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].
Since therefore the power of gravity in drawing down the body must also communicate to it the rolling motion just described; it seems evident, that the center of gravity of the body cannot be drawn down as swiftly, as when the power of gravity has no other effect to produce on the body, than merely to draw it downward. If therefore the whole matter of the body C B could be crouded into its center of gravity, so that being united into one point, this rolling motion here mentioned might give no hindrance to its descent; this center would descend faster, than it can now do. And the point, which now descends as fast, as if the whole matter or the body C B were crouded into it, will be farther removed from the point A, than the center of gravity of the body C B.
69. Again, suppose the pendulum A B (in fig. 55.) to hang obliquely. Here the power of gravity will operate less upon the ball of the pendulum, than before: but the line D E being drawn so, as to stand perpendicular to the rod A C of the pendulum; the force of gravity upon the body C B, now it is in this situation, will produce the same effect, as if the body were to glide down an inclined plane in the position of D E. But here the motion of the body, when the rod is fixed to the point A, will not be equal to the uninterrupted descent of the body down this plane; for the body will here also receive the same kind of rotation in its motion, as before; so that the motion of the center of gravity will in like manner be retarded; and the point, which here descends with that degree of swiftness, which the body would have, if not hindered by being fixed to the point A; that is, the point, which descends as fast, as if the whole body were crouded into it, will be as far removed from the point A, as before.
70. This point, by which the length of the pendulum is to be estimated, is called the center of oscillation. And the mathematicians have laid down general directions, whereby to find this center in all bodies. If the globe A B (in fig. 56.) be hung by the string C D, whose weight need not be regarded, the center of oscillation is found thus. Let the straight line drawn from C to D be continued through the globe to F. That it will pass through the center of the globe is evident. Suppose E to be this center of the globe; and take the line G of such a length, that it shall bear the same proportion to E D, as E D bears to E C. Then E H being made equal to ⅖ of G, the point H shall be the center of oscillation[65]. If the weight of the rod C D is too considerable to be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part of C D; and take K in the same proportion to C I, as the weight of the globe A B to the weight of the rod C D. Then having found H, the center of oscillation of the globe, as before, divide I K in I, so that I L shall bear the same proportion to L H, as the line C H bears to K; and L shall be the center of oscillation of the whole pendulum.
71. This computation is made upon supposition, that the center of oscillation of the rod C D, if that were to swing alone without any other weight annexed, would be the point I. And this point would be the true center of oscillation, so far as the thickness of the rod is not to be regarded. If any one chuses to take into consideration the thickness of the rod, he must place the center of oscillation thereof so much below the point I, that eight times the distance of the center from the point I shall bear the same proportion to the thickness of the rod, as the thickness of the rod bears to its length C D[66].