Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/283

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VBP, when EB is radius) as 2 CE to CB, and therefore in a given ratio.

Cor. 2. And the length of the ſemi-perimeter of the cycloid AS will be equal to a right line which is to the diameter of the wheel BV as 2 CE to CB.


Proposition L. Problem XXXIII.

To cauſe a pendulous body to oſcillate in a given cycloid.

Plate 19, Figure 3
Plate 19, Figure 3

Let there be given within the globe QVS, (Pl. 19. Fig. 3.) deſcribed with the centre C, the cycloid QRS, biſected in R, and meeting the ſuperficies of the globe with its extreme points Q and S on either hand. Let there be drawn CR biſecting the arc QS in O, and let it be produced to A in ſuch ſort that CA may be to CO as CO to CR. About the centre C, with the interval CA, let there be deſcribed an exterior globe DAF, and within this globe; by a wheel whoſe diameter is AO, let there be deſcribed two ſemi-cycloids AQ, AS, touching the interior globe in Q and S, and meeting the exterior globe in A. From that point A, with a thread APT in length equal to the line AR, let the body T depend, and oſcillate in ſuch manner between the two ſemi-cycloids AQ, AS that as often as the pendulum parts from the perpendicular AR, the upper part of the thread AP may be applied to that ſemi-cycloid APS towards which the motion tends, and fold it ſelf round that curve line, as if it were ſome ſolid obſtacle;