tion) is either at rest, or moves uniformly in a right line. Let us firstsuppose it at rest, and in s and p let there be placed two bodies, one immovable in s, the other movable in p, similar and equal to the bodies S aridP. Then let the right lines PR and pr touch the curves PQ, and pq ki Pand p, and produce CQ, and sq to R and r. And because the figuresCPRQ, sprq are similar, RQ, will be to rq as CP to sp, and therefore in agiven ratio. Hence if the force with which the body P is attracted towards the body S, and by consequence towards the intermediate point thecentre C, were to the force with which the body p is attracted towards theCentre 5. in the same given ratio, these forces would in equal times attract196 THE MATHEMATICAL PRINCIPLES |BoOK 1the bodies from the tangents PR, pr to the arcs PQ, pq, through the intervals proportional to them RQ,, rq ; and therefore this last force (tendingto s) would make the body p revolve in the curve pqv, which would becomrsimilar to the curve PQV, in which the first force obliges the body P i(revolve ; and their revolutions would be completed in the same timegBut because those forces are not to each other in the ratio of CP to sp, bu;(by reason of the similarity and equality of the bodies S and s, P and /and the equality of the distances SP, sp) mutually equal, the bodies iiequal times will be equally drawn from the tangents; and therefore tLVthe body p may be attracted through the greater interval rq, there is required a greater time, which will be in the subduplicate ratio of the intervals ; because, by Lemma X, the spaces described at the very beginning olthe motion are in a duplicate ratio of the times. Suppose, then the velocityof the body p to be to the velocity of the body P in a subduplicate ratio ofthe distance sp to the distance CP, so that the arcs pq, PQ, which are in asimple proportion to each other, may be described in times that are in nsubduplicate ratio of the distances ; and the bodies P, p, always attractedby equal forces, will describe round the quiescent centres C and s similarfigures PQV, pqv, the latter of which pqv is similar and equal to the figureivhich the body P describes round the movable body S. Q.E.I)CASE 2. Suppose now that the common centre of gravity, together withthe space in which the bodies are moved among themselves, proceeds uniformly in a right line; and (by Cor. 6, of the Laws of Motion) all the motions in this space will be performed in the same manner as before ; andtherefore the bodies will describe mutually about each other the same figures as before, which will be therefore similar and equal to the figure pqv.Q.E.D.COR. 1. Hence two bodies attracting each other with forces proportionalto their distance, describe (by Prop. X) both round their common centre olgravity, and round each other mutually concentrical ellipses ; and, viceversa, if such figures are described, the forces are proportional to the distances.COR. 2. And two bodies, whose forces are reciprocally proportional tothe square of their distance, describe (by Prop. XI, XII, XIII), both roundtheir common centre of gravity, and round each other mutually, conic sections having their focus in the centre about which the figures are described.And, vice versa, if such figures are described, the centripetal forces are reciprocally proportional to the squares of the distance.COR. 3. Any two bodies revolving round their common centre of gravitydescribe areas proportional to the times, by radii drawn both to that centreand to each other mutually>EC. XL] OF NATURAL PHILOSOPHY. 197PROPOSITION LIX. THEOREM XXII.The periodic time of two bodies S and P revolving round their commoncentre of gravity C,is to the periodic time of one of the bwlies 1? revolving round the other S remaining unmoved, and describing a figure similar and equal to those which the bodies describe about eachother mutuallyr, in a subduplicate ratio of the other body S to the sii/rnof the bodies S -f P.For, by the demonstration of the last Proposition, the times in whichany similar arcs PQ and pq are described are in a subduplicate ratio of thedistances CP and SP, or sp, that is, in a subduplicate ratio of the ody Sto the sum of the bodies S + P. And by composition of ratios, the sumsof the times in which all the similar arcs PQ and pq are described, that is,the whole times in which the whole similar figures are described are in thesame subduplicate ratio. Q.E.D.PROPOSITION LX. THEOREM XXIII.If two bodies S and P, attracting each other with forces reciprocally proportional to the squares of their distance, revolve about their commoncentre of gravity ; I say, that the principal axis of the ellipsis whicheither of the bodies, as P, describes by this motion about the other S,will be to the principal axis of the ellipsis, which the same body P maydescribe in the same periodical time about the other body S quiescent,as the sum of the two bodies S + P to the first of two mean, proportionals between that sum and the other body S.For if the ellipses described were equal to each other, their periodic timesby the last Theorem would be in a subduplicate ratio of the body S to thesum of the bodies S 4- P. Let the periodic time in the latter ellipsis bediminished in that ratio, and the periodic times will become equal ; but,by Prop. XV, the principal axis of the ellipsis will be diminished in a ratiosesquiplicate to the former ratio ; that is, in a ratio to which the ratio ofS to S 4- P is triplicate ; and therefore that axis will be to the principalaxis of the other ellipsis as the first of two mean proportionals between S-f- P and S to S 4- P. And inversely the principal axis of the ellipsis described about the movable body will be to the principal axis of that describedround the immovable as S + P to the first of two mean proportionals between S 4- P and S. Q.E.D.PROPOSITION LXI. THEOREM XXIV.If two bodies attracting each other with any kind of forces, and nototherwise agitated or obstructed, are moved in any manner whatsoever,those motions will be the same as if they did not at all attract eachother mutually, but were both attracted with the sameforces by a thirdbody placed in their common centre of gravity ; and the law of the198 THE MATHEMATICAL PRINCIPLES [BOOK Iattracting Jones will be the sam# in respect of the distance of the.bodies from, the common centre, as in respect of the distance betweenthe two bodies.For those forces with which the bodies attract each other mutually, bytending to the bodies, tend also to the common centre of gravity lying directly between them ; and therefore are the same as if they proceeded froman intermediate body. QJG.D.And because there is given the ratio of the distance of either body fromthat common centre to the distance between the two bodies, there is given,of course, the ratio of any power of one distance to the same power of the. ther distance ; and also the ratio of any quantity derived in any mannerfrom one of the distances compounded any how with given quantities, toanother quantity derived in like manner from the other distance, and asmany given quantities having that given ratio of the distances to the firstTherefore if the force with which one body is attracted by another be directly or inversely as the distance of the bodies from each other, or as anypower of that distance ; or, lastly, as any quantity derived after any manner from that distance compounded with given q-uantities ; then will thesame force with which the same body is attracted to the common centre ofgravity be in like manner directly or inversely as the distance of the attracted body from the common centre, or as any power of that distance ;or, lastly, as a quantity derived in like sort from that distance compoundedwith analogous given quantities. That is, the law of attracting force willbe the same with respect to both distances. Q,.E.D.PROPOSITION LXII. PROBLEM XXXVIII.To determine the motions of two bodies which attract each other withforces reciprocally proportional to the squares of the distance betweenthem, aflid are, let fallfrom given places.The bodies, by the last Theorem, will be moved in the same manner asif they were attracted by a third placed in the common centre of theirgravity ; and by the hypothesis that centre will be quiescent at the beginning of their motion, and therefore (by Cor. 4, of the Laws of Motion) willbe always quiescent. The motions of the bodies are therefore to be determined (by Prob. XXV) in the same manner as if they were impelled byforces tending to that centre: and then we shall have the motions of thebodies attracting each other mutually. Q.E.I.PROPOSITION LXIII. PROBLEM XXXIX.To determine the motions of two bodies attracting each other with forcesreciprocally proportional to the squares of their distance, and goingofffrom given places in, given directions with given velocities.The motions of the bodies at the beginning being given, there is givenSEC. XL] OF NATURAL PHILOSOPHY. 1%also the uniform motion of the common centre of gravity, and the motionof the space which moves along with this centre uniformly in a right line,and also the very first, or beginning motions of the bodies in respect of thisspace. Then (by Cor. 5, of the Laws, and the last Theorem) the subsequent motions will be performed in the same manner in that space, as ifthat space together with the common centre of gravity were at rest, and asif the bodies did not attract each other, but were attracted by a third bodyplaced in that centre. The motion therefore in this movable space of eachbody going off from a given place, in a given direction, with a given velocity, and acted upon by a centripetal force tending to that centre, is to bedetermined by Prob. IX and XXVI, and at the same time will be obtainedthe motion of the other round the same centre. With this motion compound the uniform progressive motion of the entire system of the space andthe bodies revolving in it, and there will be obtained the absolute motionof the bodies in immovable space. Q..E.I.PROPOSITION LXIV. PROBLEM XL.Supposingforces with which bodies mutually attract each other to increase in a simple ratio of their distances from the centres ; it is roquiredto find the motions of several bodies among themselves.Suppose the first two bodies T and Lto have their common centre of gravity inL). These, by Cor. 1, Theor. XXI, willSdescribe ellipses having their centres in D,the magnitudes of which ellipses areknown by Prob. V.J---。- ? LLet now a third body S attract the twoformer T and L with the accelerative forces ST, SL, and let it be attracted again by them. The force ST (by Cor. 2, of the Laws of Motion) isresolved into the forces SD, DT ; and the force SL into the forces SD andDL. Now the forces DT, DL. which are as their sum TL, and thereforeas the accelerative forces with which the bodies T and L attract each othermutually, added to the forces of the bodies T and L, the first to the first,and the last to the last, compose forces proportional to the distances DTand DL as before, but only greater than those former forces : and therefore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will cause thosebodies to describe ellipses as before, but with a swifter motion. The remaining accelerative forces SD and DL, by the motive forces SD X TandSD X L, which are as the bodies attracting those bodies equally and in thedirection of the lines TI, LK parallel to DS, do not at all change their situations with respect to one another, but cause them equally to approach tothe line IK ; which must be imagined drawn through the middle of thebody S, and perpendicular to the line DS. But that approach to the line200 THE MATHEMATICAL PRINCIPLES [BoOK I.IK will be hindered by causing the system of the bodies T and L on oneside, and the body S on the other, with proper velocities, to revolve roundthe common centre of gravity C. With such a motion the body S, becausethe sum of the motive forces SD X T and SD X L is proportional to thedistance OS, tends to the centre C, will describe an ellipsis round the samecentre C; and the point D, because the lines CS and CD are proportional,will describe a like ellipsis over against it. But the bodies T and L, attracted by the motive forces SD X T and SD X L, the first by the first,and the last by the last, equally and in the direction of the parallel lines TIand LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion)continue to describe their ellipses round the movable centre D, as before.Q.E.I.Let there be added a fourth body V, and, by the like reasoning, it willbe demonstrated that this body and the point C will describe ellipses aboutthe common centre of gravity B ; the motions of the bodies T, L, and Sround the centres D and C remaining the same as before ; but accelerated.Arid by the same method one may add yet more bodies at pleasure. Q..E.I.^This would be the case, though the bodies T and L attract each othermutually with accelerative forces either greater or less than those withwhich they attract the other bodies in proportion to their distance. Letall the mutual accelerative attractions be to each other as the distancesmultiplied into the attracting bodies ; and from what has gone before itwill easily be concluded that all the bodies will describe different ellipseswith equal periodical times about their common centre of gravity B, in animmovable plane. Q.E.I.PROPOSITION LXV. THEOREM XXV.Bodies, whose forces decrease in a duplicate ratio of their distances fromtheir centres, may move among" themselves in ellipses ; and by radiidrawn to the foci may describe areas proportional to the times verynearly.In the last Proposition we demonstrated that case in which the motionswill be performed exactly in ellipses. The more distant the law of theforces is from the law in that case, the more will the bodies disturb eachother s motions ; neither is it possible that bodies attracting each othermutually according to the law supposed in this Proposition should moveexactly in ellipses, unless by keepirg a certain proportion of distances fromeach other. However, in the following crises the orbits will not much differ from ellipses.CASE I. Imagine several lesser bodies to revolve about some very greatone at different distances from it, and suppose absolute forces tending torvery one of the bodies proportional to each. And because (by Cor. 4, olthe I aws) the common centre of gravity of them all is either at rest, 01iSEC. XL] OF NATURAL PHILOSOPHY. 20 imoves uniformly forward in a right line, suppose the lesser bodies so smallthat the groat body may be never at a sensible distance from that centre ;and then the great body will, without any sensible error, be either at rest,or move uniformly forward in a right line; and the lesser will revolveabout that great one in ellipses, and by radii drawn thereto will describeareas proportional to the times ;if we except the errors that may be introduced by the receding of the great body from the common centre of gravity,or by the mutual actions of the lesser bodies upon each other. But thelesser bodies may be so far diminished, as that this recess and the mutualactions of the bodies on each other may become less than any assignable;and therefore so as that the orbits may become ellipses, and the areas answer to the times, without any error that is not less than any assignable.Q.E.O.CASE 2. Let us imagine a system of lesser bodies revolving about a verygreat one in the manner just described, or any other system of two bodiesrevolving about each other to be moving uniformly forward in a right line, andin the mean time to be impelled sideways by the force ofanother vastly greaterbody situate at a great distance. And because the equal accelerative forceswith which the bodies are impelled in parallel directions do not change thesituation of the bodies with respect to each other, but only oblige the wholesystem to change its place while the parts still retain their motions amongthemselves, it is manifest that no change in those motions of the attractedbodies can arise from their attractions towards the greater, unless by theinequality of the accelerative attractions, or by the inclinations of the linestowards each other, in whose directions the attractions are made. Suppose,therefore, all the accelerative attractions made towards the great bodyto be among themselves as the squares of the distances reciprocally ; andthen, by increasing the distance of the great body till the differences of fheright lines drawn from that to the others in respect of their length, and theinclinations of those lines to each other, be less than any given, the motions of the parts of the system will continue without errors that are notless than any given. And because, by the small distance of those parts fromeach other, the whole system is attracted as if it were but one body, it willtherefore be moved by this attraction as if it were one body ; that is, itscentre of gravity will describe about the great bod/ one of the conic sections (that is, a parabola or hyperbola when the attraction is but languidand an ellipsis when it is more vigorous) ; and by radii drawn thereto, itwill describe areas proportional to the times, without any errors but thoswhich arise from the distances of the parts, which are by the suppositionexceedingly small, and may be diminished at pleasure. Q,.E.O.By a like reasoning one may proceed to more compounded cases in infinitum.COR 1. In the second Case, the nearer the very great body approaches to^0^ THE MATHEMATICAL PRINCIPLES [CoOK Ithe system of two or more revolving bodies, the greater will the perturbation be of the motions of the parts of the system among themselves; because the inclinations of the lines drawn from that great body to thoseparts become greater ; and the inequality of the proportion is also greater.COR. 2. But the perturbation will be greatest of all, if we suppose theuccelerative attractions of the parts of the system towards the greatest bodyof all are not to each other reciprocally as the squares of the distancesfrom that great body ; especially if the inequality of this proportion begreater than the inequality of the proportion of the distances from thegreat body. For if the accelerative force, acting in parallel directionsand equally, causes no perturbation in the motions of the parts of thesystem, it must of course, when it acts unequally, cause a perturbation somewhere, which will be greater or less as the inequality is greater or less.The excess of the greater impulses acting upon some bodies, and not actingupon others, must necessarily change their situation among themselves. Andthis perturbation, added to the perturbation arising from the inequalityand inclination of the lines, makes the whole perturbation greater.COR. *. Hence if the parts of this system move in ellipses or circleswithout any remarkable perturbation, it is manifest that, if they are at allimpelled by accelerative forces tending to any other bodies, the impulse isvery weak, or else is impressed very near equally and in parallel directionsupon all of them.PROPOSITION LXVL THEOREM XXVI.Tf three bodies whose forces decrease in a duplicate ratio of the distancesattract each other mutually ; and the accelerative attractions of anytwo towards the third be between themselves reciprocally as the squares,of the distances ; and the two least revolve about the greatest ; I say,that the interior of the tivo revolving bodies will, by radii drawn to theinnermost and greatest, describe round thai body areas more proportional to the times, and a figure more approaching to that of an ellipsis having its focus in the point of concourse of the radii, if that greatbody be agitated by those attractions, than it would do if lhat greatbody were not attracted at all by the lesser, but remained at rest ; orthan it would if that great body were very much more or very muchless attracted, <>r very much more or very much less agitated, by theattractions.This appears plainly enough from the demonstration of the secondCorollary of tl.e foregoing Proposition; but it may be made out afterthis manner by a way of reasoning more distinct and more universallyconvincing.CASE 1. Let the lesser bodies P and S revolve in the same plane aboutthe greatest body T, the body P describing the interior orbit PAB, and SSEC. XI.J OF NATURAL PHILOSOPHY. 203the exterior orbit ESE. Let SK be the mean distance of the bodies P andS ; and let the accelerative attraction of the body P towards S, at thatmean distance, be expressed by that line SK. Make SL to SK as theE Csquare of SK to the square of SP, and SL will be the accelerative attraction of the body P towards S at any distance SP. Join PT, and drawLM parallel to it meeting ST in M; and the attraction SL will be resolved (by Cor. 2. of the Laws of Motion) into the attractions SM, LM. Andso the body P will be urged with a threefold accelerative force. One ofthese forces tends towards T, and arises from the mutual attraction of thebodies T and P. By this force alone the body P would describe round thebody T, by the radius PT, areas proportional to the times, and anellipsis whose focus is in the centre of the body T ; and this it would dowhether the body T remained unmoved, or whether it were agitated by thatattraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor.XXI. The other force is that of the attraction LM, which, because ittends from P to T, will be superadded to and coincide with the formerforce ; and cause the areas to be still proportional to the times, by Cor. 3,Theor. XXI. But because it is not reciprocally proportional to the squareof the distance PT, it will compose, when added to the former, a forcevarying from that proportion : which variation will be the greater by howmuch the proportion of this force to the former is greater, cceteris paribus.Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force withwhich the ellipsis is described about the focus T ought to be directed tothat focus, and to be reciprocally proportional to the square of the distancePT, that compounded force varying from that proportion will make theorbit PAB vary from the figure of an ellipsis that has its focus in the pointI1; and so much the more by how much the variation from that proportionis greater ; and by consequence by how much the proportion of the secondforce LM to the first force is greater, cceteris paribus. But now the thirdforce SM, attracting the body P in a direction parallel to ST, composes withthe other forces a new force which is no longer directed from P to T : and whichvaries so much more from this direction by how much the proportion of thisthird force to the other forces is greater, cceterisparibus ; arid therefore causesthe body P to describe, by the radius TP, areas no longer proportional to thetimes : and therefore makes the variation from that proportionality so muchgreater by how much the proportion of this force to the others is greater.But this third force will increase the variation of the orbit PAB from th*THE MATHEMATICAL PRINCIPLES [BOOK 1elliptical figure before-mentioned upon two accounts ;first because thatforce is not directed from P to T ; and, secondly, because it is not reciprocally proportional to the square of the distance PT. These things beingpremised, it is manifest that the areas are then most nearly proportional tothe times, when that third force is the least possible, the rest preservingtheir former quantity ; and that the orbit PAB does then approach nearestto the elliptical figure above-mentioned, when both the second and third,but especially the third force, is the least possible; the first force remaining in its former quantity.Let the accelerative attraction of the body T towards S be expressed bythe line SN ;then if the accelerative attractions SM and SN were equal,these, attracting the bodies T and P equally and in parallel directionswould not at all change their situation with respect to each other. The motions of the bodies between themselves would be the same in that case as ifthose attractions did not act at all, by Cor. 6, of the Laws of Motion. And,by a like reasoning, if the attraction SN is less than the attraction SM, itwill take away out of the attraction SM the part SN, so that there will remain only the part (of the attraction) MN to disturb the proportionality ofthe areas and times, and the elliptical figure of the orbit. And in likemanner if the attraction SN be greater than the attraction SM, the perturbation of the orbit and proportion will be produced by the difference MNalone. After this manner the attraction SN reduces always the attractionSM to the attraction MN, the first and second attractions rema ning perfectly unchanged ; and therefore the areas and times come then nearest toproportionality, and the orbit PAB to the above-mentioned elliptical figure,when the attraction MN is either none, or the least that is possible; thatis, when the accelerative attractions of the bodies P and T approach as nearas possible to equality ; that is, when the attraction SN is neither none atall, nor less than the least of all the attractions SM, but is, as it were, amean between the greatest and least of all those attractions SM, that is,not much greater nor much less than the attraction SK. Q.E.D.CASE 2. Let now the lesser bodies P. S, revolve about a greater T in dif