When the length of the thread and the intensity of the force of gravity are given, the time of vibration depends on the length of the arc A C, or on the magnitude of the angle A O C. If, however, this angle do not exceed a certain limit of magnitude, the time of vibration will be subject to no sensible variation, however that angle may vary. Thus the time of oscillation will be the same, whether the angle A O C be 2°, or 1° 30′, or 1°, or any lesser magnitude. This property of a pendulum is expressed by the word _isochronism_. The strict demonstration of this property depends on mathematical principles, the details of which would not be suitable to the present treatise. It is not difficult, however, to explain generally how it happens that the same pendulum will swing through greater and smaller arcs of vibration in the same time. If it swing from A, the force of gravity at the commencement of its motion impels it with an effect depending on the obliquity of the lines O A and A V. If it commence its motion from _a_, the impelling effect from the force of gravity will be considerably less than at A; consequently, the pendulum begins to move at a slower rate, when it swings from _a_ than when it moves from A: the greater magnitude of the swing is therefore compensated by the increased velocity, so that the greater and the smaller arcs of vibration are moved through in the same time. (209.) To establish this property experimentally, it is only necessary to suspend a small ball of metal, or other heavy substance, by a flexible thread, and to put it in a state of vibration, the entire arc of vibration not exceeding 4° or 5°, the friction on the point of suspension and other causes will gradually diminish the arc of vibration, so that after the lapse of some hours it will be so small, that the motion will scarcely be discerned without microscopic aid. If the vibration of this pendulum be observed in reference to a correct timekeeper, at the commencement, at the middle, and towards the end of its motion, the rate will be found to suffer no sensible change. This remarkable law of isochronism was one of the earliest discoveries of Galileo. It is said, that when very young, he observed a chandelier suspended from the roof of a church in Pisa swinging with a pendulous motion, and was struck with the uniformity of the rate even when the extent of the swing was subject to evident variation.