t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1. 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 QJR.S as 1 to AR V AC' COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs have discovered concerning the vulgar cycloid. For if the diameter of the globe be infinitely increased, its sphacrical superficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of our's will become the same with the common cycloid. But in that case the length of the arc of the cycloid between that plane and the describing point will become equal to four times the versed sine of half the arc of the wheel between the same plane and the describing point, as was discovered by Sir Christopher Wren. And a pendulum between two such cycloids will oscillate in a similar and equal cycloid in equal times, as M. Huygens demonstrated. The descent of heavy bodies also in the time of one oscillation will be the same as M. Huygens exhibited. The propositions here demonstrated are adapted to the true constitution of the Earth, in so far as wheels moving in any of its great circles will de scribe, by the motions of nails fixed in their perimeters, cycloids without the globe ; and pendulums, in mines and deep caverns of the Earth, must oscil late in cycloids within the globe, that those oscillations may be performed in equal times. For gravity (as will be shewn in the third book) decreases in its progress from the superficies of the Earth ; upwards in a duplicate ratio of the distances from the centre of the Earth ; downwards in a sim ple ratio of the same. PROPOSITION LIII. PROBLEM XXXV. Granting the quadratures of curvilinear figures, it is required to find the forces with which bodies moving in given curve lines may always perform their oscillations in equal times. Let the body T oscillate in any curve line STRQ,, whose axis is AR passing through the centre of force C. Draw TX touching that curve in any place of the body T, and in that tangent TX take TY equal to the arc TR. The length of that arc is known from the common methods used