(216.) The manner in which the time of vibration of a pendulum depends on its length being explained, we are next to consider how this time is affected by the attraction of gravity. It is obvious that, since the pendulum is moved by this attraction, the rapidity of its motion will be increased, if the impelling force receive any augmentation; but it still is to be decided, in what exact proportion the time of oscillation will be diminished by any proposed increase in the intensity of the earth’s attraction. It can be demonstrated mathematically, that the time of one vibration of a pendulum has the same proportion to the time of falling freely in the perpendicular direction, through a height equal to half the length of the pendulum, as the circumference of a circle has to its diameter. Since, therefore, the times of vibration of pendulums are in a fixed proportion to the times of falling freely through spaces equal to the halves of their lengths, it follows that these times have the same relation to the force of attraction as the times of falling freely through their lengths have to that force. If the intensity of the force of gravity were increased in a four-fold proportion, the time of falling through a given height would be diminished in a two-fold proportion; if the intensity were increased to a nine-fold proportion, the time of falling through a given space would be diminished in a three-fold proportion, and so on; the rate of diminution of the time being always as the square root of the increased force. By what has been just stated this law will also be applicable to the vibration of pendulums. Any increase in the intensity of the force of gravity would cause a given pendulum to vibrate more rapidly, and the increased rapidity of the vibration would be in the same proportion as the square root of the increased intensity of the force of gravity. (217.) The laws which regulate the times of vibration of pendulums in relation to one another being well understood, the whole theory of these instruments will be completed, when the method of ascertaining the actual time of vibration of any pendulum, in reference to its length, has been explained. In such an investigation, the two elements to be determined are, 1. the exact time of a single vibration, and, 2. the exact distance of the centre of oscillation from the point of suspension. The former is ascertained by putting a pendulum in motion in the presence of a good chronometer, and observing precisely the number of oscillations which are made in any proposed number of hours. The entire time during which the pendulum swings, being divided by the number of oscillations made during that time, the exact time of one oscillation will be obtained. The distance of the centre of oscillation from the point of suspension may be rendered a matter of easy calculation, by giving a certain uniform figure and material to the pendulous body.