FUNDAMENTAL PRINCIPLES OP MATHEMATICS. 103 



and final situations and not upon the intermediate positions. For the 

 Newtonian forces, to which this principle is immediately applicable, the 

 validity of this law means the impossibility of a perpetual motion; that 

 is, the combinatiou of natural bodies in such manner as to continually 

 generate force without expenditure of work. For in the absence of this 

 law we would be able, by a selection of the method by which we caused 

 the system to return to its original condition, to save up some of the 

 work done in its displacement, and thus, by repetition of the process, to 

 generate mechanical energy out of nothing forever. The impossibility 

 of such contrivances was long known, and the law of the constancy of 

 energy established for forces of this kind; but not all the forces of 

 nature seemed thus controlled. If a system moved through the same 

 path first without, then with friction, the kinetic energy would, in the 

 latter case, be diminished, owing to the smaller velocity, and thus it 

 was necessary, to sustain the law of the constancy of energy in its gen- 

 erality, that the conception of potential energy, which had previously 

 meant only energy of position, should be extended to other forms of 

 energy, such as those which exist in heat and other natural phenomena. 



In the case just cited, the loss in mechanical energy would require to 

 be compensated by an equivalent quantity of heat energy developed 

 by the friction. R. Mayer, starting with the presumption that the 

 creation or annihilation of force is a matter lying outside the scope of 

 human conception or achievement, asserted the equivalence of heat and 

 mechanical work as the fundamental law of natural phenomena. Helm- 

 holtz, without knowledge of Mayer's researches, and including all 

 natural forces within the circle of his investigations, followed out the 

 assumption of the validity of the law of the constancy of energy, and 

 was able to show, experimentally, the impossibility of a perpetual 

 motion for a great series of physical phenomeua where heat, light, elec- 

 tricity, and chemical affinity enter as acting forces. This is equivalent 

 to the law that the work done by natural forces of all kinds during the 

 passage of a system from one condition to another is dependent solely 

 on the initial and final conditions without regard to the way in which 

 the change is affected. From this he inferred that in any closed sys- 

 tem every increase of energy involves an equal loss of energy, and thus 

 achieved the great and comprehensive result that the energy of the 

 world is constant. 



With his customary remarkable modesty he emphasized the fact that 

 it was his purpose simply to lay before physicists, in as complete form 

 as possible, the theoretical and practical importance of the law of the con- 

 stancy of energy, " whose complete verification must be regarded as one 

 of the chief objects of physics in the immediate future." It deserves 

 special emphasis that Helmholtz, in opposition to the followers of meta- 

 physical speculation, who sought to establish the law of the conservation 

 of energy from a priori considerations, declared the law, like all knowl- 

 edge of the phenomena of the actual world, to be the result of induction, 



