addition of pure numbers, and showed that the axioms of arithmetic of the equality of two numbers in respect to a third, the association law of addition, and the commutation law, can only be proved by the agree ment of the results arithmetically derived with those which can be obtained by counting of exterior numerable objects. That the objects should be numerable, certain conditions must be fulfilled concerning whose presence only experience can decide. Since that objects which in any particular respect are alike and can be numbered may be regarded as units of number, the result of their enumeration as a definite number, and the kind of units which compose it as the denomination of the number, the conception with respect to the equality of two groups containing given numbers of objects of the same denomination is given by these numbers. If we designate as quantities objects or attributes of objects which, when compared with similar ones, may be greater, equal, or less, and if we can express these quantities by known numbers, we call these numbers the values of the quantities, and the process by which we find them measurement. Thus we measure a force by the masses and displacements of systems upon which they have been exercised; or in dynamic measurements by the masses and movements of systems upon which they are working; or in the static method of measurement by bringing the forces into equilibrium with others already known. It only remains to consider under what circumstances quantities may be expressed by numbers and what is thereby attained in actual knowledge. With this purpose were instituted interesting and valuable considerations on physical equality and the commutation and association law for physical combination. Addition was regarded as a combination of quantities of the same kind, such that the result remained unchanged when the single elements were exchanged, or when the numbers were replaced by equal quantities of the same kind. In introducing irrational relations Helmholtz placed himself at the standpoint of the physicist, and in later development of the principles of mechanics he retained, as we shall see, the same point of view, showing that in geometry and physics no discontinuous functions are met with for which it is not enough to know with sufficient accuracy the bounds within which the irrational values lie. The mathematicians, however, it must be said, recognize functions of another kind, and the recent investigations of Boltzmann seem to point to a physical application of such analytical conceptions.

We now come to by far the most difficult part of our task as we attempt to describe the service of Helmholtz to analytical mechanics; for in order to understand the partial reconstruction of the science in consequence of some of his most brilliant researches it will be necessary to accurately follow him through his great series of wonderful mathematical-physical investigations and far-reaching physical discoveries in the great fields of hydrodynamics, aerodynamics, and electricity, which have contributed to the investigation of the axioms of mechanics.

Many physiologists of the time assumed, quite in defiance of the laws of mechanical natural philosophy, that through the action of the socalled life force the ordinary natural forces might be generated without limit. In his "Theory of the physiological heat phenomena” Helmholtz, 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 corresponding 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 beautiful 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 equilibrium when for each virtual displacement-that is, a displacement compatible 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 may be 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

intensity of the force divided by the mass of the material point; and (c) the actions of two bodies upon each other are always equal and take place in opposite directions. From these laws there follows, for Newtonian forces at least, and with the assumption of a rigid connection between the points of the system, the principle of d'Alembert which holds sway in the whole province of dynamics. If we designate as supplied forces those which must be made to act at each point in order that it should move if separated from the system as it actually does move, then the principle of d'Alembert asserts that all the supplied forces suffice to maintain equilibrium, and thus furnishes the mathematician a method of determining for any moment the situation of all points of the system, when the constraints of motion of the points, the forces which act upon them, and the place and velocity of one of the points are known for the moment under consideration.

The advance of mechanics in this line was accompanied by the investigation of all the forces of nature-that is to say, the investigation of all the properties of matter-for we can know nothing of these except to recognize the forces which are there in play. After the discovery of these principles of equilibrium and of motion, it was the endeavor of scientists to obtain general laws and relations of motion. One of the most important and far-reaching of these in its consequences was the principle of the conservation of the so-called vis viva. If we define as the vis viva of a material point one-half the product of its mass into the square of its velocity, and the sum total of the vis viva for all points of a system in which the separate particles are connected by ties restraining their free motion as the kinetic energy of the system, then, for any system subject to the conditions of the d'Alembert principle, the increase of the kinetic energy attending the motion of the system from one situation to another is exactly equal to the work done by the various forces during the time interval in which the displacement occurs. If now the work done by the forces of the system during the displacement is dependent only on the initial and final situations, it follows that if the system returns again from the final to the initial condition the kinetic energy returns to its original value. This law is called the law of the conservation of the kinetic energy, and systems to which it applies are called conservative systems. A simple transformation of this law leads to the most far-reaching consequences. The fact that a body by its motion from one place to another does a certain amount of work necessitates that its capacity for performing work, or, in other words, its potential energy, was in its initial situation greater than in its final situation; so that for conservative systems the law of the conservation of kinetic energy goes over into the law of the constancy of energy. This may be expressed as follows: For any conservative system the sum of the potential and kinetic energies is unchangeable. It is important to remember the supposition upon which this is based, namely, that the work done by the motion of the system is dependent only on the initial

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 combination 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 generality, that the conception of potential energy, which had Į reviously 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. Helmholtz, 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 phenomena where heat light, electricity, 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 system 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 constancy 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 metaphysical speculation, who sought to establish the law of the conservation of energy from a priori considerations, declared the law, like all knowledge of the phenomena of the actual world, to be the result of induction,

and based upon the negative issue of numerous futile attempts to construct a perpetual motion. This great general law governing the quantitative relations which must subsist during all transformations, does not, however, determine whether work can be changed into heat without reserve, and vice versa, and the same uncertainty exists with regard to light, electricity, and other forms of energy. These are questions whose answer shall later exhibit the deep and comprehensive significance of the energy conception in mathematical physics.

After Helmholtz had investigated the physical aspects of this fundamental principle of mechanics from most varied points of view, he turned his attention to physiological researches growing out of his notice on the "Theory of acoustics," and from this to very general mechanical problems and special hydrodynamic investigations. In the year 1858 appeared his famous memoir "Upon the integrals of the hydrodynamic equations which correspond to wave motions." This research formed the foundation for an entirely new conception of the motions of fluids, which was later made fruitful in various branches of physics, notably by W. Thomson (Lord Kelvin) in his theory of vortex atoms, and by other physicists as well. Upon the assumption that for a perfect fluidthat is, one in which there is no friction between the particles-the pressure is equal in all directions, Euler and Lagrange had already obtained analytical relations between the pressure in the fluid, its density, the time, the coordinates of the particle under consideration, and, on the one hand, the velocity components; on the other, the position of this particle at the beginning of the motion. Further, they had inferred the so-called continuity equation, which required that the mass of a given particle of the fluid should not change with the time, therefore that the surface of the liquid should be continually composed of the same particles.

All these equations form for the perfect fluid the analogue of the principle of d'Alembert and lead to the determination of the variable quantities through the time and the original situation as a mathematical problem whose solution, to use an expression of Kirchhoff, would describe the motion. The problem can be solved for some particular cases in which the components of the velocity of each fluid particle may be placed equal to the differential coefficients of a determined function, which Helmholtz called the velocity potential, along the corre sponding directions. This function, for incompressible fluids at least, has the same form as the potential of gravitating masses for points outside of them. But such a velocity potential does not always exist, and so Helmholtz attacked the extremely difficult problem of the forms of motion with complete generality in the memoir above referred to, which appeared in the year of his coming to Heidelberg.

First of all, he recognized that the change which an indefinitely small volume of fluid undergoes in an indefinitely small interval of time is composed of three different motions-a displacement of the