Sir Isaac Newton’s Book I.

90

circle, it will swing 11 5 times, while it ought according tothis proportion to have swung 117 times; so that, when itswings in so large an arch, it loses something less than twoswings in an hundred. If it swing through only os thecircle, it shall not lose above one vibration in 160. If itswing in 2 - o of the circle, it shall lose about one vibration in6 90. If its swing be confined to ^ os the whole circle, itshall lose very little more than one swing in 2.600. Andif it take no greater a swing than through ^ os the whole cir-cle, it shall not lose one swing in y8oo.

61. N o w it follows from hence, that, when pendulumsswing in small arches, there is very nearly a constant propor-tion observed between the time os their swing, and the time,in which a body would sail perpendicularly down throughhalf their length. And we have declared above, that thespaces, through which bodies fall, are in a two fold propor-tion of the times, which they take up in falling a . There-fore in pendulums of different lengths, swinging throug hsmallarches, the lengths of the pendulums are in a two fold orduplicate proportion of the times, they take in swinging ;so that a pendulum of sour times the length of another shalltake up twice the time in each swing, one os nine times thelength will make one swing only for three swings of theshorter, and so on.

62 . This proportion in the swings of different pendu-lums not only holds in small arches; but in large ones also,

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