical with that of the expenditure of force, and in this way I will 

 apply it in the following pages. 



How, then, can we measure this expenditure, 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 will here choose, namely that of a water-mill 

 with iron hammer, appears to be tolerably romantic, still, alas, 

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

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

 tion for 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 has at certain places small projections, thumbs, which, 

 during the rotation, lift the heavy hammer and permit it to 

 fall again. The falling hanmier belabours the mass of metal, 

 which is introduced beneath it. The work therefore done by 

 the machine consists, in 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 jilace, other circumstances 

 being equal, be proportional to the Aveight of the hammer; it 

 will, for example, be double when the weight of the hammer is 

 doubled. But tlie 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, howevei', clear that if the 

 machine, with a certain 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 Avork 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 pro- 

 duct of the weight into the space through which it ascends. And 



