THE DEVELOPMENT OF MECHANICS 271 



of development. Energy replaced force as a fundamental idea, and a 

 new system of mechanics resulted, founded on the relations between 

 space, time, mass and energy, as embodied in Hamilton's principle. 



Although a comparatively short time has elapsed since the estab- 

 lishment of energetics as the basis of mechanics, a third stage of de- 

 velopment is already clearly marked. To characterize each stage by a 

 single word, the first may be called constructive, the second deductive 

 and the third, or present stage, critical. To the founders of the first 

 two systems the concepts of force and mass, although more artificial 

 than the intuitive ideas of space and time, were probably no less axio- 

 matic. With the growth of modern scientific criticism, however, came 

 the desire to go back of intuition, if that be possible, and subject the 

 foundations of science to the last analysis. As the result of this tend- 

 ency the foundations of the first two systems were found open to certain 

 objections, which have been admirably expressed by the late Heinrich 

 Hertz. The chief objection to the first system is in relation to the 

 idea of force, any definition of which seems to involve its author in 

 certain logical difficulties somewhat similar to those encountered in 

 attempting to define a straight line. In the second system, criticism 

 is aimed not at the fundamental concepts, but at the relation between 

 them as expressed in Hamilton's principle, the objections to which are 

 twofold : namely, that it has no simple, natural interpretation, and that 

 it seems to endow matter with the attributes of thought and volition. 

 A further objection is made to both systems on the ground of a certain 

 redundancy in the fundamental ideas, three fundamental concepts being 

 both necessary and sufficient, according to Kirchhoff, for the develop- 

 ment of a complete system of mechanics. 



In view of these and other objections, Hertz and his followers have 

 outlined an ideal system of mechanics based upon three elements only : 

 namely, space, time and mass. To supplement the deficiency caused 

 by the lack of a fourth element without increasing the number of funda- 

 mental concepts, Hertz has introduced the idea of concealed motions 

 acting in connection with those visible to the senses. This idea was 

 originated by Lord Kelvin in his theory of vortex atoms, and was 

 further developed by Maxwell in his attempt to explain electro- 

 magnetic action. The first complete treatment of concealed motions, 

 however, was given by Helmholtz, and in the hands of his pupil Hertz 

 it has proved a powerful instrument in establishing mechanics upon 

 a more satisfactory basis. 



What the future of mechanics may be it is of course impossible 

 to predict. However, the brief review of its development that has 

 just been given suggests that the foundations have reached bed rock, 

 and that future effort must be directed toward the enlargement of the 

 superstructure and its adaptation to the growing needs of humanity. 



