Finding the orbits from the focus given

PROPOSITION 47 THEOREM 15

Supposing the centripetal force to be proportional to t/ie distance of the body from the centre ; all bodies revolving in any planes whatsoever will describe ellipses, and complete their revolutions in equal times ; and those which move in right lines, running backwards and forwards going and returning alternately will complete ttieir several periods of in the same times. , For letting all things stand as in the foregoing Proposition, the force SV, with which the body Q, revolving in any plane PQ,R is attracted to wards the centre S, is as the distance SO. and therefore because SV and SQ,, TV and CQ, are proportional, the force TV with which the body is ; attracted towards the given point C in the plane of the orbit is as the dis tance CQ,. Therefore the forces with which bodies found in the plane PQ,R are attracted towaitis the point O, are in proportion to the distances with which the same bodies are attract-ed every way to equal to the forces wards the centre S and in the same ; and therefore the bodies will move in the same times, figures, in any plane PQR about the point C. n* they would do X, ai in free spaces about the centre S

and therefore (by Cor. 2, Prop. 2, Prop. XXXVIII.) they will in equal times either describe that plane about the centre C, or move to and fro in right lines ; d Cor. m ellipses passing through the centre C in that plane; completing the same periods of time in all cases.

SCHOLIUM

The rise and fall of bodies in curve superficies has a near relation motions we have been speaking of.

Imagine curve lines to be de to these scribed on any plane, and to revolve about any given axes passing through the centre of force, and and by that revolution to describe curve superficies that the bodies move in such sort that their centres m may ; be always found 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 were generated. In those cases, therefore, consider thp motion in those curve lines. superficies it

Proposition 48 Theorem 16

If a wheel stands upon the outside of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle, the length 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 of 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.