THE MATHEMATICAL PRINCIPLES [BOOK I. would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop. X, ai d Cor. 2, Prop. XXXVIII.) they will in equal times either describe ellipses m that plane about the centre C, or move to and fro in right lines passing through the centre C in that plane; completing 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 de scribed 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 m those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through tiie 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 thp motion in those curve lines. PROPOSITION XL VIII. THEOREM XVI. If a wheel stands npon the outside of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle, the length of lite curvilinear path which any point, given in the perimeter of the wheel, hath described, since, the time that it touched the globe (which curvilinear path w~e may call the cycloid, or epicycloid), will be to double the versed sine of half the arc which since that time has touched the globe in passing over it, as the sn,m of the diameters of the globe and the wheel to the semi-diameter of the globe. PROPOSITION XLIX. THEOREM XVII. ff a wheel stand upon the inside of a concave globe at right angles there to, and revolving about its own axis go forward in one of the great circles of the globe, the length of the curvilinear path which any point, given in the perimeter of the wheel^ hath described since it toncJied the globe, imll be to the double of the versed sine of half the arc which in all that time has touched the globe in passing over it, as the difference of the diameters of the globe and the wheel to the semi-diameter of the globe. Let ABL be the globe. C its centre, BPV the wheel insisting thereon, E the centre of the wheel, B the point of contact, and P the given point in the perimeter of the wheel. Imagine this wheel to proceed in the great circle ABL from A through B towards L, and in its progress to revolve in such a manner that the arcs AB, PB may be always equal one to the other, :if;d the given point P in the peri meter of the wheel may describe in thf