same be bisected in S, the lenc th of the part PS will be to the length PV(which is the double of the sine of the angle YBP, when EB is radius) as2CE to CB, and therefore in a given ratio.COR. 2. And the length of the semi-perimeter of the cycloid AS will beequal to a right line which is to the dumeter of the wheel BY as 2CFtoCB.PROPOSITION L. PROBLEM XXXIII.To cause a pendulous body to oscillate in a given cycloid.Let there be given within the globe QYS describedwith the centre C, the cycloid QRS, bisected in R, and meeting the superficies of theglobe with its extreme points Q and S on eitherhand. Let there be drawn CR birxcting the arcQS in O, and let it be produced to A in suchsort that CA may be to CO as CO to CR.About the centre C, with the interval CA, letthere be described an exterior globe DAF ; andwithin this globe, by a wheel whose diameter isAO, let there be described two semi-cycloids AQ,,AS, touching the interior globe in Q, and S, and meeting the exterior globein A. From that point A, with a thread APT in length equal to the lineAR, let the body T depend, and oscillate in such manner between the twoSlCC. X.J OF NATURAL PHILOSOPHY. 187semi-cycloids AQ, AS, that, as often as the pendulum parts from the perpendicular AR, the upper part of the thread AP may be applied to thatsemi-cycloid APS towards which the motion tends, and fold itself roundthat curve line, as if it were some solid obstacle, the remaining part of thesame thread PT which has not yet touched the semi-cycloid continuingstraight. Then will the weight T oscillate in the given cycloid QRS.Q.E.F.For let the thread PT meet the cycloid QRS in T, and the circle QOSm V, and let 0V be drawn j and to the rectilinear part of the thread PTfrom the extreme points P and T let there be erected the perpendicularsBP, TW, meeting the right line CV in B and W. It is evident, from theconstruction and generation of the similar figures AS, SR, that those perpendiculars PB, TVV, cut off from CV the lengths VB, VVV equal thediameters of the wheels OA, OR. Therefore TP is to VP (which is double the sine of the angle VBP when ^BV is radius) as BYV to BV, or AO-f-OR to AO, that is (since CA and CO, CO and CR; and by division AOand OR are proportional), as CA + CO to CA, or, if BV be bisected in E,as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of therectilinear part of the thread PT is always equal to the arc of the cycloidPS, and the whole thread APT is always equal to the half of the cycloidAPS, that is (by Cor. 2, Prop. XLIX), to the length AR. And therefore contrariwise, if the string remain always equal to the length AR, thepoint T will always move in the given cycloid QRS. Q.E.D.COR. The string AR is equal to the semi-cycloid AS, and therefore hasthe same ratio to AC the semi-diameter of the exterior globe as the likesemi-cycloid SR has to CO the semi-diameter of the interior globe.PROPOSITION LI. THEOREM XVIII.If a centripetal force tending on all sides to the centre C of a globe, be inall places as the distance of the place from the centre, and by thisforcealone acting upon it, the body T oscillate (in the manner above described] in the perimeter of the cycloid QRS ; / say, that all the oscillations, how unequal soever in tfiemselves, will be performed in equaltimes.For upon the tangent TW infinitely produced let fall the perpendicularCX, and join CT. Because the centripetal force with which the body Tis impelled towards C is as the distance CT, let this (by Cor. 2, of theI ,aws) be resolved into the parts CX, TX, of which CX impelling thebody directly from P stretches the thread PT, and by the resistance therhread makes to it is totally employed, producing no other effect; but the3ther part TX, impelling the body transversely or towards X, directlyaccelerates the motion in the cycloid. Then it is plain that the acceleration of the body, proportional to this accelerating force, will bo every188 THE MATHEMATICAL PRINCIPLES [BOOK 1moment as the length TX, that is (because CV。WV, and TX, TW proportional to them are given),as the length TW, that is (by Cor. 1, Prop. XLIX)as the length of the arc of the cycloid TR. If therefore two pendulums APT, Apt, be unequally drawnaside from the perpendicular AR, and let fall together,their accelerations will be always as the arcs to be described TR, tR. But the parts described at thebeginning of the motion are as the accelerations, thaiis, as the wholes that are to be described at the beginning, and therefore the parts which remain to bedescribed, and the subsequent accelerations proportional to those parts, arealso as the wholes, and so on. Therefore the accelerations, and consequentlythe velocities generated, and the parts described with those velocities, andthe parts to be described, are always as the wholes ; and therefore the partsto be described preserving a given ratio to each other will vanish together,that is, the two bodies oscillating will arrive together at the perpendicular AR.And since on the other hand the ascent of the pendulums from the lowest placeR through the same cycloidal arcs with a retrograde motion, is retarded inthe several places they pass through by the same forces by which their descent was accelerated : it is plain that the velocities of their ascent and descent through the same arcs are equal, and consequently performed in equaltimes ; and, therefore, since the two parts of the cycloid RS and RQ lyingon either side of the perpendicular are similar and equal, the two pendulums will perform as well the wholes as the halves of their oscillations inthe same times. Q.E.D.COR. The force with which the body T is accelerated or retarded in anyplace T of the cycloid, is to the whole weight of the same body in thehighest place S or Q, as the arc of the cycloid TR is to the arc SR or QRPROPOSITION LIL PROBLEM XXXIV.To define the velocities of the pendulums in the several places, and thetimes in which both the entire oscillations, and the several parts of themare performed.About any centre G, with the interval GH equal tothe arc of the cycloid RS, describe a semi-circle HKMbisected by the semi-diameter GK. And if a centripetal force proportional to the distance of the places fromthe centre tend to the centre G, and it be in the perimeter HIK equal to the centripetal force in the perimeter of the globe Q,OS tending towards its centre, and atthe same time that the pendulum T is let fall from thehighest place S, a body, as L, is let fall from H to G ; then because th<SEC. X.J OF NATURAL PHILOSOPHY. 189forces which act upon the bodies are equal at the beginning, and always proportional to the spaces to bedescribed TR, LG, and therefore if TR and LG areequal, are also equal in the places T and L, it is plainthat those bodies describe at the beginning equal spacesMST, HL, and therefore are still acted upon equally, and continue to describeequal spaces. Therefore by Prop. XXXVIII, the time in which the bodydescribes the arc ST is to the time of one oscillation, as the arc HI the timein which the body H arrives at L, to the semi-periphery HKM, the timein which the body H will come to M. And the velocity of the pendulousbody in the place T is to its velocity in the lowest place R, that is, thevelocity of the body H in the place L to its velocity in the place G, or themomentary increment of the line HL to the momentary increment of theline HG (the arcs HI, HK increasing with an equable flux) as the ordinatoLI to the radius GK. or as v/SR2 Til2 to SR. Hence, since in unequaloscillations there are described in equal time arcs proportional to the entire arcs of the oscillations, there are obtained from the times given, boththe velocities and the arcs described in all the oscillations universally.Which was first required.Let now any pendulous bodies oscillate in different cycloids describedwithin different globes, whose absolute forces are also different ; and if theabsolute force of any globe Q.OS be called V, the accelerative force withwhich the pendulum is acted on in the circumference of this globe, when itbegins to move directly towards its centre, will be as the distance of thependulous body from that centre and the absolute force of the globe conjunctly,that is, as CO X V. Therefore the lineola HY, which is as thisaccelerated force CO X V, will be described in a given time : and if therebe erected the perpendicular YZ meeting the circumference in Z, the nascentarc HZ will denote that given time. But that nascent arc HZ is in thesubduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X VWhence the time of an entire oscillation in the cycloid Q,RS (it being asthe semi-periphery HKM, wrhich denotes that entire oscillation, directly :and as the arc HZ which in like manner denotes a given time inversely)will be as GH directly and v/GH X CO X V inversely ; that is, becauseGH and SR are equal, as VnUrU, . or (by Cor. Prop. L,) as X/-TTVT- X V AO X VTherefore the oscillations in all globes and cycloids, performed with whatabsolute forces soever, are in a ratio compounded of the subduplicate ratio ofthe length of the string directly, and the subduplicate ratio of the distancebetween the point of suspension and the centre of the globe inversely, andthe subduplicate ratio of the absolute force of the globe inversely alsoQ.E.I.t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1.COR. 1. Hence also the times of oscillating, falling, and revolving bodiesmay be compared among themselves. For if the diameter of the wheelwith which the cycloid is described within the globe is supposed equal tothe semi-diameter of the globe, the cycloid will become a right line passingthrough the centre of the globe, and the oscillation will be changed into adescent and subsequent ascent in that right line. Whence there is givenboth the time of the descent from any place to the centre, and the time equalto it in which the body revolving uniformly about the centre of the globeat any distance describes an arc of a quadrant For this time (byCase 2) is to the time of half the oscillation in any cycloid QJR.S as 1 toARV ACCOR. 2. Hence also follow what Sir Christopher Wren and M. Huygevshave discovered concerning the vulgar cycloid. For if the diameter of theglobe be infinitely increased, its sphacrical superficies will be changed into aplane, and the centripetal force will act uniformly in the direction of linesperpendicular to that plane, and this cycloid of our s will become the samewith the common cycloid. But in that case the length of the arc of thecycloid between that plane and the describing point will become equal tofour times the versed sine of half the arc of the wheel between the sameplane and the describing point, as was discovered by Sir Christopher Wren.And a pendulum between two such cycloids will oscillate in a similar andequal cycloid in equal times, as M. Huygens demonstrated. The descentof 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 constitutionof the Earth, in so far as wheels moving in any of its great circles will describe, by the motions of nails fixed in their perimeters, cycloids without theglobe ; and pendulums, in mines and deep caverns of the Earth, must oscillate in cycloids within the globe, that those oscillations may be performedin equal times. For gravity (as will be shewn in the third book) decreasesin its progress from the superficies of the Earth ; upwards in a duplicateratio of the distances from the centre of the Earth ; downwards in a simple ratio of the same.PROPOSITION LIII. PROBLEM XXXV.Granting the quadratures of curvilinear figures, it is required to findthe forces with which bodies moving in given curve lines may alwaysperform their oscillations in equal times.Let the body T oscillate in any curve line STRQ,, whose axis is ARpassing through the centre of force C. Draw TX touching that curve inany place of the body T, and in that tangent TX take TY equal to thearc TR. The length of that arc is known from the common methods usedSEC. X. OF NATURAL PHILOSOPHY. 191for the quadratures of figures. From the point Ydraw the right line YZ perpendicular to the tangent.Draw CT meeting that perpendicular in Z, and thecentripetal force will be proportional to the right lineTZ. Q.E.I.For if the force with which the body is attractedfrom T towards C be expressed by the right line TZtaken proportional to it, that force will be resolvedinto two forces TY, YZ, of which YZ drawing thebody in the direction of the length of the thread PT,docs not at all change its motion ; whereas the otherforce TY directly accelerates or retards its mction in the curve STRQ.Wherefore since that force is as the space to be described TR, the accelerations or retardations of the body in describing two proportional parts (ugreater arid a less) of two oscillations, will be always as those parts, andtherefore will cause those parts to be described together. But bodies whichcontinually describe together parts proportional to the wholes, will describethe wholes together also. Q,.E.l).COR. 1. Hence if the body T, hanging by a rectilinear threadAT from the centre A, describe the circular arc STRQ,,and in the mean time be acted on by any force tendingdownwards with parallel directions, which is to the uniform force of gravity as the arc TR to its sine TN, thetimes of the several oscillations will be equal. For becauseTZ, AR are parallel, the triangles ATN, ZTY are similar ; and therefore TZ will be to AT as TY to TN ; that is, if the uniform force ofgravity be expressed by the given length AT, the force TZ. by which theoscillations become isochronous, will be to the force of gravity AT, as thearc TR equal to TY is to TN the sine of that arc.COR. 2. And therefore in clocks, if forces were impressed by some machine upon the pendulum which preserves the motion, and so compoundedwith the force of gravity that the whole force tending downwards shouldbe always as a line produced by applying the rectangle under the arc TRand the radius AR to the sine TN, all the oscillations will becomeisochronous.PROPOSITION LIV. PROBLEM XXXYI.Granting the quadratures of curvilinear figures, it is required to findthe times in which bodies by means of any centripetal force will descendor ascend in any curve lines described in, a plane passing through thecentre of force.Let the body descend from any place S, and move in any curve ST/Rgiven in a plane passing through the centre of force C. Join CS, and lei192 THE MATHEMATICAL PRINCIPLES [BOOK 1Q it be divided into innumerable equal parts, and letDd be one of those parts. From the centre C, withthe intervals CD, Cd, let the circles DT, dt be described, meeting the curve line ST*R in T and t.And because the law of centripetal force is given.and also the altitude CS from which the body atfirst fell, there will be given the velocity of the bodyin any other altitude CT (by Prop. XXXIX). Butthe time in which the body describes the lineola Ttis as the length of that lineola, that is, as the secantof the angle /TC directly, and the velocity inversely.Lei, the ordinate DN, proportional to this time, be made perpendicular tothe right line CS at the point D, and because Dd is given, the rectangleDd X DN, that is, the area DNwc?, will be proportional to the same time.Therefore if PN/?, be a curve line in which the point N is perpetually found,and its asymptote be the right line SQ, standing upon the line CS at rightangles, the area SQPJN D will be proportional to the time in which the bodyin its descent hath described the line ST ; and therefore that area beingfound, the time is also given. Q.E.I.PROPOSITION LV. THEOREM XIX.If a body move in any curve superficies, whose axis passes through thecentre offorce, and from the body a perpendicular be let fall iipon theaxis 。 and a line parallel and equal thereto be drawn from any givenpoint of the axis ; I say, that this parallel line will describe an areaproportional to the time.Let BKL be a curve superficies, T a bodyrevolving in it, STR a trajectory which thebody describes in the same, S the beginningof the trajectory, OMK the axis of the curvesuperficies, TN a right line let fall perpendicularly from the body to the axis ; OP a lineparallel and equal thereto drawn from thegiven point O in the axis ; AP the orthographic projection of the trajectory described bythe point P in the plane AOP in which therevolving line OP is found : A the beginningof that projection, answering to the point S ;TO a right line drawn from the body to the centre ; TG a part thereofproportional to the centripetal force with which the body tends towards thecentre C ; TM a right line perpendicular to the curve superficies ; TI apart thereof proportional to the force of pressure with which the body urgesSEC. X.] OF NATURAL PHILOSOPHY. 193the superficies, and therefore with which it is again repelled by the superficies towards M ; PTF a right line parallel to the axis and passing throughthe body, and OF, IH right lines let fall perpendicularly from the pointsG and I upon that parallel PHTF. I say, now. that the area AGP, described by the radius OP from the beginning of the motion, is proportionalto the time. For the force TG (by Cor. 2, of the Laws of Motion) is resolved into the forces TF, FG ; and the force TI into the forces TH, HI ;but the forces TF, TH, acting in the direction of the line PF perpendicularto the plane AOP, introduce no change in the motion of the body but in a direction perpendicular to that plane. Therefore its motion, so far as it hasthe same direction with the position of the plane, that is, the motion of thepoint P, by which the projection AP of the trajectory is described in thatplane, is the same as if the forces TF, TH were taken away, and the bodywei e acted on by the forces FG, HI alone ; that is, the same as ,f the bodywere to describe in the plane AOP the curve AP by means of a centripetalforce tending to the centre O, and equal to the sum of the forces FG andHI. But with such a force as that (by Prop. 1) the area AOP will be described proportional to the time. Q.E.D.COR. By the same reasoning, if a body, acted on by forces tending totwo or more centres in any the same right line CO, should describe in afree space any curve line ST, the area AOP would be always proportionalto the time.PROPOSITION LVI. PROBLEM XXXVII.Granting the quadratures of curvilinear figures, and supposing thatthere are given both the law of centripetal force tending to a given centre, and the curve superficies whose axis passes through that centre;it is required to find the trajectory which a body will describe in thatsuperficies, when going offfrom a given place with a given velocity,and in a given direction in that superficies.The last construction remaining, let thebody T go from the given place S, in the direction of a line given by position, and turninto the trajectory sought STR, whose orthographic projection in the plane BDO is AP.And from the given velocity of the body inthe altitude SC, its velocity in any other altitude TC will be also given. With thatvelocity, in a given moment of time, let thebody describe the particle Tt of its trajectory,and let P/? be the projection of that particledescribed in the plane AOP. Join Op, anda little circle being described upon the curve superficies about the centre T13194 THE MATHEMATICAL PRINCIPLES [BOOK Iwith the interval TV let the projection of that little circle in the plane AOPbe the ellipsis pQ. And because the magnitude of that little circle T/, andTN or PO its distance from the axis CO is also given, the ellipsis pQ, willbe given both in kind and magnitude, as also its position to the right linePO. And since the area PO/? is proportional to the time, and thereforegiven because the time is given, the angle POp will be given. And thencewill be given jo the common intersection of the ellipsis and. the right lineOp, together with the angle OPp, in which the projection APp of the trajectory cuts the line OP. But from thence (by conferring Prop. XLI, withUs 2d Cor.) the mariner of determining the curve APp easily appears.Then from the several points P of that projection erecting to the planeAOP, the perpendiculars PT meeting the curve superficies in T, there willbe iven the several points T of the trajectory. Q.E.I.SECTION XLf f the motions of bodies tending to each other with centripetal forces.I have hitherto been treating of the attractions of bodies towards an immovable centre; though very probably there is no such thing existent innature. For attractions are made towards bodies, and the actions of thebodies attracted and attracting are always reciprocal and equal, by Law III;BO that if there are two bodies, neither the attracted nor the attracting bodyis truly at rest, but both (by Cor. 4, of the Laws of Motion), being as itwere mutually attracted, revolve about a common centre of gravity. Andif there be more bodies, which are either attracted by one single one whichis attracted by them again, or which all of them, attract each other mutually , these bodies will be so moved among themselves, as that their commoncentre of gravity will either be at rest, or move uniformly forward in aright line. I shall therefore at present go on to treat of the motion ofbodies mutually attracting each other ; considering the centripetal forcesas attractions ; though perhaps in a physical strictness they may more trulybe called impulses. But these propositions are to be considered as purelymathematical; and therefore, laying aside all physical considerations, Imake use of a familiar way of speaking, to make myself the more easilyunderstood by a mathematical reader.PROPOSITION LVII. THEOREM XX.Two bodies attracting each other mutually describe similarfigures abouttheir common centre of gravity, and about each other mutually.For the distances of the bodies from their common centre of gravity areleciprocally as the bodies; and therefore in a given ratio to each other:*nd thence, bv composition of ratios, in a given ratio to the whole distanceSEC. XL] OF NATURAL PHILOSOPHY. 195between the bodies. Now these distances revolve about their common termwith an equable angular motion, because lying in the same right line theynever change their inclination to each other mutually But right linesthat are in a given ratio to each other, and revolve about their terms withan equal angular motion, describe upon planes, which either rest withthose terms, or move with any motion not angular, figures entirely similarround those terms. Therefore the figures described by the revolution otthese distances are similar. Q.E.D.PROPOSITION LVIll. THEOREM XXI.If two bodies attract each other mutually with forces of any kind, andin the mean time revolve about the common centre of gravity ; I say,that, by the same forces, there may be described round either body unmoved ajigure similar and equal to the figures ivhich the bodies somoving describe round each other mutually.Let the bodies S and P revolve about their common centre of gravityC, proceeding from S to T, and from P to Q,. From the given point s lotthere be continually drawn sp, sq, equal and parallel to SP, TQ, ; and the;urve pqv, which the point p describes in its revolution round the immovablepoint s, will be similar and equal to the curves which the bodies S and Pdescribe about each other mutually ; and therefore, by Theor. XX, similarto the curves ST and PQ,V which the same bodies describe about theircommon centre of gravity C and that because the proportions of the linesSC. CP, and SP or sp, to each other, are given.CASE 1. The common centre of gravity C (by Cor. 4, of the Laws of Mo