10. _Controversy of the Measure of Force._--In the History of Mechanics[36\3], we have given an account of the controversy which, for some time, occupied the mathematicians of Europe, whether the forces of bodies in motion should be reckoned proportional to the velocity, or to the square of the velocity. We need not here recall the events of this dispute; but we may remark, that its history, as a metaphysical controversy, is remarkable in this respect, that it has been finally and completely settled; for it is now agreed among mathematicians that both sides were right, and that the results of mechanical action may be expressed with equal correctness by means of _momentum_ and of _vis viva_. It is, in one sense, as D'Alembert has said[37\3], a dispute about words; but we are not to infer that, on that account, it was frivolous or useless; for such disputes are one principal means of reducing the principles of our {261} knowledge to their utmost simplicity and clearness. The terms which are employed in the science of mechanics are now liberated for ever, in the minds of mathematicians, from that ambiguity which was the battleground in the war of the _vis viva_. [Note 36\3: B. vi. c. v. sect. 2.] [Note 37\3: D'Alembert has also remarked (_Dynamique_, Pref. xxi.) that this controversy 'shows how little justice and precision there is in the pretended axiom that causes are proportional to their effects.' But this reflection is by no means well founded. For since both measures are true, it appears that causes may be _justly_ measured by their effects, even when very different kinds of effects are taken. That the axiom does not point out one _precise_ measure, till illustrated by experience or by other considerations, we grant: but the same thing occurs in the application of other axioms also.] But we may observe that the real reason of this controversy was exactly that tendency which we have been noticing;--the disposition of man to assume in his speculations certain general propositions as true, and to fix the sense of terms so that they shall fall in with this truth. It was agreed, on all hands, that in the mutual action of bodies the same quantity of force is always preserved; and the question was, by which of the two measures this rule could best be verified. We see, therefore, that the dispute was not concerning a definition merely, but concerning a definition combined with a general proposition. Such a question may be readily conceived to have been by no means unimportant; and we may remark, in passing, that such controversies, although they are commonly afterwards stigmatized as quarrels about words and definitions, are, in reality, events of considerable consequence in the history of science; since they dissipate all ambiguity and vagueness in the use of terms, and bring into view the conditions under which the fundamental principles of our knowledge can be most clearly and simply presented.