Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/291

as the ſemi-periphery HKM which denotes that entire oſcillation, directly; and as the arc HZ which in like manner denotes a given time inverſely) will be as GH directly and, $\scriptstyle {\sqrt {GH\times CO\times V}}$ inverſely, that is, becauſe GH and SR are equal, as, $\textstyle {\sqrt {\frac {SR}{CO\times V}}}$, or (by cor. prop. 50.) as $\textstyle {\sqrt {\frac {AR}{AC\times V}}}$. Therefore the oſcillations in all globes and cycloids, performed with what abſolute forces ſoever, are in a ratio compounded of the ſubduplicate ratio of the length of the ſtring directly, and the ſubduplicate ratio of the diſtance between the point of fufpenſion and the centre of the globe inverſely, and the ſubduplicate ratio of the abſolute force of the globe inverſely alſo. Q. E. I.

Cor. 1. Hence alſo the times of oſcillating, falling, and revolving bodies may be compared among themſelves. For if the diameter of the wheel with which the cycloid is deſcribed within the globe is ſuppoſed equal to the ſemi-diameter of the globe, the cycloid will become a right line passing through the centre of the globe, and the oſcillation will be changed into a deſcent and ſubſequent aſcent in that right line. Whence there is given both the time of the deſcent from any place to the centre, and the time equal to it in which the body revolving uniformly about the centre of the globe at any diſtance deſcribes an arc of a quadrant. For this time (by caſe 2.) is to the time of half the oſcillation in any cycloid QRS as 1 to $\textstyle {\sqrt {\frac {AR}{AC}}}$.

Cor. 2. Hence alſo follow what Sir Chriſtopher Wren and M. Huygens have difcovered concerning