"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 through the arc BD to the height of the horizontal CD. Now gentlemen, you will be pleased to see the ball rise to the horizontal line at the point G, and the same thing also happen if the nail be placed lower as at F, in which case the ball would describe the arc BJ, always terminating its ascent precisely at the line CD. If the nail be placed so low that the length of thread below it does not reach to the height of CD (which would happen if F were nearer B than to the intersection of AB with the horizontal CD), then the thread will wind itself about the nail. This experiment leaves no room for doubt as to the truth of the supposition. For as the two arcs CB, DB are equal and similarly situated, the momentum acquired in the descent of the arc CB is the same as that acquired in the descent of the arc DB; but the momentum acquired at B by the descent through the arc CB is capable of driving up the same moving body through the arc BD; hence also the momentum acquired in the descent DB is equal to that which drives the same moving body through the same arc from B to D, so that in general every momentum acquired in the descent of an arc is equal to that which causes the same moving body to ascend through the same arc; but all the momenta which cause the ascent of all the arcs BD, BG, BJ, are equal since they are made by the same momentum acquired in the descent CB, as the experiment shows: therefore all the momenta acquired in the descent of the arcs DB, GB, JB are equal."
Fig. 43.
The remark relative to the pendulum may be applied to the inclined plane and leads to the law of inertia. We read on page 124:[45]