270 POPULAR SCIENCE MONTHLY 



mechanics verifies this statement, for since the time of d'Alembert no 

 essentially new principle has been discovered, and Gauss may be quoted 

 as authority for saying that none ever can be. 



The second stage of development was characterized by the elabora- 

 tion of the system of mechanics formulated by Archimedes, Galileo, 

 Newton and d'Alembert. In the course of this process a new view of 

 the fundamental ideas underlying the subject was attained, which 

 resulted in establishing mechanics upon an entirely different basis. 

 The first step in this direction was made by Euler, and consisted in 

 replacing the geometrical methods of Newton and his predecessors by 

 those of analysis. Euler thus laid the foundation for a system of 

 analytical mechanics which was brought to its perfection by Lagrange 

 in his generalized equations of motion. 



This new representation of mechanics was followed by the estab- 

 lishment in the early part of the last century of two great principles ; 

 the principle of least action and the principle of the conservation of 

 energy. It is important to note in this connection, however, that each 

 of these principles is deducible from that of d'Alembert, and, conse- 

 quently, that their establishment did not increase the number of 

 independent fundamental postulates. 



The first of these principles dates back to the attempt of Maupertuis 

 to establish on theological grounds a principle of similar nature but 

 of much more limited scope. This attempt, although fruitless in itself, 

 served to direct thought in a new channel, and finally led Gauss to 

 the statement of his ' Principle of Least Constraint.' This in turn 

 led investigators to the idea that all natural phenomena present a 

 maximum or a minimum, and induced Euler and Jacobi to seek ex- 

 pressions whose conditions for a minimum would give the equations of 

 motion. From this it was but a step to the establishment of Hamilton's 

 principle, which consists in the analytical statement that the variations 

 of work and energy vanish for the initial and final configurations. As 

 Hamilton's principle includes both conservative and non-conservative 

 systems, it constitutes a generalization of the principle of least action. 



This second principle, like the first, was the product of evolution, 

 as the ideas underlying it had been the subject of investigation from 

 the time of Leibnitz and Descartes. The principle did not assume 

 definite form, however, until the middle of the nineteenth century, 

 when it was stated by several investigators almost simultaneously as 

 the law of the conservation of energy. The names most closely asso- 

 ciated with this principle are those of Mayer, Joule and Helmholtz, 

 and it is curious to note that each of these scientists arrived at his 

 results by a different process ; Mayer by philosophical reasoning, Joule 

 by experimentation and Helmholtz by mathematical analysis. 



The establishment of this law marked the close of the second stasre 



