There is also another condition of moving bodies which may be expected to afford a simple and general measure of their force. When a moving body it opposed by pres sure, by a vis mortv,a, or a resistance like that of gravity, the quantity of such resist ance required to extinguish the motion, and reduce the body to rest, must serve to measure the force of that body. It is a force which, by repeated impulses, has anni hilated another, and these impulses, when properly collected into one sum, must evi dently be equal to the force which they have extinguished. It happens, however, that there are two ways of computing the amount of these retarding forces, which lead to different results, both of them just, and neither of them to be assumed to the exclu sion of the other.
Suppose the body, the force of which is to be measured, to be projected perpendi cularly upward with any velocity, then, if we would compute the quantity of the force of gravity which is employed in reducing it to rest, we may either inquire into the re tardation which that force produces during a given time, or while the body is moving over a given space. In other words, we may either inquire how long the motion will continue, or how far it will carry the body before it be entirely exhausted. If the length of the time that the uniform resistance must act before it reduce the body to rest be taken for the effect, and consequently for the measure of the force of the body, that force must be proportional to the velocity, for to this the time is confessedly proportional. If, on the other hand, the length of the line which the moving body describes, while sub jetted to this uniform resistance, be taken for the effect and the measure of the force, the force must be as the square of the velocity, because to that quantity the line in question is known to be proportional. Here, therefore, are two results, or two values of the same thing, the force of a moving body, which are quite different from one another ; an inconsistency which evidently arises from this, that the thing denoted by the term force, is too vague and indefinite to be capable of measurement, unless some farther condition be annexed. This condition is no other than a specification of the work to be performed, or of the effect to be produced by the action of the moving body. Thus, when to the question concerning the force of the moving body, you add that it is to be employed in putting in motion another body, which is itself free to move, no doubt remains that the force is as the velocity multiplied into the quantity of matter. So also, if the force of the moving body is to be opposed by a resistance like
that of gravity, the length of time that the motion may continue is one of its measur able effects, and that effect is, like the former, proportional to the velocity. There is a third effect to be considered, and one which always occurs in such an experi ment as the last,—the height to which the moving body will ascend. This limitation gives to the force a definite character, and it is now measured by the square of the velocity. In fact, therefore, it is not a precise question to ask, What is the measure of the force of a moving body ? You must, in addition, say, How is the moving body to be employed, or in which of its different capacities is it that you would measure its ef fect ? In this state of the question there is no ambiguity, nor any answer to be given but one. Hence it was that the mathematicians and philosophers who differed so much about the general question of the force of moving bodies, never differed about the particular applications of that force. It was because the condition necessary for limiting the vagueness and ambiguity of the data, in all such cases, was fully sup plied.
In the. argument, therefore, so strenuously maintained on the force of moving bodies, both sides were partly in the right and both partly in the wrong. Each produced a measure of force which was just in certain circumstances, and thus far had truth on his side : but each argued that his was the only true measure, so that all others ought to be rejected; and here each of them was in error. Hence, also, it is not an accurate account of the controversy to say that it was about words merely ; the disputants did indeed misunderstand one another, but their error lay in ascribing ge nerality to propositions that were true only in particular cases, to which indeed the am biguity and vagueness of the word force materially contributed. It does not appear, however, that any good would now accrue from changing the language of dyna mics. If, as has been already said, to the question, How are we to measure the force of a moving body ? be added the nature of the effect which is to be produced, all am. biguity will be avoided.