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

 as the emi-periphery HKM which denotes that entire ocillation, directly; and as the arc HZ which in like manner denotes a given time inverely) will be as GH directly and, $$\scriptstyle \sqrt {GH \times CO \times V}$$ inverely, that is, becaue 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 ocillations in all globes and cycloids, performed with what abolute forces oever, are in a ratio compounded of the ubduplicate ratio of the length of the tring directly, and the ubduplicate ratio of the ditance between the point of fufpenion and the centre of the globe inverely, and the ubduplicate ratio of the abolute force of the globe inverely alo. Q. E. I.

Hence alo the times of ocillating, falling, and revolving bodies may be compared among themelves. For if the diameter of the wheel with which the cycloid is decribed within the globe is uppoed equal to the emi-diameter of the globe, the cycloid will become a right line passing through the centre of the globe, and the ocillation will be changed into a decent and ubequent acent in that right line. Whence there is given both the time of the decent 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 ditance decribes an arc of a quadrant. For this time (by cae 2.) is to the time of half the ocillation in any cycloid QRS as 1 to $$\textstyle \sqrt {\frac {AR}{AC}}$$.

Hence alo follow what Sir Chritopher Wren and M. Huygens have difcovered concerning