116 FUNDAMENTAL PEINCIPLES OF MATHEMATICS. 



forces of attraction were common to all natural forces, while on the 

 other hand it had been doubted whether, for example, the application 

 of the principle of the equality of action and reaction was generally 

 permissible. As already pointed out, the hypothesis had been made that 

 action at a distance was resolvable into continuous dynamic reactions 

 in an invisible intervening medium, thus establishing an analogy with 

 the role of a spring or cord in the transmission of force. Since, how- 

 ever, it is the province of physics to refer the phenomena of nature to 

 the simple laws of mechanics, the question arises first of all what con- 

 stitute the first principles of mechanics, and what are, as Hertz has 

 said, the final and simplest laws which each natural motion must obey, 

 which no motion can ignore whose presence in nature is determined by 

 our everyday experience, and from which as the fundamental principles 

 of mechanics the whole science may be deduced without further refer- 

 ence to observation. 



Until the pioneer researches of Helmholtz on the conservation of 

 energy, mechanics, as has been remarked, following Galileo's concep- 

 tion of the inertia of masses, had been developed by application of 

 the three laws of Newton. When, however, the whole structure was 

 systematically and critically examined, a want of clearness was appar- 

 ent in the definition of mechanical quantities, and the proofs of funda- 

 mental laws of statics, such as the laws of the jiarallelogram of forces 

 and of virtual velocities, were found to be not altogether rigorous. 

 Knowledge of the action of forces at a distance and of molecular, 

 chemical, electrical, and magnetic forces w T as purely empirical. 



The discovery of the principle of the conservation of energy made 

 possible a consistent development of theoretical mechanics. The idea 

 of force became less prominent, while mass and energy came forward as 

 the indestructible physical quantities. Energy is present in two great 

 divisions, of which the one — kinetic energy — is in all cases given by a 

 constant function of the velocity of masses, while the other — the poten- 

 tial energy — is determined by the relative position of the masses, but 

 must be derived in each case with consideration of their particular 

 nature. The discussion of the different forms of energy, as well as of 

 their mutual transformation, forms the subject of both physics and 

 chemistry. In expressing the progress of phenomena as a function of 

 the time, Helmholtz did not, like most of his x>redecessors, make the 

 equations of motion the starting point from which to derive the genera 

 principles of mechanics, because in this method it becomes necessary 

 to make certain assumptions regarding the forces operating and 

 regarding the limiting conditions of the problem, and these limi- 

 tations exclude from the consideration a large number of possible 

 motions. He proceeded, on the other hand, from the principle of least 

 action, and by this means brought into the discussion many examples 

 of relations between forces found in nature, but not occurring in 

 treatises in which the former method is pursued, 



