Accordingly, that part of the Third Law of Motion which we are now considering, that the velocity generated is as the force, was obtained, in fact, by a separate train of research. The first exemplification of this {251} law which was studied by mathematicians, was the motion of bodies upon inclined planes: for the force which urges a body down an inclined plane is known by statics, and hence the velocity of its descent was to be determined. Galileo originally[30\3] in his attempts to solve this problem of the descent of a body down an inclined plane, did not proceed from the principle which we have stated, (the determination of the force which acts down the inclined plane from statical considerations,) obvious as it may seem; but assumed, as we have already seen, a proposition apparently far more precarious;--namely, that a body sliding down a smooth inclined plane acquires always the same velocity, so long as the _vertical_ height fallen through is the same. And this conjecture (for at first it was nothing more than a conjecture) he confirmed by an ingenious experiment; in which bodies acquired or lost the same velocity by descending or ascending through the same height, although their paths were different in other respects. [Note 30\3: _Dial. della Sc. Nuov._ iii. p. 96. See _Hist. Ind. Sci._ b. vi. c. ii. sect. 5.] This was the form in which the doctrine of the motion of bodies down inclined planes was at first presented in Galileo's _Dialogues_ on the Science of Motion. But his disciple Viviani was dissatisfied with the assumption thus introduced; and in succeeding editions of the _Dialogues_, the apparent chasm in the reasoning was much narrowed, by making the proof depend upon a principle nearly identical with the third law of motion as we have just stated it. In the proof thus added, 'We are agreed,' says the interlocutor[31\3], 'that in a moving body the impetus, energy, momentum, or propension to motion, is as great as is the force or least resistance which suffices to sustain it;' and the impetus or momentum, in the course of the proof, being taken to be as the velocity produced in a given time, it is manifest that the principle so stated amounts to this; that the velocity produced is as the statical force. And thus this law of motion appears, {252} in the school of Galileo, to have been suggested and established at first by experiment, but afterwards confirmed and demonstrated by _à priori_ considerations. [Note 31\3: _Dialogo_, p. 104.]