FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 101 



Many physiologists of the time assumed, quite in defiance of the laws 

 of mechanical natural philosophy, that through the action of the so- 

 called life force the ordinary natural forces might be generated without 

 limit. In his "Theory of the physiological heat phenomena " Helm- 

 holtz, starting from the proposition based on mechanical laws that 

 a given quantity of a moving force can never by any complication 

 of mechanism produce more than a definite correspouding quantity of 

 motion, proceeded to discuss the question of the source of animal 

 heat — so weighty in the theoretical consideration of life processes. 

 The results of this investigation and of that simultaneously instituted 

 " Upon the evolution of heat attending muscular action" gave him the 

 verification of the great law of the conservation of energy which formed 

 the subject of a lecture before the Physical Society of Berlin in the 

 year 1847. It was certainly an interesting moment in the history of 

 the sciences when thirty years ago to-day one of the most distinguished 

 physicists of this century, Gustav Kirchhoff, in the course of his beauti- 

 ful and luminous discourse " On the goal of the natural sciences," 

 delivered from this spot, and in the presence of Helmholtz, declared 

 the discovery of this law undoubtedly the most momentous which has 

 been made in the province of natural science during the present century. 

 Hertz also, in his posthumous work, " The Principles of Mechanics," 

 asserts that physics at the end of our century has turned its preference 

 to an entirely new method of thought, and, influenced by the tremendous 

 impression made by Helmholtz's discovery of the constancy of energy, 

 it is now preferred to refer all phenomena in their analyses to the laws 

 of transformation of energy. For the sake of a proper appreciation 

 of this great discovery of Helmholtz, as well as of his later fundamental 

 researches upon the principles of mechanics, I must here briefly review 

 the historical development of theoretical mechanics. 



From the early investigations of the lever, the pulley, and the 

 inclined plane, there were soon developed the general conceptions 

 which are the basis of the science of equilibrium. With the definition 

 of work as the product of a force by the infinitely small displacement 

 of a material particle along the direction in which the force is measured 

 arose the principle of virtual velocities, upon which rests the theory of 

 statics. According to this principle, a material system is in equilib- 

 rium when for each virtual displacement — that is, a displacement com- 

 patible with the connection of the system of points — the total work 

 done within the system is equal to zero. After the discovery of the 

 inertia of masses by Galileo, and the conception of gravitation by 

 Newton, the development of mechanics was founded upon the three 

 famous laws of Newton. These maybe stated as follows: (a) Every 

 body remains in a state of rest or of uniform motion in a straight line 

 unless compelled by outside force to change that state; (b) the accele- 

 ration of a material point by the action of a constraining force takes 

 place in the direction in which the force acts and is equal to the 



