59. THE increase of velocity, which the body acquires in falling from a greater height, has such an effect, that, if straight lines be drawn from A to C and E, the body would fall through the longer straight line C A just in the same time, as through the shorter straight line E A. This is demonstrated by the geometers, who prove, that if any circle, as A B C D (fig. 49.) be placed in a perpendicular situation; a body shall fall obliquely through every line, as A B drawn from the lowest point A in the circle to any other point in the circumference just in the same time, as would be imployed by the body in falling perpendicularly down through the diameter C A. But the time in which the body will descend through the arch, is different from the time, which it would take up in falling through the line A B.
60. IT has been thought by some, that because in very small arches this correspondent straight line differs but little from the arch itself; therefore the descent through this straight line would be performed in such small arches nearly in the same time as through the arches themselves: so that if a pendulum were to swing in small arches, half the time of a single swing would be nearly equal to the time, in which a body would fall perpendicularly through twice the length of the pendulum. That is, the whole time of the swing, according to this opinion, will be four fold the time required for the body to fall through half the length of the pendulum; because the time of the body’s falling down twice the length of the pendulum is half the time required for the fall through one quarter of this space, that is through half the pendulum’s length. However there is here a mistake; for the whole time of the swing, when the pendulum moves through small arches, bears to the time required for a body to fall down through half the length of the pendulum very nearly the same proportion, as the circumference of a circle bears to its diameter; that is very nearly the proportion of 355 to 113, or little more than the proportion of 3 to 1. If the pendulum takes so great a swing, as to pass over an arch equal to one sixth part of the whole circumference of the circle, it will swing 115 times, while it ought according to this proportion to have swung 117 times; so that, when it swings in so large an arch, it loses something less than two swings in an hundred. If it swing through 1/10 only of the circle, it shall not lose above one vibration in 160. If it swing in 1/20 of the circle, it shall lose about one vibration in 690. If its swing be confined to 1/40 of the whole circle, it shall lose very little more than one swing in 2600. And if it take no greater a swing than through 1/60 of the whole circle, it shall not lose one swing in 5800.