64. NOW in this pendulum, as all the swings, whether long or short, will be performed in the same time; so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the circumference of a circle bears to its diameter.
65. IT may from hence be understood in some measure, why, when pendulums swing in circular arches, the times of their swings are nearly equal, if the arches are small, though those arches be of very unequal lengths; for if with the semidiameter L K the circular arch O K P be described, this arch in the lower part of it will differ very little from the line C K H.
66. IT may not be amiss here to remark, that a body will fall in this line C K H (fig. 53.) from C to any other point, as Q or R in a shorter space of time, than if it moved through the straight line drawn from C to the other point; or through any other line whatever, that can be drawn between these two points.
67. BUT as I have observed, that the time, which a pendulum takes in swinging, depends upon its length; I shall now say something concerning the way, in which this length of the pendulum is to be estimated. If the whole ball of the pendulum could be crouded into one point, this length, by which the motion of the pendulum is to be computed, would be the length of the string or rod. But the ball of the pendulum must have a sensible magnitude, and the several parts of this ball will not move with the same degree of swiftness; for those parts, which are farthest from the point, whereon the pendulum is suspended, must move with the greatest velocity. Therefore to know the time in which the pendulum swings, it is necessary to find that point of the ball, which moves with the same degree of velocity, as if the whole ball were to be contracted into that point.
68. THIS point is not the center of gravity, as I shall now endeavour to shew. Suppose the pendulum A B (in fig. 54.) composed of an inflexible rod A C and ball C B, to be fixed on the point A, and lifted up into an horizontal situation. Here if the rod were not fixed to the point A, the body C B would descend directly with the whole force of its weight; and each part of the body would move down with the same degree of swiftness. But when the rod is fixed at the point A, the body must fall after another manner; for the parts of the body must move with different degrees of velocity, the parts more remote from A descending with a swifter motion, than the parts nearer to A; so that the body will receive a kind of rolling motion while it descends. But it has been observed above, that the effect of gravity upon any body is the same, as if the whole force were exerted on the body’s center of gravity[64].