Again, if we withdraw any of the pendulums from its position of rest and let it swing, we shall find that in any stated period of time, say 1 minute, it executes a certain definite number of oscillations which is peculiar to itself. You might imagine that, by withdrawing it more or less from its position of rest, and making it swing over a larger or smaller distance, you could make these swings per minute more or less as you please. But you would find, on trying the experiment, that this is not the case, and that, provided the arc of vibration is not too great, the time of one complete swing to and fro is the same whether the swing be large or small. In scientific language this is called the _isochronism of the pendulum_, and is said to have been discovered by Galileo in the Cathedral at Pisa, when watching the swings of a chandelier die away, whilst counting their number by the beats of his pulse. This periodic time of vibration, which is independent of the amplitude of vibration, provided the latter is small, is called the natural time of vibration of the pendulum, or its _free periodic time_. In the case of the simple pendulum the free periodic time is proportional to the square root of the length of the pendulum. Accordingly, a short pendulum makes more swings per minute than a long one, and this rate of swinging is quite independent of the weight of the bob. We can, of course, take hold of the bob with our hand and force it to vibrate in any period we please, and thus produce a _forced vibration_; but a _free_ vibration, or one which is unforced, has a natural time-period of its own. In order that any body may vibrate when displaced and then set free, two conditions must exist. In the first place, there must be a controlling force tending to make the substance return to its original position when displaced. In the second place, the thing moved must have mass or inertia, and when displaced and allowed to return it must in consequence overshoot the mark, and acquire a displacement in an opposite direction. In the case of the pendulum the elastic control or restoring force is the weight of the bob, which makes it always try to occupy the lowest position. We can, however, make a pendulum of another kind. Here, for instance, is a heavy ball suspended by a spiral spring (see Fig. 53). If I pull the ball down a little, and then let it go, it jumps up and down, and executes vertical vibrations. The elastic control here is the spring which resists extension. In this instance, also, there is a natural free time of vibration, independent of the extent of the motion, but dependent upon the weight of the ball and the stiffness of the spring. [Illustration: FIG. 53.]