IV. The exertion of power is called action, and its intensity, in the material world, depends on the distance of the agent from the patient.
V. The amount of the exertion, or the quantity of the action, is measured by its true effect, which is the only true exponent or representative of the degree of the exertion; for, all matter being equally indifferent to receive motion, the amount of its passion must always agree with the amount of the action received; and thus the one is the natural and necessary measure of the other.
VI. The amount of the exertion, as measured by the effect it is able to produce, is what in the scientific language can be styled force properly.
VII. The amount of the effect, as measuring the amount of the exertion from which it arises, or by which it is neutralized, is again called force, but improperly, and only in a technical sense, as it is in fact a mere measure of force.
These propositions are so logically connected with one another that, the first of them being admitted, all the others must follow. I might, therefore, dispense with all discussion with regard to them; yet, to help the scientific reader to form a philosophical notion of forces, I will endeavor to throw some additional light on my sixth and seventh propositions.
And, first, I observe that since forces can only be measured by their effects, the mathematical expression of a force always exhibits the quantity of the effect which such a force is competent to cause; and as such an effect is a certain quantity of movement, hence forces are mathematically expressed in terms of movement. So long as physicists preserved their old philosophical traditions, a distinction was kept up between force and movement. A quantity of movement was indeed called a force, inasmuch as it was the true measure of the action from which it had originated, or by which it could be destroyed; but such a force was not confounded with the action itself. The action was called vis motrix, a motive force, whilst the quantity of movement was called vis simply, and was not considered as having any efficient causality. Thus before Dr. Mayer's invention of "potential energies," the word force was used with proper discrimination: 1st, as a quantity of action actually producing movement; 2d, as a quantity of action actually opposed by a resistance sufficient to prevent the production of movement; 3d, as a quantity of movement and a measure of action.
A quantity of action followed by movement was called a dynamical force, and was measured by the quantity of movement imparted in the unit of time. Its mathematical expression in rational mechanics was, and is still, a differential coefficient of the second order representing the product of the mass acted on into the velocity which the action, if continued for a unit of time, would communicate to it. As instances of dynamical force, we may mention the action of the sun on the planets, of the planets on their satellites, of the earth on a pendulum, on a drop of rain, etc.
A quantity of action not followed by movement was called a statical force, and was measured by the quantity of movement into which it would develop, if no obstacle existed. Its mathematical expression in rational mechanics is a differential coefficient of the first order representing the product of the mass, whose movement is neutralized into its virtual velocity. By virtual velocity we mean the velocity which the mass would acquire in a unit of time, if all resistance to the movement were suddenly suppressed. As instances of statical force, we may mention the action of a weight on the string from which it hangs, or on the table on which it lies.
A quantity of movement, or the dynamical effect of all the actions to which a body has been subjected for any length of time, was called a kinetic force. As kinetic forces cannot be destroyed except by actions producing equal and opposite quantities of movement, hence every kinetic force can be taken as a measure, not only of the amount of action from which it has resulted, but also of the amount of action by which it can be checked. The mathematical expression of a kinetic force is the product of the moving mass into its actual velocity. As instances of this force, we may mention the momentum of a cannon-ball, of a hammer, of wind, falling water, etc.
To obviate the many abuses which this notion of kinetic force has engendered, and to cut the ground from under the feet of those blundering theorists who reduce all forces to movement, it is important to remark that kinetic force could be defined as "that quantity of action which a moving body can exercise against an obstacle until its velocity is exhausted." This definition would change nothing in the mathematical expression of kinetic forces; for the quantity of the action which a moving body can exercise against the obstacle is exactly equal to the quantity of movement, or momentum, by which the body is animated. The only change would be in the terminology, which, instead of technical, would become philosophical. As instances of kinetic force thus defined, we might mention the quantity of action of a cannon-ball, of the hammer on the anvil, of the wind on the sails, etc.