Principia Mathematica (Motte Translation, 1848) — part 14 of 45

For let the thread PT meet the cycloid QRS in T, and the circle dOS in V, and let CV be drawn ; and to the rectilinear part of the thread PT from the extreme points P and T let there be erected the perpendiculars BP, T W, meeting the right line CY in B and W. It is evident, from the construction and generation of the similar figures AS, SR, that those per- pendiculars PB, TW, cut off from CY the lengths YB, YW equal the diameters of the wheels OA, OR. Therefore TP is to YP (which is dou- ble the sine of the angle YBP when |BY is radius) as BW to BY, or AO -hOR to AO, that is (since CA and CO, CO and CR, and by division AO and OR are proportional), as CA + CO to CA, or, if BY be bisected in E, as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the rectilinear part of the thread PT is always equal to the arc of the cycloid PS, and the whole thread APT is always equal to the half of the cycloid APS, that is (by Cor. 2, Prop. XliX), to the length AR. And there- fore contrariwise, if the String remain always equal to the length AR, the point T will always move in the given cycloid Q.RS. Q.E.D.

Cor. The string AR is equal to the semi-cycloid AS, and therefore has the same ratio to AC the semi-diameter of the exterior globe as the like semi-cycloid SR has to CO the semi-diameter of the interior globe.

PROPOSITION LI. THEOREM XYIII.

If a centripetal force tending on all sides to the centre C of a globe, be in all places as the. distance of the place from the centre, and by this force alone acting iipon it, the body T oscillate [in the 'manner above de- scribed) in the perimeter of the cycloid Q.RS ; I say, that all the oscil- lations, hoio unequal soever in themselves, loill be performed in equal times.

For upon the tangent TW infinitely produced let fall the perpendicular CX, and join CT. Because the centripetal force with which the body T is impelled towards C is as the distance CT, let this (by Cor. 2, of the Laws) be resolved into the parts CX, TX, of which CX impelling the body directly from P stretches the thread PT, and by the resistance the thread makes to it is totally employed, producing no other effect ; but the other part TX, impelling the body transversely or towards X, directly accelerates the motion in the cycloid. Then it is plain that the accelera- tion of the body, proportional to this accelerating force, will be every

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moment 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. ], Prop. XLIX), as the length of the arc of the cycloid Til. If there- fore two pendulums APT, kfpt^ be unequally drawn aside from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be de- scribed TR, ^R. But the parts described at the beginning of the motion are as the accelerations, that is, as the wholes that are to be described at the be- ginning, and therefore the parts which remain to be described, and the subsequent accelerations proportional to those parts, are also as the wholes, and so on. Therefore the accelerations, and consequently the velocities generated, and the parts described with those velocities, and the parts to be described, are always as the wholes ; and therefore the parts to 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 place R through the same cycloidal arcs with a retrograde motion, is retarded in the several places they pass through by the same forces by which their de- scent was accelerated ; it is plain that the velocities of their ascent and de- scent through the same arcs are equal, and consequently performed in equal times ; and, therefore, since the two parts of the cycloid RS and RQ, lying on either side of the perpendicular are similar and equal, the two pendu- lums will perform as well the wholes as the halves of their oscillations in the same times. Q.E.D.

Cor. The force with which the body T is accelerated or retarded in any place T of the cycloid, is to the whole weight of the same body in the highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR.

PROPOSITION LII. PROBLEM XXXIV.

Tb define the velocities of the 'pendulums in the several places^ and the times in lohich both the entire oscillations, and the several parts of them, are performed.

About any centre G, with the interval GH equal to the arc of the cycloid RS, describe a semi-circle HKM bisected by the semi-diameter GK. And if a centripe- tal force proportional to the distance of the places from the centre tend to the centre G, and it be in the peri- meter HIK equal to the centripetal force in the perime- ter of the globe Q,OS tending toAvards its centre, and at the same time that the pendulum T is let fall from the highest place S, a body, as L, is let fall from H to G ; then because the

Sec. X.] OF natural philosophy. 1S9

forces which act upon the bodies are equal at the be- ginning, and always proportional to the spaces to be described TR, LG, and therefore if TR and LG are equal, are also equal in the places T and L, it is plain that those bodies describe at the beginning equal spaces '^'^ ST, HL, and therefore are still acted upon equally, and continue to describe equal spaces. Therefore by Prop. XXXYIII, the time in which the body describes the arc ST is to the time of one oscillation, as the arc HI the time in which the body H arrives at L, to the semi-periphery HKM, the time in which the body H will come to M. And the velocity of the pendulous body in the place T is to its velocity in the lowest place R, that is, the velocity of the body H in the place L to its velocity in the place G, or the momentary increment of the line HL to the momentary increment of the line HG (the arcs HI, HK increasing with an equable flux) as the ordinate LI to the radius GK, or as ^/SW — TR^ to SR. Hence, since in unequal oscillations there are described in equal time arcs proportional to the en- tire arcs of the oscillations, there are obtained from the times given, both the velocities and the arcs described in all the oscillations universally. Which was first required.

Let now any pendulous bodies oscillate in different cycloids described within different globes, whose absolute forces are also different ; and if the absolute force of any globe Q,OS be called Y, the accelerative force with which the pendulum is acted on in the circumference of this globe, when it begins to move directly towards its centre, will be as the distance of the pendulous body from that centre and the absolute force of the globe con- junctly, that is, as CO X Y. Therefore the lineola HY, which is as this accelerated force CO X V, will be described in a given time; and if there be erected the perpendicular YZ meeting the circumference in Z, the nascent arc HZ will denote that given time. But that nascent arc HZ is in the

subduplicate ratio of the rectangle GHY, and therefore as ^/GH X CO X Y Whence the time of an entire oscillation in the cycloid QRS (it being as the semi-periphery HKM, which 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 y'GH X CO X Y inversely ; that is, because

GH and SR are equal, as VpTTTpF; oi* (by Cor. Prop. L,) as V-Tn — v*

Therefore the oscillations in all globes and cycloids, performed with what absolute forces soever, are in a ratio compounded of the subduplicate ratio of the length of the string directly, and the subduplicate ratio of the distance between the point of suspension and the centre of the globe inversely, and the subduplicate ratio of the absolute force of the globe inversely also. Q.E.L

190 THE MATHEMATICAL PRINCIPLES [Bg^^ I.

Cor. 1. Hence also the times of oscillating, falling, and revolving bodies may be compared among themselves. For if the diameter of the wheel with which the cycloid is described within the globe is supposed equal to the semi-diameter of the globe, the cycloid will become a right line passing throuo'h the centre of the eiobe, and the oscillation will be chan2;ed into a descent and subsequent ascent in that right line. Whence there is given both the time of the descent from any place to the centre, and the time equal to it in which the body revolving uniformly about the centre of the globe at any distance describes an arc of a quadrant For this time (by Case 2) is to the time of half the oscillation in any cycloid QRS as 1 to

AR ^ AC-

Cor. 2. Hence also follow what Sir Christopher Wren and M. Hmjgens have discovered concerning the vulgar cycloid. For if the diameter of the globe be infinitely increased, its sphaerical superficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of our's will become the same with the common cycloid. But in that case the length of the arc of the cycloid bet^veen that plane and the describing point will become equal to four times the versed sine of half the arc of the wheel between the same plane and the describing point, as was discovered by Sir Christopher Wren. And a pendulum between two such cycloids will oscillate in a similar and equal cycloid in equal times, as M. Huygens demonstrated. The descent of 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 constitution of the Earth, in so far as wheels moving in any of its great circles will de- scribe, by the motions of nails fixed in their perimeters, cycloids without the globe ; and pendulums, in mines and deep caverns of the Earth, must oscil- late in cycloids within the globe, that those oscillations may be performed in equal times. For gravity (as will be shewn in the third book) decreases in its progress from the superficies of the Earth ; upwards in a duplicate ratio of the distances from the centre of the Earth ; downwards in a sim- ple ratio of the same.

PROPOSITION LIII. PROBLEM XXXY.

Granting the quadratures of curvilinear figures, it is required to find

the forces luith lohich bodies moving in given curve lines may always

perform their oscillations iji equal times. - .

Let the body T oscillate in any curve line STRQ., whose axis is AR

passing through the centre of force C. Draw TX touching that curve in

any place of the body T, and in that tangent TX take TY equal to the

arc TR. The length of that arc is known from the common methods used

Sec. X.

OF NATURAL PHILOSOPHY.

191

for the quadratures of figures. From the point Y draw the right line YZ perpendicular to the tangent. Draw GT meeting that perpendicular in Z, and the centripetal force will be proportional to the right line TZ. aE.L

For if the force with which the body is attracted from T towards C be expressed by the right line TZ taken proportional to it, that force will be resolved into two forces TY, YZ, of which YZ drawing the body in the direction of the length of the thread PT, does not at all change its motion ; whereas the other force TY directly accelerates or retards its motion in the curve STRQ,. Wherefore since that force is as the space to be described TK, the acceler- ations or retardations of the body in describing two proportional parts (a greater and a less) of two oscillations, will be always as those parts, and therefore will cause those parts to be described together. But bodies which continually describe together parts proportional to the wholes^ will describe the wholes together also. Q..E.D.

Cor. 1. Hence if the body T, hanging by a rectilinear thread A

AT from the centre A, describe the circular arc STRQ,, l^ g

and in the mean time be acted on by any force tending ^^v in I \rT downwards with parallel directions, which is to the uni- ^^y^ form force of gravity as the arc TR to its sine TN, the ^^\

times of the several oscillations will be equal. For because ^z

TZ, AR are parallel, the triangles ATN, ZTY are similar ; and there- fore TZ will be to AT as TY to TN ; that is, if the uniform force of gravity be expressed by the given length AT, the force TZ, by which the oscillations become isochronous, will be to the force of gravity AT, as the arc TR equal to TY is to TN the sine of that arc.

Cor. 2. And therefore in clocks, if forces were impressed by some ma- chine upon the pendulum which preserves the motion, and so compounded with the force of gravity that the whole force tending downwards should be always as a line produced by applying the rectangle under the arc TR and the radius AR to the sine TN, all the oscillations will become isochronous.

PROPOSITION Liy. PROBLEM XXXYI.

Granting the quadratures of curvilinear figures, it is required to find the times in which bodies by means of any centripetal force loill descend or ascend in any curve lines described m a plane passing through the centre of force. Let the body descend from any place S, and move in any curve ST^R

given in a plane passing through the centre of force C. Join CS, and let

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THE MATHEMATICAL PRINCIPLES

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Q it be divided into innumerable equal parts, and let Dd be one of those parts. From the centre C, witii the intervals CD, Cd, let the circles DT, dt be de- scribed, 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 at first fell, there will be given the velocity of the body in any other altitude CT (by Prop. XXXIX). But the time in which the body describes the lineola T^ is as the length of that lineola, that is, as the secant of the angle ^TC directly, and the velocity inversely. Let the ordinate DN, proportional to this time, be made perpendicular to the right line CS at the point D, and because Dd is given, the rectangle Dd X DN, that is, the area DlSnd, will be proportional to the same time. Therefore if PN?i be a curve line in which the point N is perpetually found, and its asymptote be the right line SO. standing upon the line CS at right angles, the area SQPND will be proportional to the time in which the body in its descent hath described the line ST ; and therefore that area being found, the time is also given. Q.E.I.

PROPOSITION LV. THEOREM XIX.

If a body move in any curve superficies, whose axis passes through the centre of force, and from the body a perpendicidar be let fall upon the axis] and a line parallel and equal thereto be drawn from any given point of the axis ; I say, that this parallel line will describe an area proportional to the time.

Let BKL be a curve superficies, T a body revolving in it, STR a trajectory which the body describes in the same, S the beginning of the trajectory, OMK the axis of the curve superficies, TN a right line let fall perpendic- ularly from the body to the axis ; OP a line parallel and equal thereto drawn from the given point O in the axis ; AP the orthogra- phic projection of the trajectory described by the point P in the plane AOP in which the revolving line OP is found : A the beginning of that projection, answering to the point S ; TO a right line drawn from the body to the centre ; TG a part thereof proportional to the centripetal force with which the body tends towards the centre C ; TM a right line perpendicular to the curve superficies ; TI a part thereof proportional to the force of pressure with which the body urges

Sec. X.]

OF NATURAL PHILOSOPHY.

193

the superficies, and tlierefore "vvith -vYhicli it is again repelled by the super- ficies towards M ; PTF a right line parallel to the axis and passing through the body, and GF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I say, now, that the area AOP, de- scribed by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re- solved 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 perpendicular to the plane AOP, introduce no change in the motion of the body but in a di- rection perpendicular to that plane. Therefore its motion, so far as it has the same direction with the position of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is described in that plane, is the same as if the forces TF, TH were taken away, and the body were acted on by the forces FG, HI alone ; that is, the same as if the body were to describe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the sum of the forces FG and HI. But with such a force as that (by Prop. 1) the area AOP will be de- scribed proportional to the time. Q,.E.D.

Cor. By the same reasoning, if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST, the area AOP would be alwa-ys proportional to the time.

PROPOSITION LYI. PROBLEM XXXVIL

Granting the quadratures of curvilinear figures, and supposing that there are given both the law of centripetal force tending to a given cen- tre, and the curve superficies whose axis passes through that centre ; it is required to find the trajectory ivhich a body luill describe in that superficies, %ohen going ofi" from a given "place with a given velocity^ and in a given direction in that superficies. The last construction remaining, let the body T go from the given place S, in the di- rection of a line given by position, and turn into the trajectory sought STR, whose ortho- graphic projection in the plane BDO is AP. And from the given velocity of the body in the altitude SC, its velocity in any other al- titude TC will be also given. With that velocity, in a given moment of time, let the body describe the particle T^ of its trajectory, and let Vp be the projection of that particle described in the plane AOP. Join Op, and a little circle being described upon the curve superficies about the centre T

13

194 THE MATHEMATICAL PRINCIPLES [BoOK 1.

with the interval Tt let the projection of that little circle in the plane AOP be the ellipsis pQ,. And because the magnitude of that little circle Tt, and TN or PO its distance from the axis CO is also given, the ellipsis /)Q, -vyill be given both in kind and magnitude, as also its position to the right line PO. And since the area POp is proportional to the time, and therefore given because the time is given, the angle FOp will be given. And thence will be given jo the common intersection of the ellipsis and the right line Op, together with the angle OFp, in which the projection APp of the tra- jectory cuts the line OP. But from thence (by conferring Prop. XLI, with its 2d Cor.) the manner of determining the curve APp easily appears. Then from the several points P of that projection erecting to the plane AOP, the perpendiculars PT meeting the curve superficies in T, there will be given the several points T of the trajectory. Q,.E.I.

SECTION XL Of the motions of bodies tending- to each other with centripetal forces, I have hitherto been treating of the attractions of bodies towards an im- movable centre ; though very probably there is no such thing existent in nature. For attractions are made towards bodies, and the actions of the bodies attracted and attracting are always reciprocal and equal, by Law III ; BO that if there are two bodies, neither the attracted nor the attracting body is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it were mutually attracted, revolve about a common centre of gravity. And if there be more bodies, which are either attracted by one single one which is attracted by them again, or which all of them, attract each other mutu- ally , these bodies will be so moved among themselves, as that their common centre of gravity will either be at rest, or move uniformly forward in a right line. I shall therefore at present go on to treat of the motion of bodies mutually attracting each other ; considering the centripetal forces as attractions ; though perhaps in a physical strictness they may more truly be called impulses. But these propositions are to be considered as purely mathematical ; and therefore, laying aside all physical considerations, I make use of a familiar way of speaking, to make myself the more easily understood by a mathematical reader.

PROPOSITION LVn. THEOREM XX.

Tioo bodies attracting each other nnutually describe similar figures about their common centre of gravity, and about each other miitually. For the distances of the bodies from their common centre of gravity are

reciprocally as the bodies ; and therefore in a given ratio to each other ;

and thence, by composition of ratios, in a given ratio to the whole distance

Sec. XL] of natural philosophy. 195

between the bodies. Now these distances revolve about their common term with an equable angular motion, becatise lying in the same right line they never change their inclination to each other mutually. But right lines that are in a given ratio to each other, and revolve about their terms with an equal angular motion, describe upon planes, which either rest with those terms, or move with any motion not angular, figures entirely similar round those terms. Therefore the figures described by the revolution of these distances are similar. Q.E.D.

PROPOSITION LYIIL THEOREM XXI.

If two bodies attract each other mutually with forces of airy kind, and in the mean time revolve about the common centre of gravity ; I say, that, by the sam,e forces, there may be described round either body un- moved a figure similar and equcd to the figures which the bodies so Wjoving describe round each other ^nutually. Let the bodies S and P revolve about their common centre of gravity

C, proceeding from S to T, and from P to Q,. From the given point 5 let

there be continually drawn sp, sq, equal and parallel to SP, TQ ; and the curve j^^-Vj which the point p describes in its revolution round the immovable point s, will be similar and equal to the curves which the bodies S and P describe about each other mutually ; and therefore, by Theor. XX, similar to the curves ST and PQ.V which the same bodies describe about their common centre of gravity C ; and that because the proportions of the lines SO, 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- tion) is either at rest, or moves uniformly in a right line. Let us first suppose it at rest, and in 5 and p let there be placed two bodies, one im- movable in s, the other movable in p, similar and equal to the bodies S and P. Then let the right lines PR and pr touch the curves PQ, and pq in P and p, and produce CQ, and sq to R and r. And because the figures CPRQ., sprq are similar, RQ. will be to rq as CP to sp, and therefore in a given ratio. Hence if the force with which the body P is attracted to- wards the body S, and by consequence towards the intermediate point the centre C, were to the force with which the body p is attracted towards the centre s, in the same given ratio, these forces would in equal times attract

196 THE MATHEMATICAL PRINCIPLES [BoOK I.

the bodies from tlie tangents PR; pr to the arcs PQ,, pq, through the in- tervals proportional to them RQ., rq ; and therefore this last force (tending to s) would make the body p revolve in the curve pqv, which would become similar to the curve PQY, in which the first force obliges the body P tc revolve; and their revolutions would be completed in the same times. But because those forces are not to each other in the ratio of CP to sp, bu4 (by reason of the similarity and equality of the bodies S and s, P and p^ and the equality of the distances SP, sp) mutually equal, the bodies ip equal times will be equally drawn from the tangents ; and therefore tha.* the body p may be attracted through the greater interval rq^ there is re- quired a greater time, which will be in the subduplicate ratio of the inter- vals ; because, by Lemma X, the spaces described at the very beginning ot the motion are in a duplicate ratio of the times. Suppose, then the velocity of the body p to be to the velocity of the body P in a subduplicate ratio of the distance sp to the distance CP, so that the arcs pq^ PQ,, which are in a simple proportion to each other, may be described in times that are in a Bubduplicate ratio of the distances ; and the bodies P, p, always attracted by equal forces, will describe round the quiescent centres C and s similar figures PQV, pqv^ the latter of which jo^'V is similar and equal to the figure which the body P describes round the movable body S. Q.E.D.

Case 2. Suppose now that the common centre of gravity, together with the space in which the bodies are moved among themselves, proceeds uni- formly in a right line ; and (by Cor. 6, of the Laws of Motion) all the mo-, tions in this space will be performed in the same manner as before ; and therefore the bodies will describe mutually about each other the same fig- ures as before, which will be therefore similar and equal to the figure pqv.

aE.D.

Cor. 1. Hence two bodies attracting each other with forces proportional to their distance, describe (by Prop. X) both round their common centre of gravity, and round each other mutually concentrical ellipses ; and, vice versa, if such figures are described, the forces are proportional to the dis- tances.

Cor. 2. And two bodies, whose forces are reciprocally proportional to the square of their distance, describe (by Prop. XI, XII, XIII), both round their common centre of gravity, and round each other mutually, conic sec- tions having their focus in the centre about which the figures are described. And, vice versa, if such figures are described, the centripetal forces are re- ciprocally proportional to the squares of the distance.

Cor. 3. Any two bodies revolving round their common centre of gravity describe areas proportional to the timeS; by radii drawn both to that centre ^nd to each other mutually.

Sec. XL] of natural philosophy. 197

PROPOSITION LIX. THEOREM XXII.

The periodic time of two bodies S and P revolving round their common centre of gravity C, is to the periodic time of one of the bodies P re- volving round the other S remaining unmoved, and describing a fig- ure similar and equal to those lohich the bodies describe about each other ?}uitimlli/, in a subdwplicate ratio of the other body S to the sum of the bodies S + P.

For, by the demonstration of the last Proposition, the times in which any similar arcs PQ. and pq are described are in a subduplicate ratio of the distances CP and SP, or sp, that is, in a subdnplicate ratio of the body S to the sum of the bodies S + P. And by composition of ratios, the sums of 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 the same subduplicate ratio. Q,.E.D.

PROPOSITION LX. THEOREM XXIII.

If two bodies S and P, attracting each other with forces reciprocally pro- portional to the squares of their distance^ revolve about their common ce7itre of gravity ; I say, that the principal axis of the ellipsis which either of the bodies, as P, describes by this onotion about the other S, will be to the principal axis of the ellipsis, which the same body P may describe in the same periodical time about the other body S quiescent, as the sum of the tioo bodies S -1- P ^o the first of tioo mean propor- tionals betiueen that sum and the other body S.

For if the ellipses described were equal to each other, their ioerioplic times by the last Theorem would be in a subduplicate ratio of the body S to the sum of the bodies S + P. Let the periodic time in the latter ellipsis be diminished 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 ratio sesquiplicate to the former ratio ; that is, in a ratio to which the ratio of S to S + P is triplicate ; and therefore that axis will be to the principal axis of the other ellipsis as the first of two mean proportionals between S -f- P and S to S + P. And inversely the principal axis of the ellipsis de- scribed about the movable body will be to the principal axis of that described round the im_movable as S + P to the first of two mean proportionals be- tween S -{- P and S. Q.E.D.

PROPOSITION LXI. THEOREM XXIV.

If two bodies attracting each other with any kind of forces, and not othenvise agitated or obstructed, are moved in any tnanner whatsoever, those 'motions will be the same as if they did not at all attract each other mutually, but ivere both attracted with the same forces by a third body placed in their common centre of gravity ; and the law of the

19S THE MATHEMATICAL PRINCIPLES [BoOK I.

attracting forces will be the saiiie in respect of the distance of the

bodies from the conimon centre^ as in respect of the distance between

the tioo bodies.

For those forces with which the bodies attract each other mutually, by tending to the bodies, tend also to the common centre of gravity lying di- rectly between them ; and therefore are the same as if they proceeded from an intermediate body. Q,.E.D.

And because there is given the ratio of the distance of either body from that 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 other distance ; and also the ratio of any quantity derived in any manner from one of the distances compounded any how with given quantities, to another quantity derived in like manner from the other distance, and as many given quantities having that given ratio of the distances to the first. Therefore if the force with which one body is attracted by another be di- rectly or inversely as the distance of the bodies from each other, or as any power of that distance ; or, lastly, as any quantity derived after any man- ner from that distance compounded with given quantities ; then will the same force with which the same body is attracted to the common centre of gravity be in like manner directly or inversely as the distance of the at- tracted body from the common centre, or as any power of that distance ; or, lastly, as a quantity derived in like sort from that distance compounded with analogous given quantities. That is, the law of attracting force will be the same with respect to both distances. Q.E.D.

PROPOSITION LXII. PROBLEM XXXVIII.

Tb determine the motions of two bodies which attract each other with forces reciprocally proportional to the squares of the distance between them, and ore let fall from given places.

The bodies, by the last Theorem, will be moved in the same manner as if they were attracted by a third placed in the common centre of their gravity ; and by the hypothesis that centre will be quiescent at the begin- ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will be always quiescent. The motions of the bodies are therefore to be deter- mined (by Prob. XXV) in the same manner as if they were impelled by forces tending to that centre ; and then we shall have the motions of the bodies attracting each other mutually. Q.EJ.

PROPOSITION LXIIL PROBLEM XXXIX.

To determiiie the motions of tioo bodies attracting each other luith forces reciprocally proportional to the squares of their distance, and going off from given places in given directio7is loith given velocities. The motions of the bodies at the beginning being given, there is given

Sec. XL] of natural philosophy. 199

also the uniform motion of the common centre of gravity, and the motion of 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 this space. Then (by Cor. 5, of the Laws, and the last Theorem) the subse- quent motions will be performed in the same manner in that space, as if that space together with the common centre of gravity were at rest, and as if the bodies did not attract each other, but were attracted by a third body placed in that centre. The motion therefore in this movable space of each body going oiF from a given place, in a given direction, with a given velo- city, and acted upon by a centripetal force tending to that centre, is to be determined by Prob. IX and XXYI, and at the same time will be obtained the motion of the other round the same centre. With this motion com- pound the uniform progressive motion of the entire system of the space and the bodies revolving in it, and there will be obtained the absolute motion of the bodies in immovable space. Q.E.L

PROPOSITION LXIY. PROBLEM XL.

Supposing forces 7vith ivhich bodies 'miihially attract each other to irp- crease in a simple ratio of their distances fro'm the centres ; it is TO' quired to find the ^notions of several bodies among themselves. Suppose the first two bodies T and L j 2

to have their common centre of gravity in I ^ '• t>

D. These, by Cor. 1, Theor. XXI, will ^^ ' '■

describe ellipses having their centres in D, the magnitudes of which ellipses are

known by Prob. V. I -V iii

Let now a third body S attract the two 0v