But here, as in other cases, the question occurs again; Is experiment really requisite for the proof of this law? If we look to authorities, we shall be not a little embarrassed to decide. D'Alembert is against the necessity of experimental proof. 'Why,' says he[32\3], 'should we have recourse to this principle employed, at the present day, by everybody, that the force is proportional to the velocity? ... a principle resting solely upon this vague and obscure axiom, that the effect is proportional to the cause. We shall not examine here,' he adds, 'if this principle is necessarily true; we shall only avow that the proofs which have hitherto been adduced do not appear to us unexceptionable: nor shall we, with some geometers, adopt it as a purely contingent truth; which would be to ruin the certainty of mechanics, and to reduce it to be nothing more than an experimental science. We shall content ourselves with observing,' he proceeds, 'that certain or doubtful, clear or obscure, it is useless in mechanics, and consequently ought to be banished from the science.' Though D'Alembert rejects the third law of motion in this form, he accepts one of {254} equivalent import, which appears to him to possess axiomatic certainty; and this procedure is in consistence with the course which he takes, of claiming for the science of mechanics more than mere experimental truth. On the contrary, Laplace considers this third law as established by experiment. 'Is the force,' he says'[33\3], 'proportioned to the velocity? This,' he replies, 'we cannot know _à priori_, seeing that we are in ignorance of the nature of moving force: we must therefore, for this purpose, recur to experience; for all which is not a necessary consequence of the few data we have respecting the nature of things, is, for us, only a result of observation.' And again he says[34\3], 'Here, then, we have two laws of motion,--the law of inertia [the first law of motion], and the law of the force proportional to the velocity,--which are given by observation. They are the most natural and the most simple laws which we can imagine, and without doubt they flow from the very nature of matter; but this nature being unknown, they are, for us, only observed facts: the only ones, however, which Mechanics borrows from experience.' [Note 32\3: _Dynamique_, Pref. p. x.] [Note 33\3: _Méc. Cél._ p. 15.] [Note 34\3: p. 18.]