Galileo, as we know, arrived at the law of uniformly accelerated motion by _a priori_ considerations, as that law which was the "simplest and most natural," after having first assumed a different law which he was compelled to reject. To verify his law he executed experiments with falling bodies on inclined planes, measuring the times of descent by the weights of the water which flowed out of a small orifice in a large vessel. In this experiment he assumes as a fundamental principle, that the velocity acquired in descent down an inclined plane always corresponds to the vertical height descended through, a conclusion which for him is the immediate outcome of the fact that a body which has fallen down one inclined plane can, with the velocity it has acquired, rise on another plane of any inclination only to the same vertical height. This principle of the height of ascent also led him, as it seems, to the law of inertia. Let us hear his own masterful words in the _Dialogo terzo_ (_Opere_, Padova, 1744, Tom. III). On page 96 we read: "I take it for granted that the velocities acquired by a body in descent down planes of different inclinations are equal if the heights of those planes are equal."[43] Then he makes Salviati say in the dialogue:[44] "What you say seems very probable, but I wish to go further and by an experiment so to increase the probability of it that it shall amount almost to absolute demonstration. Suppose this sheet of paper to be a vertical wall, and from a nail driven in it a ball of lead weighing two or three ounces to hang by a very fine thread _AB_ four or five feet long. (Fig. 43.) On the wall mark a horizontal line _DC_ perpendicular to the vertical _AB_, which latter ought to hang about two inches from the wall. If now the thread _AB_ with the ball attached take the position _AC_ and the ball be let go, you will see the ball first descend through the arc _CB_ and passing beyond _B_ rise through the arc _BD_ almost to the level of the line _CD_, being prevented from reaching it exactly by the resistance of the air and of the thread. From this we may truly conclude that its impetus at the point _B_, acquired by its descent through the arc _CB_, is sufficient to urge it through a similar arc _BD_ to the same height. Having performed this experiment and repeated it several times, let us drive in the wall, in the projection of the vertical _AB_, as at _E_ or at _F_, a nail five or six inches long, so that the thread _AC_, carrying as before the ball through the arc _CB_, at the moment it reaches the position _AB_, shall strike the nail _E_, and the ball be thus compelled to move up the arc _BG_ described about _E_ as centre. Then we shall see what the same impetus will here accomplish, acquired now as before at the same point _B_, which then drove the same moving body