Finding the orbits from the focus given

Proposition 49 Theorem 17

If 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 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 circles 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.

ABL Let E be the globe. the centre of the wheel, C its centre, BPV the wheel insisting thereon, B the point of contact, and P the given point in the perimeter of the wheel. circle ABL from A through B the given point P this wheel to proceed in the great its progress to revolve in be always equal one to the other, in the peri meter of the wheel may describe in thf such a manner that the arcs :if;d Imagine towards L, and in AB, PB may185 SEC. X.I s H AP be the whole curvilinear Let the curvilinear path AP. the globe in A, and the length cf touched wheel the described since path will be to twice the versed sine of the arc |PB as 20 E to this path mean time AP let the right line CE (produced if need be) meet the wheel in V, and join CP, BP, EP, VP produce CP, and let fall thereon the perpen Let PH, VH, meeting in H, touch the circle in P and V, dicular VF. and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK. CB. For ; From the centre C with any interval CP in n AP in m ; cutting the right line the curvilinear path Vo let there t let there be described the circle wow, BP in o, and the perimeter of the wheel with the interval and from the centre V be described a circle cutting VP produced in q. progress always revolves about the point of con tact B. it is manifest that the right line BP is perpendicular to that curve line which the point P of the wheel describes, and therefore that the right Because the wheel in its AP VP will touch this curve in the point P. Let the radius of the circle nmn be gradually increased or diminished so that at last it become equal to the distance CP and by reason of the similitude of the evanescent figure line ; Pnn-mq, and the figure Pra, P//, Po, that P<y, AP, the right line PFGVI, is, the ultimate ratio of the evanescent lined ae the ratio of the CP, the circular arc momentary mutations of the curve BP, and the right line VP, will

PV, PF, PG, PI, respectively. But since VF is and VH to CV, and therefore the angles HVG, VCF OF, as of the lines perpendicular to VHG (because the angles of the quadrilateral figure equal= and the angle are right in and P) is equal to the angle CEP, the triangles V HG, will be similar and thence it will come to pass that as EP is HVEP V CEP to CE so is HG to HV or HP, and so KI to KP, and by composition or division as CB to CE so is PI to PK, and doubling the consequents asCB ; to 2CE line so PI to VP, that curve line is, AP is PV, and so is Pq to Pm. the increment of the line in a given ratio of CB Therefore the decrement of the BY VP to to the increment of the 2CE, and therefore (by Cor. BY YP and AP, generated by those increments, arc BY be radius, YP is the cosine of the angle BYP YP is the versed sine of the same angle, and or -*BEP, and therefore BY therefore in this wheel, whose radius is ^BV, BY YP will be double the versed sine of the arc ^BP. Therefore AP is to double the versed sine oi the arc ^BP as 2CE to CB.

The line AP in the former of these Propositions we shall name the cy Lena. IV) the lengths in the same ratio. But if cloid without the globe, the other in the latter Proposition the cycloid within the globe, for distinction sake. COR. 1. Hence if there be described the same be bisected in S, the len c th of the part (which 2CE COR. PS the double of the sine of the angle CB, and therefore in a given ratio. is to 2. And ASL, and the will be to the length PV entire cycloid YBP, when EB is radius) as the length of the semi-perimeter of the cycloid AS will be which is to the dumeter of the wheel BY as 2CF equal to a right line toCB.