Principia Mathematica (Motte Translation, 1848) — part 16 of 45

one case, collect very nearly the errors and times of the errors in any other case. But this may be done more expeditiously by the following method. The forces NM, ML, other things remaining unaltered, are as the radius TP ; and their periodical effects (by Cor. 2, Lem. X) are as the forces and the square of the periodical time of the body P conjunctly. These are the linear errors of the body P ; and hence the angular errors as they appear from the centre T (that is, the motion of the apsides and of the nodes, and all the apparent errors as to longitude and latitude) are in each revolution of the body P as the square of the time of the revolution, very nearly. Let these ratios be compounded with the ratios in Cor. 14, and in any system of bodies T, P, 8, where P revolves about T very near to it, and T re- volves about S at a great distance, the angular errors of the body P, ob- served from the centre T, will be in each revolution of the body P as the square of the periodical time of the body P directly, and the square of the periodical time of the body T inversely. And therefore the mean motion of the line of the apsides will be in a given ratio to the mean motion of the nodes ; and both those motions will be as the periodical time of the body P directly, and the square of the periodical time of the body T in- versely. The increase or diminution of the eccentricity and inclination of the orbit PAB makes no sensible variation in the motions of the apsides and nodes, unless that increase or diminution be very great indeed.

Cor. 17. Since the line LM becomes sometimes greater and sometimes less than the radius PT, let the mean quantity of the force LM be expressed (E C li

-— ^-M

by that radius PT ; and then that mean force will be to the mean force SK or SN (which may be also expressed by ST) as the length PT to the length ST. But the mean force SN or ST, by which the body T is re- tained in the orbit it describes about S, is to the force with which the body P is retained in its orbit about T in a ratio compounded of the ratio of the radius ST to the radius PT, and the duplicate ratio of the periodical time of the body P about T to the periodical time of the body T about S. And, ea: CEquo, the mean force LM is to the force by which the body P is retain- ed in its orbit about T (or by which the same body P might revolve at the distance PT in the same periodical time about any immovable point T) in the same duplicate ratio of the periodical times. The periodical times therefore being given, together with the distance PT, the mean force LM is also given ; and that force being given, there is given also the force MN, very nearly, by the analogy of the lines PT and MN.

212 THE MATHEMATICAL PRINCIPLES [BoOK I.

Cor. 18. By tlie same laws by which the body P revolves about the body T, let ns suppose many fluid bodies to move round T at equal dis- tances from it ; and to be so numerous^ that they may all become contiguous to each other, so as to form a fluid annulus, or ring, of a round figure, and concentrical to the body T; and the several parts of this annulus, perform- ing their motions by the same law as the body P, will draw nearer to the body T, and move swifter in the conjunction and opposition of themselves and the body S, than in the quadratures. And the nodes of this annulus, or its intersections with the plane of the orbit of the body S or T, will rest at the syzygies ; but out of the syzygies they will be carried backward, or iii antecedentia ; with the greatest swiftness in the quadratures, and more slowly in other places. The inclination of this annulus also will vary, and its axis will oscillate each revolution, and when the revolution is completed will return to its former situation, except only that it will be carried round a little by the precession of the nodes.

Cor. 19. Suppose noAV the sph83rical body T, consisting of some matter not fluid, to be enlarged, and to extend itself on every side as far as that annulus, and that a channel were cut all round its circumference contain- ing water ; and that this sphere revolves uniformly about its own axis in the same periodical time. This water being accelerated and retarded by turns (as in the last Corollary), will be swifter at the syzygies, and slower at the quadratures, than the surface of the globe, and so will ebb and flow in its channel after the manner of the sea. If the attraction of the body S were taken away, the water would acquire no motion of flux and reflux by revolv- ing round the quiescent centre of the globe. The case is the same of a globe moving uniformly forwards in a right line, and in the mean time revolving about its centre (by Cor. 5 of the Laws of Motion), and of a globe uni- formly attracted from its rectilinear course (by Cor. 6, of the same Laws). But let the body S come to act upon it, and by its unequable attraction the water will receive this new motion ; for there will be a stronger attraction upon that part of the water that is nearest to the body, and a weaker upon that part which is more remote. And the force LM will attract the water downwards at the quadratures, and depress it as far as the syzygies ; and the force KL will attract it upwards in the syzygies, and withhold its descent, and make it rise as far as the quadratures ; except only in so far as the motion of flux and reflux may be directed by the channel of the water, and be a little retarded by friction.

Cor. 20. If, now, the annulus becomes hard, and the globe is diminished, the motion of flux and reflux will cease ; but the oscillating motion of the inclination and the praecession of the nodes will remain. Let the globe have the same axis with the annulus, and perform its revolutions in the same times, and at its surface touch the annulus within, and adhere to it ; then the globe partaking of the motion of the annulus, this whole compages

Sec. XL of natural philosophy. 213

will oscillate, and the nodes will go backward, for the globe, as we shall shew presently, is perfectly indifferent to the receiving of all impressions. The greatest angle of the inclination of the annulus single is when the nodes are in the syzygies. Thence in the progress of the nodes to the quadratures, it endeavours to diminish its inclination, and by that endea- vour impresses a motion upon the whole globe. The globe retains this motion impressed, till the annulus by a contrary endeavour destroys that motion, and impresses a new motion in a contrary direction. And by this means the greatest motion of the decreasing inclination happens when the nodes are in the quadratures, and the lea^st angle of inclination in the octants /E C li ,

— -M

D

after the quadratures ; and, again, the greatest motion of reclination happens when the nodes are in the syzygies ; and the greatest angle of reclination in the octants following. And the case is the same of a globe without this an- nulus, if it be a little higher or a little denser in the equatorial than in the polar regions ; for the excess of that matter in the regions near the equator supplies the place of the annulus. And though we should suppose the cen- tripetal force of this globe to be any how increased, so that all its parts were to tend downwards, as the parts of our earth gravitate to the centre, yet the phsenomena of this and the preceding Corollary would scarce be al- tered ; except that the places of the greatest and least height of the water will be different ; for the water is now no longer sustained and kept in its orbit by its centrifugal force, but by the channel in which it flows. And, besides, the force LM attracts the water downwards most in the quadra- tures, and the force KL or NM — LM attracts it upwards most in the syzygies. And these forces conjoined cease to attract the water downwards, and begin to attract it upwards in the octants before the syzygies ; and cease to attract the water upwards, and begin to attract the water down- wards in the octants after the syzygies. And thence the greatest height of the water m.ay happen about the octants after the syzygies ; and the least height about the octants after the quadratures ; excepting only so far as the motion of ascent or descent impressed by these forces may by the vis msita of the water continue a little longer, or be stopped a little sooner by impe- diments in its channel.

Cor. 21. For the same reason that redundant matter in the equatorial regions of a globe causes the nodes to go backwards, and therefore by the increase of that matter that retrogradation is increased, by the diminution is diminished, and by the removal quite ceases ; it follows, that, if more than

214 THE MATHEMATICAL PRINCIPLES [BoOK I.

that redundant matter be taken away, that is, if the globe be either more depressed, or of a more rare consistence near the equator than near the poles, there will arise a motion of the nodes iii consequentia.

Cor. 22. And thence from the motion of the nodes is known the consti- tution of the globe. That is, if the globe retains unalterably the same poles, and the motion (of the nodes) be in antecedentia, there is a redundance of the matter near the equator; but if in consequentia, a deficiency. Sup- pose a uniform and exactly sphaerical globe to be first at rest in a free space ; then by some impulse made obliquely upon its superficies to be driven from its place, and to receive a motion partly circular and partly right forward. Because this globe is perfectly indifferent to all the axes that pass through its centre, nor has a greater propensity to one axis or to one situation of the axis than to any other, it is manifest that by its own force it will never change its axis, or the inclination of it. Let now this globe be impelled obliquely by a new impulse in the same part of its superficies as before ; and since the effect of an impulse is not at all changed by its coming sooner or later, it is manifest that these two impulses, successively impressed, will produce the same motion as if they were impressed at the same time ; that is, the same motion as if the globe had been impelled by a simple force compounded of them both (by Cor. 2, of the Laws), that is, a simple motion about an axis of a given inclination. And the case is the same if the sec- ond impulse were made upon any other place of the equator of the first motion ; and also if the first impulse were made upon any place in the equator of the motion which would be generated by the second impulse alone ; and therefore, also, when both impulses are made in any places whatsoever ; for these impulses will generate the same circular motion as if they were impressed together, and at once, in the place of the intersec- tions of the equators of those motions, which would be generated by each of them separately. Therefore, a homogeneous and perfect globe will not retain several distinct motions, but will unite all those that are impressed on it, and reduce them into one ; revolving, as far as in it lies, always with a simple and uniform motion about one single given axis, with an inclina- tion perpetually invariable. And the inclination of the axis, or the velocity of the rotation, will not be changed by centripetal force. For if the globe be supposed to be divided into two hemispheres, by any plane whatsoever passing through its own centre, and the centre to which the force is direct- ed, that force will always urge each hemisphere equally ; and therefore will not incline the globe any way as to its motion round its own axis. But let there be added any where between the pole and the equator a heap of new matter like a mountain, and this, by its perpetual endeavour to recede from the centre of its motion, will disturb the motion of the globe, and cause its poles to wander about its superficies, describing circles about themselves and their opposite points. Neither can this enormous evagation

Sec. XL] of natural philosophy. 215

of the poles be corrected, unless by placing that mountain either in one of the poles ; in which case, by Cor. 21, the nodes of the equator will go for- wards ; or in the equatorial regions, in which case, by Cor. 20, the nodes will go backwards ; or, lastly, by adding on the other side of the axis a new quantity of matter, by which the mountain may be balanced in its motion; and then the nodes will either go forwards or backwards, as the mountain and this newly added matter happen to be nearer to the pole or to the equator.

PROPOSITION LXYII. THEOREM XXVII. The same Imos of attraction being supposed, I say, that the exterior body S does, by radii drawn to the point O, the common centre of gravity of the interior bodies P and T, describe round that centre areas more proportional to the times, and an orbit more approaching to the form of an ellipsis having its focus in that centre, than it can describe round the innermost and greatest body T by radii drawn to that body.

For the attractions of the body S towards T and / y^ \

P compose its absolute attraction, which is more / i^^^-. o \

directed towards O, the common centre of gravity ^T r -^ 1

of the bodies T and P, than it is to the greatest \ \ y

body T ; and which is more in a reciprocal proper- ^~ — ^

tion to the square of the distance SO, than it is to the square of the distance ST ; as will easily appear by a little consideration.

PROPOSITION LXYIII. THEOREM XXVIII.

The same laws of attraction supposed^ I say, that the exterior body S loill, by radii drawn to O, the common centre of gravity of the interior bodies P and T, describe round that centre areas m^ore propor- tiojial to the times, and an orbit more approaching to the form, of an ellipsis having its focus in that centre, if the innerfnost and greatest body be agitated by these attractions as well as the rest, than it woidd do if that body were either at rest as not attracted, or were inuch more or mjuch less attracted, or m^uch m^ore or much less agitated. This may be demonstrated after the same manner as Prop. LXVI, but by a more prolix reasoning, which I therefore pass over. It will be suf- ficient to consider it after this manner. From the demonstration of the last Proposition it is plain, that the centre, towards which the body S is urged by the two forces conjunctly, is very near to the common centre of gravity of those two other bodies. If this centre were to coincide with that common centre, and moreover the common centre of gravity of all the three bodies were at rest, the body S on one side, and the common centre of gravity of the other two bodies on the other side, would describe true ellip-

216 THE MATHEMATICAL PRINCIPLES [BoOK I.

ses about that quiescent common centTe. This appears from Cor. 2, Prop. LYIII, compared with what was. demonstrated in Prop. LXIV, and LXV. Now this accurate elliptical motion Avill be disturbed a little by the dis- tance of the centre of the two bodies from the centre towards which the third body S is attracted. Let there be added, moreover, a motion to the common centre of the three, and the perturbation will be increased yet / y^ ^ more. Therefore the perturbation is least when the

/ / ---. \ common centre of the three bodies is at rest ; that

^t" ' r -^ lis, when the innermost and greatest body T is at-

\ \ / tracted acccordinp; to the same law as the rest are ;

^- — -^ and is always greatest when the common centre of the three, by the diminution of the motion of the body T, begins to be moved, and is more and more agitated.

Cor. And hence if more lesser bodies revolve about the great one, it may easily be inferred that the orbits described will approach nearer to ellipses ; and the descriptions of areas will be more nearly equable, if all the bodies mutually attract and agitate each other with accelerative forces that are as their absolute forces directly, and the squares of the distances inversely : and if the focus of each orbit be placed in the common centre of gravity of all the interior bodies (that is, if the focus of the first and in- nermost orbit be placed in the centre of gravity of the greatest and inner- most body ; the focus of the second orbit in the common centre of gravity of the two innermost bodies ; the focus of the third orbit in the common centre of gravity of the three innermost ; and so on), than if the innermost body were at rest, and was made the common focus of all the orbits.

PROPOSITION LXIX. THEOREM XXIX.

In a system of several bodies A, B, C, D, Sfc, if any one of those bodies, as A, attract all the rest, B, C, D, Sfc.^ioith accelerative forces that are reciprocally as the squares of the distances fro^m the attracting body ; and another body, as B, attracts also the rest, A, C, D, Sj^c.,ioith forces that are reciprocally as the squares of the distances from the attr act- ing body ; the absolute forces of the attracting bodies A and B will be to each other as those very bodies A and B to which those forces belong.

For the accelerative attractions of all the bodies B, C, D, towards A, are by the supposition equal to each other at equal distances ; and in like manner the accelerative attractions of all the bodies towards B are also equal to each other at equal distances. But the absolute attractive force of the body A is to the absolute attractive force of the body B as the ac- celerative attraction of all the bodies towards A to the accelerative attrac- tion of all the bodies towards B at equal distances ; and so is also the ac- -celerative attraction of the body B towards A to the accelerative attraction

Sec. XL] of natural philosoppiy. SIT

of the body A towards B. But the accelerative attraction of the body B towards A is to the accelerative attraction of the body A towards B as the mass of the body A to the mass of the body B ; because* the motive forces which (by the 2d, 7th, and 8th Definition) are as the accelerative forces and the bodies attracted conjunctly are here equal to one another by the third Law. Therefore the absolute attractive force of the body A is to the absolute attractive force of the body B as the mass of the body A to the mass of the body B. Q..E.D.

Cor. 1. Therefore if each of the bodies of the system A, B, C, D, <fec. does singly attract all the rest with accelerative forces that are reciprocally as the squares of the distances from the attracting body, the absolute forces of all those bodies will be to each other as the bodies themselves.

OoR. 2. By a like reasoning, if each of the bodies of the system A, B, C, D, (fcc, do singly attract all the rest with accelerative forces, which are either reciprocally or directly in the ratio of any power whatever of the distances from the attracting body ; or which are defined by the distances from each of the attracting bodies according to any common law ; it is plain that the absolute forces of those bodies are as the bodies themselves.

CoR. 3. In a system of bodies whose forces decrease in the duplicate ra- tio of t^e distances, if the lesser revolve about one very great one in ellip- ses, having their common focus in the centre of that great body, and of a figure exceedingly accurate ; and moreover by radii drawn to that great body describe areas proportional to the times exactly ; the absolute forces of those bodies to each other will be either accurately or very nearly in the ratio of the bodies. And so on the contrary. This appears from Cor. of Prop. XLVIII, compared with the first Corollary of this Prop.

SCHOLIUM. These Propositions naturally lead us to the analogy there is between centripetal forces, and the central bodies to which those forces used to be directed ; for it is reasonable to suppose that forces which are directed to bodies should depend upon the nature and quantity of those bodies, as we see they do in magnetical experiments. And when such cases occur, we are to compute the attractions of the bodies by assigning to each of their particles its proper force, and then collecting the sum of them all. I here use the word attraction in general for any endeavour, of wliPtlT^nd soever, made by bodies to approach to each other ; whether that endeavour arise from the action of the bodies themselves, as tending mutually to or agita- ting each other by spirits emitted ; or whether it arises from the action of the jether or of the air, or of any medium whatsoever, whether corporeal or incorporeal, any how impelling bodies placed therein towards each other. In the same general sense I use the word impulse, not defining in this trea- tise the species or physical qualities of forces, but investigating the quantities

218 THE MATHEMATICAL PRINCIPLES [BoOK T.

and matlieinatical proportions of them ; as I observed before in the Defi- nitions. In mathematics we are to investigate the quantities of forces with their proportions consequent upon any conditions supposed ; then, when we enter upon physics, we compare those proportions with the phse- nomena of Nature, that we may know what conditions of those forces an- swer to the several kinds of attractive bodies. And this preparation being made, we argue more safely concerning the physical species, causes, and proportions of the forces. Let us see, then, with what forces sphaerical bodies consisting of particles endued with attractive powers in the manner above spoken of must act mutually upon one another ; and what kind of motions will follow from thence.

SECTION XII.

Of the attractive forces of sphcerical bodies.

PROPOSITION LXX. THEOREM XXX.

If to every point of a sphcerical surface there tend equal centripetal forces decreasing in the duplicate ratio of the distances from those points ; I say, that a corpuscle placed within that superficies will not be attract- ed by those forces any ivay.

Let HIKL, be that sphcerical superficies, and P a corpuscle placed within. Through P let there be I^L-____,^^^ y drawn to this superficies to two lines HK, IL, inter- cepting very small arcs HI, KL ; and because (by Cor. 3, Lem. YII) the triangles HPI,LPK are alike, those arcs will be proportional to the distances HP, LP ; and any particles at HI and KL of the sphseri- cal superficies, terminated by right lines passing through P, will be in the duplicate ratio of those distances. Therefore the forces of these particles exerted upon the body P are equal between themselves. For the forces are as the particles directly, and the squares of the distances inversely. And these two ratios compose the ratio of equality. The attractions therefore, beino' made equally towards contrary parts, destroy each other. ' And by a like reasoning all the attractions through the whole sphasrical superficies are destroyed by contrary attractions. Therefore the body P will not be any way impelled by those attractions. Q.E.D.

PROPOSITION LXXL THEOREM XXXL

The same thino-s supposed as above, I say, that a corpuscle placed with- out the sphmrical superficies is attracted toivards the centre of the sphere ivith a force reciprocally proportional to the square of its dis- tance from that centre. Let AHKB, ahkb, be two equal sphasrical superficies described about

Sec. XII.] OF natural philosophy. 219

the centre S, s ; their diameters AB, ab ; and let P and p be two corpus- cles situate without the spheres in those diameters produced. Let there

be drawn from the corpuscles the lines PHK^ PIL, phk, pil, cutting oif from the great circles AHB, ahb, the equal arcs HK, hk, IL, it ; and to those lines let fall the perpendiculars SD, sd, SE, se, IR, ir ; of which let SD, sd, cut Fhjpl, in F and/. Let fall also to the diameters the perpen- diculars IQ,; iq. Let now the angles DPE, dpe, vanish; and because DS and ds, ES and es are equal, the lines PE, PF, and pe, pf, and the lineola3 DF, df may be taken for equal ; because their last ratio, when the angles DPE, dpe vanish together, is the ratio of equality. These things then supposed, it will be, as PI to PF so is ill to DF, and as pf to pi so is df or DF to ri ; and, ea; cequo, as PI X jq/" to PF X pi so is RI to ri, thaf is (by Cor. 3, Lem. YII), so is the arc IH to the arc ih. Again, PI is to PS as IQ. fo SE, and ps to pi as se or SE to iq ; and, ex cequo, PI X ps to PS X pi as IQ, to iq. And compounding the ratios PI^ X pf X ps isio pi^ X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super- ficies which is described by the arc IH, as the semi-circle AKB revolves about the diameter AB, is to the circular superficies described by the arc ih as the semi^circle akh revolves about the diameter ab. And the forces with which these superficies attract the corpuscles P and p in the direction of lines tending to those supjjrficies are by the hypothesis as the superficies themselves directly, and the squares of the distances of the superficies from those corpuscles inversely ; that is, 2.^ pf X ps to PF XPS. And these forces again are to the oblique parts of them which (by the resolution of forces as in Cor. 2, of the Laws) tend to the centres in the directions of the lines PS, ps, as PI to PQ, and pi to pq ; that is (because of the like trian- gles PIQ and PSF, piq and psf^, as PS to PF and ps to pf. Thence ex cequo, the attraction of the corpuscle P towards S is to the attraction of

. T , . PF X^/X jf?5. ^ jo/X PF X PS . .

the corpuscle 10 towards s as —^ is to , that is,

^ ^ PS ps

as ps^ to PS". And, by a like reasoning, the forces with which the su- perficies described by the revolution of the arcs KL, kl attract those cor- puscles, will be as ^9^2 to PS^. And in the same ratio will be the forces of all the circular superficies into which each of the sphaarical superficies may be divided by taking sd always equal to SD, and se equal to SE. And therefore, by composition, the forces of the entire spherical superficies ex- erted upon those corpuscles will be in the same ratio. Q.E.D.

220 THE MATHEMATICAL PRINCIPLES [BoOK 1.

PROPOSITION LXXIL THEOREM XXXII.

If to the several points of a sphere there tend equal centripetal forces de- creasing ill a duplicate ratio of the distances from those points ; and there be given both the density of the sphere and the ratio of the di- ameter of the sphere to the distance of the corpuscle from its centre ; I say, that the force with which the corpuscle is attracted is propor- tional to the semi-diameter of the sphere.

For conceive two corpuscles to be severally attracted by two spheres, one by one, the other by the other, and their distances from the centres of the spheres to be proportional to the diameters of the spheres respectively ; and the spheres to be resolved into like particles, disposed in a like situation to the corpuscles. Then the attractions of one corpuscle towards the sev- eral particles of one sphere will be to the attractions of the other towards as many analogous particles of the other sphere in a ratio compounded of the ratio of the particles directly, and the duplicate ratio of the distances inversely. But the particles are as the spheres, that is, in a triplicate ra- tio of the diameters, and the distances are as the diameters ; and the first ratio directly with the last ratio taken twice inversely, becomes the ratio of diameter to diameter. Q.E.D.

Cor, 1. Hence if corpuscles revolve in circles about spheres composed of matter equally attracting, and the distances from the centres of the spheres be proportional to their diameters, the periodic times will be equal. Cor. 2. And, vice versa, if the periodic times are equal, the distances will be proportional to the diameters. These two Corollaries appear from Cor. 3, Prop. IV.

Cor. 3. If to the several points of any two solids whatever, of like fig- ure and equal density, there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points, the forces, with which corpuscles placed in a like situation to those two solids will be attracted by them, will be to each other as the diameters of the solids.

PROPOSITION LXXIII. THEOREM XXXHI.

If to the several points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from the poiiits ; I say, that a corpuscle placed icithin the sphere is attracted by a force proportional to its distance from the centre.

-^ In the sphere ABCD, described about the centre S,

let there be placed the corpuscle P ; and about the same centre S, with the interval SP, conceive de- A( |P s ^j ]^ scribed an interior sphere PEQ.F. It is plain (by Prop. LXX) that the concentric sphserical superficies, of which the difference AEBF of the spheres is com- D posed, have no effect at all upon the ^ody P, their at-

i

Sec. XIL] of natural philosophy. 221

tractions being destroyed by contrary attractions. There remains, there- fore; only the attraction of the interior sphere PEQF. And (by Prop. LXXII) this is as the distance PS. Q.E.D.

SCHOLIUM. By the superficies of which I here imagine the solids composed, I do not mean superficies purely mathematical, but orbs so extremely thin, that their thickness is as nothing ; that is, the evanescent orbs of which the sphere will at last consist, when the number of the orbs is increased, and their thickness diminished without end. In like manner, by the points of which lines, surfaces, and solids are said to be composed,-» are to be understood equal particles, whose magnitude is perfectly inconsiderable.

PROPOSITION LXXIV. THEOREM XXXIV. The same things supposed, I sai/, that a corpuscle situate vnthout the

sphere is attracted ivith a force reciprocally proportional to the square

of its distance from the centre.

For suppose the sphere to be divided into innumerable concentric sphag- rical superficies, and the attractions of the corpuscle arising from the sev- eral superficies will be reciprocally proportional to the square of the dis- tance of the corpuscle from the centre of the sphere (by Prop. LXXI). And, by composition, the sum of those attractions, that is, the attraction of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D.

Cor. 1. Hence the attractions of homogeneous spheres at equal distances from the centres will be as the spheres themselves. For (by Prop. LXXII) if the distances be proportional to the diameters of the spheres, the forces will be as the diameters. Let the greater distance be diminished in that ratio ; and the distances now being equal, the attraction will be increased in the duplicate of that ratio ; and therefore will be to the other attraction in the triplicate of that ratio ; that is, in the ratio of the spheres.

Cor. 2. At any distances whatever the attractions are as the spheres applied to the squares of the distances.

Cor. 3. If a corpuscle placed without an homogeneous sphere is attract- ed by a force reciprocally proportional to the square of its distance from the centre, and the sphere consists of attractive particles, the force of every particle will decrease in a duplicate ratio of the distance from each particle.

PROPOSITION LXXY. THEOREM XXXV.

If to the several points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from, the points ; I say, that another similar sphere will be attracted by it with a force recip- rocally proportional to the square of the distance of the centres. For the attraction of every particle is reciprocally as the square of its

222

THE MATHEMATICAL PRINCIPLES

OOOK

distance from the centre of the attracting sphere (by Prop. LXXIY), and is therefore the same as if that whole attracting force issued from one sin- gle corpuscle placed in the centre of this sphere. But this attraction is as great as on the other hand the attraction of the same corpuscle would be, if that were itself attracted by the several particles of the attracted sphere with the same force with which they are attracted by it. But that attrac- tion of the corpuscle would be (by Prop. LXXIY) reciprocally propor- tional to the square of its distance from the centre of the sphere ; therefore the attraction of the sphere, equal theretO; is also in the same ratio. Q..E.D.

Cor. 1. The attractions of spheres towards other homogeneous spheres are as the attracting spheres applied to the squares of the distances of their centres from the centres of those which they attract.

Cor. 2. The case is the same when the attracted sphere does also at- tract. For the several points of the one attract the several points of the other with the same force with which they themselves are attracted by the others again ; and therefore since in all attractions (by Law III) the at- tracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining.

Cor. 3. Those several truths demonstrated above concerning the motion of bodies about the focus of the conic sections will take place when an attracting sphere is placed in the focus, and the bodies move without the sphere.

Cor. 4. Those things which were demonstrated before of the motion of bodies about the centre of the conic sections take place when the motions are performed within the sphere.

PROPOSITION LXXVL THEOREM XXXVI.

If spheres be however dissimilar {as to density of matter and attractive force) in the same ratio omoard from the centre to the circumference; hut every where similar, at every given distance from the centre, on all sides round about ; and the attractive force of every point decreases in the duplicate ratio of the distance of the body attracted ; I say, that the ivhole force loith lohich one of these spheres attracts the other will be reciprocally proportional to the square of the distance of the centres.