(210.) It has been stated in (117.) that the attraction of gravity affects all bodies equally, and moves them with the same velocity, whatever be the nature or quantity of the materials of which they are composed. Since it is the force of gravity which moves the pendulum, we should therefore expect that the circumstances of that motion should not be affected either by the quantity or quality of the pendulous body. And we find this, in fact, to be the case; for if small pieces of different heavy substances such as lead, brass, ivory, &c., be suspended by fine threads of equal length, they will vibrate in the same time, provided their weights bear a considerable proportion to the atmospherical resistance, or that they be suspended _in vacuo_. (211.) Since the time of vibration of a pendulum, which oscillates in small arcs, depends neither on the magnitude of the arc of vibration nor on the quality or weight of the pendulous body, it will be necessary to explain the circumstances on which the variation of this time depends. The first and most striking of these circumstances is the length of the suspending thread. The rudest experiments will demonstrate the fact, that every increase in the length of this thread will produce a corresponding increase in the time of vibration; but according to what law does this increase proceed? If the length of the thread be doubled or trebled, will the time of vibration also be increased in a double or treble proportion? This problem is capable of exact mathematical solution, and the result shows that the time of vibration increases not in the proportion of the increased length of the thread, but as the square root of that length; that is to say, if the length of the thread be increased in a four-fold proportion, the time of vibration will be augmented in a two-fold proportion. If the thread be increased to nine times its length, the time of vibration will be trebled, and so on. This relation is exactly the same as that which was proved to subsist between the spaces through which a body falls freely, and the times of fall. In the table, page 89, if the figures representing the height be understood to express the length of different pendulums, the figures immediately above them will express the corresponding times of vibration. This law of the proportion of the lengths of pendulums to the squares of the time of vibration may be experimentally established in the following manner:--