prop. 38) they will in equal times either deſcribe ellipſes in that plane about the centre C, or move to and fro in right lines paſſing through the centre C in that plane; compleating the ſame periods of time in all caſes. Q. E. D.
Scholium.
The aſcent and deſcent of bodies in curve ſuperficies has a near relation to theſe motions we have been ſpeaking of. Imagine curve lines to be deſcribed on any plane, and to revolve about any given axes paſſing through the centre of force, and by that revolution to deſcribe curve ſuperficies; and that the bodies move in ſuch ſort that their centres may be always found in thoſe ſuperficies. If thoſe bodies reciprocate to and fro with an oblique aſcent and deſcent; their motions will be performed in planes paſſing through the axis, and therefore in the curve lines by whoſe revolution thoſe curve ſuperficies were generated. In thoſe caſes therefore it will be ſufficient to conſider the motion in thoſe curve lenes.
Proposition XLVIII. Theorem XVI.
If a wheel ſands upon the out-ſide of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle; the length of the curvilinear path which any point, given in the perimeter of the wheel,
