Finding the orbits from the focus given

PROPOSITION 52 PROBLEM 34

Define the velocities of the pendulums in the several places, and the times in which both the entire oscillations, and the several parts of them are performed.

About any centre G, with the interval equal to the arc of the cycloid RS, describe a semi-circle

If a centripetal bisected by the semi-diameter GK. HKM proportional to the distance of the places from the centre tend to the centre G, and it be in the peri meter equal to the centripetal force in the perime tal force HIK Q,OS tending towards its centre, and at is let fall from the the same time that the pendulum ter of the globe highest place S, a body, as L, is let fall from H to T G ; then because which act upon the bodies are equal at the be to the spaces to be ginning, and always proportional described TR, LG, and therefore if TR and LG are in the places T and L, it is plain equal, are also equal forces M that those bodies describe at the beginning equal spaces ST, HL, and therefore are still acted upon equally, and continue to describe Therefore by Prop. XXXVIII, the time in which the body equal spaces. describes the arc ST is to the time of one oscillation, as the arc HI the time H arrives at L, to the semi-periphery HKM, the time H will come to M. And the velocity of the pendulous the is to its velocity in the lowest place R, that T in the place body L to its velocity in the place G, or the velocity of the body H in the place momentary increment of the line HL to the momentary increment of the line HG (the arcs HI, HK increasing with an equable flux) as the ordinato which the body in which the body in is, LI to the radius GK. or as v/SR Til 2 2 to SR.

Hence, since in unequal oscillations there are described in equal time arcs proportional to the en obtained from the times given, both the velocities and the arcs described in all the oscillations universally. tire arcs of the oscillations, there are Which was first required. Let now any pendulous bodies oscillate in different cycloids described within different globes, whose absolute forces are also different and if the absolute force of any globe Q.OS be called V, the accelerative force with ; is acted on in the circumference of this globe, when it directly towards its centre, will be as the distance of the from that centre and the absolute force of the globe con- which the pendulum move begins to pendulous body CO X V. Therefore the lineola HY, which is as this CO X V, will be described in a given time and if there be erected the perpendicular YZ meeting the circumference in Z, the nascent But that nascent arc HZ is in the arc HZ will denote that given time. subduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X V junctly, that is, as accelerated force : Whence the time of an entire oscillation in the cycloid Q,RS (it being as the semi-periphery w r hich denotes that entire oscillation, directly in like manner denotes a given time inversely) and as the arc which HZ will be as GH and GH SR HKM, directly : and are equal, as v/GH X V nr UU X Therefore the oscillations in , all . V CO X V inversely ; that is, because or (by Cor. Prop. L,) as X/-TTVT- globes and AO X cycloids, V performed with what absolute forces soever, are in a ratio compounded of the subduplicate ratio of the length of the string directly, and the subduplicate ratio of the distance between the point of suspension and the centre of the globe inversely, and the subduplicate ratio of the absolute force of the globe inversely also

1. Hence also the times of oscillating, falling, and revolving bodies be may 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

Corollary

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 For this time (by any distance describes an arc of a quadrant Case 2) is to the time of half the oscillation in any cycloid QJR.S as 1 to at AR V AC

Corollary 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 But in that case the length of the arc of the with the common cycloid. 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 The descent equal cycloid in equal times, as M. Huygens demonstrated. 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 within the globe, that those oscillations may be performed late in cycloids For gravity (as will be shewn in the third book) decreases in equal times. in its progress from the superficies of the Earth upwards in a duplicate downwards in a sim ratio of the distances from the centre of the Earth ; ; ple ratio of the same.