body performing its circuits in a revolving ellipsis will describe in a quiescent plane. By this collation of the terms, these orbits are made similar ;not universally, indeed, but then only when they approach very near to acircular figure. A body, therefore revolving with an uniform centripetal180force in an orbit nearly circular, will always describe an angle of deg/, orv/o103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to thelower apsis when it has once described that angle, and thence returning tothe upper apsis when it has described that angle again ; and so on in infinitwn.EXAM. 2. Suppose the centripetal force to be as any power of the alti-Antude A, as, for example, An 3, or-r^ ; where n 3 and n signify any in-A.dices of powers whatever, whether integers or fractions, rational or surd,affirmative or negative. That numerator An or T X| nbeing reduced toan indeterminate series by my method of converging series, will becomeTn>/XTn T + ^XXTn 2, &c. And conferring these termswith the terms of the other numerator RGG RFF + TFF FFX, itbecomes as RGG RFF 4- TFF to Tn, so FF to ?/.Tn r + ?~^XTn 2, &c. And taking the last ratios where the orbits approach tocircles, it becomes as RGG to T 1, so FF to nT-1 T, or as GG toT", so FF to ?*Tn; and again, GG to FF, so Tn l to nT"1, thatis, as 1 to n ; and therefore G is to F, that is the angle VCp to the angleVCP, as 1 to ^/n. Therefore since the angle VCP, described in the descent of the body from the upper apsis to the lower apsis in an ellipsis, isof 180 deg., the angle VC/?, described in the descent of the body from theupper apsis to the lower apsis in an orbit nearly circular which a body describes with a centripetal force proportional to the power An 3, will be equalISOto an angle of -deg., and this angle being repeated, the body will re-。/titurn from the lower to the upper apsis, and so on in infinitum. As if thecentripetal force be as the distance of the body from the centre, that is, as A,A4or -p, n will be equal to 4, and ^/n equal to 2 ; and thereLre the angleIX.] OF NATURAL PHILOSOPHY. IT9ISObetween the upper and the lower apsis will be equal to deg., or 90 deg.Therefore the body having performed a fourth part of one revolution, willarrive at the lower apsis, and having performed another fourth part, willarrive at the upper apsis, and so on by turns in infiuitum. This appearsalso from Prop. X. For a body acted on by this centripetal force will revolve in an immovable ellipsis, whose centre is the centre of force. If the1 A 2centripetal force is reciprocally as the distance, that is, directly as or A A"ji will be equal to 2 ; and therefore the angle between the upper and lower180apsis will be -deg., or 127 deg., 16 min., 45 sec. ; and therefore a body rev/2volving with such a force, will by a perpetual repetition of this angle, movealternately from the upper to the lower and from the lower to the upperapsis for ever. So. also, if the centripetal force be reciprocally as thebiquadrate root of the eleventh power of the altitude, that is, reciprocallyas A , and, therefore, directly as -r-fp or asTs>n wil* ^e etl ual f。>an(14 A^- A1 Of)-deg. will be equal to 360 deg. ; and therefore the body parting fromv/ nthe upper apsis, and from thence perpetually descending, will arrive at thelower apsis when it has completed one entire revolution ; and thence ascending perpetually, when it has completed another entire revolution, itwill arrive again at the upper apsis ; and so alternately for ever.EXAM. 3. Taking m and n for any indices of the powers of the altitude, and b and c for any given numbers, suppose the centripetal force6Ara + cA" b into T X> -f- c into T Xto be as r^ that is, as A3 A3or (by the method of converging series above-mentioned) asbTm + cTnm6XT" - 1 //cXTn mm m vvrpm un n~~2--0A.A1 ^t-XXT"2, fcc.T$~~ and comparing the terms of the numerators, there willarise RGG IIFF -f TFF to ^Tm + cT" as FF to mbTm i" - + 2" mbXT" - * + "^pcXTn - .fee. And takingthe last ratios that arise when the orbits come to a circular form, therewill come forth GG to 6Tm l -f cTn 1 as FF to mbTm l + ncT"J;and again, GG to FF as 6Tm + cTn to mbTn 1-f ncTn 。This proportion, by expressing the greatest altitude CV or T arithmetically by unity, becomes, GG to FF as b -{- c to mb -。- ?/c, and therefore as I(80 THE MATHEMATICAL PRINCIPLES [BOOK 1tub ~h ncto -y7 Whence G becomes to P, that is, the angle VCjo to the anf)~T~ Cgle VCP. as 1 to >/-.- -. And therefore since the angle VCP betweenthe upper and the lower apsis, in an immovable ellipsis, is of 180 deg., thrangle VC/? between the same apsides in an orbit which a body describesb A m I c A nwith a centripetal force, that is. as - r , will be equal to an angle ofA.ISO v/ 1~TT~; deg. And y tne same reasoning, if the centripetal forcebe as -73 , the angle between the apsides will be found equal tofi f*18o V - -deg. After the same manner the Problem is solved innib >icmore difficult cases. The quantity to which the centripetal force is proportional must always be resolved into a converging series whose denominator is A*. Then the given part of the numerator arising from thatoperation is to be supposed in the same ratio to that part of it which is notgiven, as the given part of this numerator RGG RFF -f TFF FFX.is to that part of the same numerator which is not given. And takingaway the superfluous quantities, and writing unity for T, the proportionof G to F is obtained.COR. 1 . Hence if the centripetal force be as any power of the altitude,that power may be found from the motion of the apsides ; and so contrariwise. That is, if the whole angular motion, with which the body returnsto the same apsis, be to the angular motion of one revolution, or 360 deg.,MS any number as m to another as n, and the altitude called A ; the forcennwill be as the power A HSii3 of the altitude A; the index of which power is- 3. This appears by the second example. Hence it is plain thatthe force in its recess from the centre cannot decrease in a greater than atriplicate ratio of the altitude. A body revolving with such a force, andparting from the apsis, if it once begins to descend, can never arrive at thelower apsis or least altitude, but will descend to the centre, describing thecurve line treated of in Cor. 3, Prop. XLL But if it should, at its parting from the lower apsis, begin to ascend never so little, it will ascend inirtfimtifm, and never come to the upper apsis ;but will describe the curveline spoken of in the same Cor., and Cor. 6, Prop. XLIV. So that wherethe force in its recess from the centre decreases in a greater than a triplicate ratio of the altitude, the body at its parting from the apsis, will eitherdescend to the centre, or ascend in iiiftnitum, according as it descends orAscends at the beginning of its motion. But if the force in its recess from"SEC. IX.J OF NATURAL PHILOSOPHY. ISithe centre either decreases in a less than a triplicate ratio of the altitude,or increases in any ratio of the altitude whatsoever, the body will neverdescend to the centre, but will at some time arrive at the lower apsis ; and,on the contrary, if the body alternately ascending and descending from oneapsis to another never comes to the centre, then either the force increasesin the recess from the centre, or it decreases in a less than a triplicate ratioof the altitude; and the sooner the body returns from one apsis to another,the farther is the ratio of the forces from the triplicate ratio. As if thebody should return to and from the upper apsis by an alternate descent andascent in 8 revolutions, or in 4, or 2, or 。。 that is, if m should be to n as 8,or 4, or 2, or H to 1. and therefore ---3, be g。 3,or TV~3, or imm 3, or3I - 3; then the force will be as A~ ? or AT "~3j or A*~~ 3j or A""Gthat is. it will be reciprocally as A 3 C4 or A 3 T ^ or A 34 or A 3""If the body after each revolution returns to the same apsis, and the apsisnn _remains unmoved, then m will be to n as 1 to 1, and therefore A"will be equal to A 2, or -; and therefore the decrease of the forces will AAbe in a duplicate ratio of the altitude ; as was demonstrated above. If thebody in three fourth parts, or two thirds, or one third, or one fourth partof an entire revolution, return to the same apsis ; m will be to n as or ?n n i_6 _ 39 _ 3oor ^ or i to 1, and therefore Amm 3 is equal to A 9 or A4 or A_ 3 1 6 _ 3 l_lor A ; and therefore the force is either reciprocally as A fl or3 613A 4 or directly as A or A . Lastly if the body in its progress from theupper apsis to the same upper apsis again, goes over one entire revolutionand three deg. more, and therefore that apsis in each revolution of the bodymoves three deg. in consequentia ; then m will be to u as 363 deg. to360 deg. or as 121 to 120, and therefore Amm will be equal to2 9_ 5_ 2_ JJ A "and therefore the centripetal force will be reciprocally as^T4"6TT> or recip rocally as A 2 ^ 4 ^very nearly. Therefore the centripetalforce decreases in a ratio something greater than the duplicate ; but approaching 59f times nearer to the duplicate than the triplicate.COR. 2. Hence also if a body, urged by a centripetal force which is reciprocally as the square of the altitude, revolves in an ellipsis whose focusis in the centre of the forces ; and a new and foreign force should be addedto or subducted from this centripetal force, the motion of the apsides arisingfrom that foreign force may (by the third Example) be known ; and so onthe contrary. As if the force with which the body revolves in the ellipsis182 THE MATHEMATICAL PRINCIPLES [BOOK Ioe as -r-r-A ; and the foreign force subducted as cA, and therefore the remain- .A.^ c^4ing force as-^ ; then (by the third Example) b will be equal to 1.m equal to 1, and n equal to 4 ; and therefore the angle of revolution be1 ctween the apsides is equal to 180 <*/- deg. Suppose that foreign forceto be 357.45 parts less than the other force with which the body revolvesin the ellipsis : that is, c to be -3 }y j ; A or T being equal to 1; and thenl8(Vl~4cwill be 18<Vfff Jf or 180.7623, that is, 180 deg., 45 min.,44 sec. Therefore the body, parting from the upper apsis, will arrive atthe lower apsis with an angular motion of 180 deg., 45 min., 44 sec , andthis angular motion being repeated, will return to the upper apsis ; andtherefore the upper apsis in each revolution will go forward 1 deg., 31 min.,28 sec. The apsis of the moon is about twice as swiftSo much for the motion of bodies in orbits whose planes pass throughthe centre of force. It now remains to determine those motions in eccentrical planes. For those authors who treat of the motion of heavy bodiesused to consider the ascent and descent of such bodies, not only in a perpendicular direction, but at all degrees of obliquity upon any given planes ;and for the same reason we are to consider in this place the motions ofbodies tending to centres by means of any forces whatsoever, when thosebodies move in eccentrical planes. These planes are supposed to beperfectly smooth and polished, so as not to retard the motion of the bodiesin the least. Moreover, in these demonstrations, instead of the planes uponwhich those bodies roll or slide, and which are therefore tangent planes tothe bodies, I shall use planes parallel to them, in which the centres of thebodies move, and by that motion describe orbits. And by the same methodI afterwards determine the motions of bodies performer1 in curve superficies.SECTION X.Of the motion of bodies in given superficies, and of the reciprocal motionoffnnependulous bodies.PROPOSITION XLVI. PROBLEM XXXII.Any kind of centripetal force being supposed, and the centre offorce,atfftany plane whatsoever in which the body revolves, being given, and tintquadratures of curvilinear figures being allowed; it is required to determine the motion of a body going off from a given place., with agiven velocity, in the direction of a given right line in, that plane.SEC. X.J OF NATURAL PHILOSOPHY- 183Let S be the centre of force, SC theleast distance of that centre from the givenplane, P a body issuing from the place Pin the direction of the right line PZ, Q,the same body revolving in its trajectory,and PQ,R the trajectory itself which isrequired to be found, described in thatgiven plane. Join CQ, Q.S, and if in Q,Swe take SV proportional to the centripetalforce with which the body is attracted towards the centre S, and draw VT parallelto CQ, and meeting SC in T ; then will the force SV be resolved intotwo (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ;ofwhich ST aMracting the body in the direction of a line perpendicular tothat plane, does not at all change its motion in that plane. But the actionc f the other force TV, coinciding with the position of the plane itself, attracts the body directly towards the given point C in that plane ; adt icreftre causes the body to move in this plane in the same manner as ifthe force S F were taken away, and the body were to revolve in free spaceabout the centre C by means of the force TV alone. But there being giventhe centripetal force TV with which the body Q, revolves in free spaceabout the given centre C, there is given (by Prop. XLII) the trajectoryPQ.R which the body describes ; the place Q,, in which the body will befound at any given time ; and, lastly, the velocity of the body in that placeQ,. And so e contra. Q..E.I.PROPOSITION XLV1L THEOREM XV.Supposing the centripetal force to be proportional to t/ie distance of thebody from the centre ; all bodies revolving in any planes whatsoeverwill describe ellipses, and complete their revolutions in equal times ;and those which move in right lines, running backwards andforwardsalternately, will complete ttieir several periods of going and returningin the same times.For letting all things stand as in the foregoing Proposition, the forceSV, with which the body Q, revolving in any plane PQ,R is attracted towards the centre S, is as the distance SO. ; and therefore because SV andSQ,, TV and CQ, are proportional, the force TV with which the body isattracted towards the given point C in the plane of the orbit is as the distance CQ,. Therefore the forces with which bodies found in the planePQ,R are attracted towaitis the point O, are in proportion to the distancesequal to the forces with which the same bodies are attract-ed every way towards the centre S ; and therefore the bodies will move in the same times,and in the same figures, in any plane PQR about the point C. n* theyTHE 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 describeellipses m that plane about the centre C, or move to and fro in right linespassing through the centre C in that plane; completing the same periodsof time in all cases. Q.E.D.SCHOLIUM.The ascent and descent of bodies in curve superficies has a near relationto these motions we have been speaking of. Imagine curve lines to be described on any plane, and to revolve about any given axes passing throughthe centre of force, and by that revolution to describe curve superficies ; andthat the bodies move in such sort that their centres may be always foundm those superficies. If those bodies reciprocate to and fro with an obliqueascent and descent, their motions will be performed in planes passing throughtiie axis, and therefore in the curve lines, by whose revolution those curvesuperficies were generated. In those cases, therefore, it will be sufficient toconsider thp motion in those curve lines.PROPOSITION XLVIII. THEOREM XVI.If a wheel stands npon the outside of a globe at right angles thereto, andrevolving about its own axis goes forward in a great circle, the lengthof lite curvilinear path which any point, given in the perimeter of thewheel, hath described, since, the time that it touched the globe (whichcurvilinear path w~e may call the cycloid, or epicycloid), will be to doublethe versed sine of half the arc which since that time has touched theglobe in passing over it, as the sn,m of the diameters of the globe andthe 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 thereto, and revolving about its own axis go forward in one of the greatcircles of the globe, the length of the curvilinear path which any point,given in the perimeter of the wheel^ hath described since it toncJied theglobe, imll be to the double of the versed sine of half the arc which inall that time has touched the globe in passing over it, as the differenceof the diameters of the globe and the wheel to the semi-diameter of theglobe.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 pointin the perimeter of the wheel. Imagine this wheel to proceed in the greatcircle ABL from A through B towards L, and in its progress to revolve insuch 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 thfSEC. X.I OF NATURAL PHILOSOPHY.s185Hmean time the curvilinear path AP. Let AP be the whole curvilinearpath described since the wheel touched the globe in A, and the length cfthis path AP will be to twice the versed sine of the arc |PB as 20E toCB. For 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 perpendicular VF. Let PH, VH, meeting in H, touch the circle in P and V,and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK.From the centre C with any interval let there be described the circle wow,cutting the right line CP in nt the perimeter of the wheel BP in o, andthe curvilinear path AP in m ; and from the centre V with the intervalVo let there be described a circle cutting VP produced in q.Because the wheel in its progress always revolves about the point of contact B. it is manifest that the right line BP is perpendicular to that curve lineAP which the point P of the wheel describes, and therefore that the rightline VP will touch this curve in the point P. Let the radius of the circle nmnbe gradually increased or diminished so that at last it become equal to thedistance CP ; and by reason of the similitude of the evanescent figurePnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined aePra, P//, Po, P<y,that is, the ratio of the momentary mutations of the curveAP, the right line CP, the circular arc BP, and the right line VP, will <iSS THE MATHEMATICAL PRINCIPLES [BOOK 1.the same as of the lines PV, PF, PG, PI, respectively. But since VF isperpendicular to OF, and VH to CV, and therefore the angles HVG, VCFequal: and the angle VHG (because the angles of the quadrilateral figureHVEP are right in V and P) is equal to the angle CEP, the trianglesV HG, CEP will be similar ; and thence it will come to pass that as EP isto CE so is HG to HV or HP, and so KI to KP, and by composition ordivision as CB to CE so is PI to PK, and doubling the consequents asCBto 2CE so PI to PV, and so is Pq to Pm. Therefore the decrement of theline VP, that is, the increment of the line BY VP to the increment of thecurve line AP is in a given ratio of CB to 2CE, and therefore (by Cor.Lena. IV) the lengths BY YP and AP, generated by those increments, arcin the same ratio. But if BY be radius, YP is the cosine of the angle BYPor -*BEP, and therefore BY YP is the versed sine of the same angle, andtherefore in this wheel, whose radius is ^BV, BY YP will be double theversed sine of the arc ^BP. Therefore AP is to double the versed sine oithe arc ^BP as 2CE to CB. Q.E.D.The line AP in the former of these Propositions we shall name the cycloid without the globe, the other in the latter Proposition the cycloid withinthe globe, for distinction sake.COR. 1. Hence if there be described the entire cycloid ASL, and the