Principia Mathematica (Motte Translation, 1848) — part 17 of 45

Imagine several concentric similar spheres, AB, CD, EF, &c., the inner- most of which added to the outermost may compose a matter more dense to- wards the centre, or subducted from them may leave the same more lax and rare. Then, by Prop. LXXV, these spheres will attract other similar con-

Sec. XILJ of natural philosophy. 223

centric spheres GH. IK^ LM, (fee, each the other, with forces reciprocally proportional to the square of the distance SP. And, by composition or division, the sum of all those forces, or the excess of any of them above the others ; that is, the entire force with which the whole sphere AB (com- posed of any concentric spheres or of their differences) will attract the whole sphere GH (composed of any concentric spheres or their differences) in the same ratio. Let the number of the concentric spheres be increased ifi infinitum, so that the density of the matter together with the attractive force ma}^, in the progress from the circumference to the centre, increase or decrease according to any given law ; and by the addition of matter not at- tractive, let the deficient density be supplied, that so the spheres may acquire any form desired ] and the force with which one of these attracts the other will be still, by the former reasoning, in the same ratio of the square of the distance inversely. Q.E.D.

Cor. 1. Hence if many spheres of this kind, similar in all respects, at- tract each other mutually, the accelerative attractions of each to each, a,t any equal distances of the centres, will be as the attracting spheres.

Cor. 2. And at any unequal distances, as the attracting spheres applied to the squares of the distances between the centres.

Cor. 3. The motive attractions, or the weights of the spheres towards one another, will be at equal distances of the centres as the attracting and attracted spheres conjunctly ; that is, as the products arising from multi- plying the spheres into each other.

Cor. 4. And at unequal distances, as those products directly, and the squares of the distances between the centres inversely.

Cor. 5. These proportions take place also when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.

CoR. 6. If spheres of this kind revolve about others at rest, each about each ; and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of thfe quiescent bodies ; the peri- odic times will be equal.

CoR. 7. And, again, if the periodic times are equal, the distances will be proportional to the diameters.

Cor. 8. All those truths above demonstrated, relating to the motions of bodies about the foci of conic sections, will take place when an attract- ing sphere, of any form and condition like that above described, is placed in the focus.

Cor. 9. And also when the revolving bodies are also attracting spheres of any condition like that above described.

224

THE MATHEMATICAL PRINCIPLES

[Book I.

PROPOSITION LXXVII. THEOREM XXXYII.

If to the several points of spheres there tend centripetal forces propor- tional to the distances of the points from the attracted bodies ; I saij, that the compounded forc^e luith which two spheres attract each other mutually is as the distance between the centres of the spheres.

Case 1. Let AEBF be a sphere ; S its centre ; P a corpuscle attracted ; PA SB the axis of the sphere passing through the centre of the corpuscle : EF, e/'two planes cutting the sphere, and perpendicular to the axis, and equi-distant, one on one side, the other on the other, from the centre of the sphere ; G and g the intersections of the planes and the axis : and H any point in the plane EF. The centri- petal force of the point H upon the corpuscle P, exerted in the direction of the line PH, is as the distance PH ; and (by Cor. 2, of the Laws) the same exerted in the direction of the line PG, or towards the centre S, is as the length PG. Therefore the force of all the points in the plane EF (that is, of that whole plane) by which the corpuscle P is attracted towards the centre S is as the distance PG multiplied by the number of those points, that is, as the solid contained under that plane EF and the distance PG. And in like manner the force of the plane ef by which the corpuscle P is attracted towards the centre S, is as that plane drawn into its distance P^, or as the equal plane EF drawn into that distance Fg ; and the sum of the forces of both planes as the plane EF drawn into the sum of the distances PG + Pg', that is, as that plane drawn into twice the distance PS of the centre and the corpuscle ; that is, as twice the plane EF drawn into the dis- tance PS, or as the sum of the equal planes EF + ef drawn into the same distance. And, by a like reasoning, the forces of all the planes in the whole sphere, equi-distant on each side from the centre of the sphere, are as the sum of those planes drawn into the distance PS, that is, as the whole sphere and the distance PS conjunctly. Q,.E.D.

Case 2. Let now the corpuscle P attract the sphere AEBF. And, by the same reasoning, it will appear that the force with which the sphere is attracted is as the distance PS. Q.E.D.

, Case 3. Imagine another sphere composed of innumerable corpuscles P ; and because the force with which every corpuscle is attracted is as the dis- tance of the corpuscle from the centre of the first sphere, and as the same sphere conjunctly, and is therefore the same as if it all proceeded from a single corpuscle situate in the centre of the sphere, the entire force with which all the corpuscles in the second sphere are attracted, that is, with which that whole sphere is attracted, will be the same as if that sphere

Sec. XIL] op natural philosophy. 225

were attracted by a force issuing from a single corpuscle in the centre of the first sphere; and is therefore proportional to the distance between the centres of the spheres. Q,.E.D.

Case 4. Let the spheres attract each other mutually, and the force will be doubled, but the proportion will remain. Q.E.D.

Case 5. Let the corpuscle p be placed within the sphere AEBP ; and because the force of the plane ef upon the corpuscle is as the solid contain- ed under that plane and the distance jog- ; and the contrary force of the plane EF as the solid con- tained under that plane and the distance pG ; the force compounded of both will be as the difference of the solids, that is, as the sum of the equal planes drawn into half the difference of the distances ; that is, as that sum drawn into j^S, the distance of the corpuscle from the centre of the sphere. And, by a like reasoning, the attraction of all the planes EF, ef, throughout the whole sphere, that is, the attraction of the whole sphere, is conjunctly as the sum of all the planes, or as the whole sphere, and as ^S, the distance of the corpuscle from the centre of the sphere. Q.E.D.

Case 6. And if there be composed a new sphere out of innumerable cor- puscles such as p, situate within the first sphere AEBF, it may be proved, as before, that the attraction, whether single of one sphere towards the other, or mutual of both towards each other, will be as the distance ^S of the centres. Q,.E.D.

PROPOSITION LXXVIIL THEOREM XXXVIIL

If spheres in the progress from the centre to the circumference behow&cer dissimilar and unequable, hut similar on every side round about at all given distances from the centre; and the attractive force cf every point be as the distance of the attracted body ; I say, that the entire force with tvhich tioo spheres of this kind attract each other mutually is proportional to the distance betiveen the centres of the spheres. This is demonstrated from the foregoing Proposition, in the same man- ner as Proposition LXXVI was demonstrated from Proposition LXXY.

Cor. Those things that were above demonstrated in Prop. X and LXIY, of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphaerical bodies of the condi- tion above described, and the attracted bodies are spheres of the same kind.

SCHOLIUM.

I have now explained the two principal cases of attractions ; to wit, when the centripetal forces decrease in a duplicate ratio of the distances, or increase in a simple ratio of the distances, causing the bodies in both

15

226 THE MATHEMATICAL PRINCIPLES [BoOK I.

cases to revolve in conic sections, and composing sphaerical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do ; which is very remarkable. It would be tedious to run over the other cases, whose con- clusions are less elegant and important, so particularly as I have done these. I choose rather to comprehend and determine them all by one gen- eral method as follows.

LEMMA XXIX. If about the centre S there be described any circle as AEB, and about the centre P there be also described tico circles EF, ef, cutting the first in E and e, and the line PS in F and f ; and there be let fall to PS the perpendiculars ED, ed; I say, that if the distance of the arcs EF, ef be supposed to be infinitely diminished, the last ratio of the evanscent line hd to the evanescent line Ff is the sa?Jie as that of the line PE to the Hue PS. For if the line Pe cut the arc EF in q ; and the right line Ee, which

coincides with the evanescent arc Ee, be produced, and meet the right line PS in T ; and there be let fall from S to PE the perpendicular SG ; then, because of the like triangles DTE, d'Te, DES, it will be as Dd to Ee so DT to TE, or DE to ES ; and because the triangles, Eeq, ESG (by Lem. yni, and Cor. 3, Lem. YII) are similar, it will be as Ee to eq or Ff so ES to SG ; and, ex cequo, as Dd to F/ so DE to SG ; that is (because of the similar triangles PDF, PGS), so is PE to PS. Q.E.D.

PROPOSITION LXXIX. THEOREM XXXIX.

Suppose a superficies as EFfe to have its breadth infinitely dimi?iishedj and to be just vanishing ; and that the same superficies by its rei^olu- tio7i round the axis PS describes a sphcerical co?icavo-convex solid, to the several equal particles of which there tend equal centripetal forces ; I say, that the force loith lohich that solid attracts a corpuscle situate in P is in a ratio compounded of the ratio of the solid DE^ x Ff and the ratio of the force loith lohich the given particle in the place Ff tvould attract the same corpuscle. For if we consider, first, the force of the sphssrical superficies FE which

Sec. XIL] of natural philosophy. 227

is generated by the revolution of the arc FE, and is cut any where, as in r, by the line de, the annular part of the superficies generated by the revolution of the arc r'E will be as the lineola Del, the radius of the sphere PE re-^L maining the same; as Archimedes has de- \ monstrated in his Book of the Sphere and Cylinder. And the force of this super- ficies exerted in the direction of the lines PE or Pr situate all round in the conical superficies, will be as this annular superficies itself; that is as the lineola J}d, or, which is the same, as the rectangle under the given radius PE of the sphere and the lineola Dd ; biit that force, exerted in the direction of the line PS tending to the centre S, will be less in the ratio PD to PE, and therefore will be as PD X Dc?. Suppose now the line DF to be divided into innumerable little equal par- ticles, each of which call Dd, and then the superficies FE will be divided into so many equal annuli, whose forces will be as the sum of all the rec- tangles PD X Dd, that is, as ^PF^ _|PD^ and therefore as DE'\ Let now the superficies FE be drawn into the altitude Yf; and the force of the solid EF/e exerted upon the corpuscle P will be as DE^ x F/"; that is, if the force be given which any given particle as ¥f exerts upon the corpuscle P at the distance PF. But if that force be not given, the force of the solid EF/e will be as the solid DE^ x F/ and that force not given, conjunctly. Q,.E.D.

PROPOSITION LXXX. THEOREM XL.

If to the several equal parts of a sphere ^BE described about the centre S there tend equal centripetal forces ; a7id from the several points B in the axis of the sphere AB m tohich a corpuscle, as P, is placed^ there be erected the perpendiculars DE 'meeting the sphere in E, and if in those perpendicidars the lengths DN be taken as the quantity

DE2 X PS

^^^ , and as the force which a 'particle of the sphere situate in

the axis exerts at the distance PE upon the corpuscle P conjunctly ; 1 say, that the lohole force with lohich the corpuscle P is attracted to- wards the sphere is as the area ANB, comprehended under the axis of the sphere AB, aiid the curve line ANB, the locus of the point N, For supposing the construction in the last Lemma and Theorem to stand, conceive the axis of the sphere AB to be divided into innumerable equal particles Dd, and the whole sphere to be divided into so many sphge- rical concavo-convex laminae EF/e ; and erect the perpendicular dn. By the last Theorem, the force with which the lamina EF/e attracts the cor- puscle P is as DE^ X F/ and the force of one particle exerted at the

228

THE MATHEMATICAL PRINCIPLES

[Book I.

Ee

distance PE or PF, conjunctly. But (by the last Lemma) Dd is to F/ as PE to PS, and therefore F/

IS equal to — :^:^ — ; and DE^ x

PE

F/ is equal to J)d X

DE^XJPS PE

and therefore the force of the la- mmaEF/e is as Bd X ^^^

and the force of a particle exerted at the distance PF conjunctly ; that i&, by the supposition, as DN X Dd, or as the evanescent area DlSnd, Therefore the forces of all the lamina exerted upon the corpuscle P are as all the areas DN/ic/, that is, the whole force of the sphere will be as the whole area ANB. Q.E.D.

Cor. 1. Hence if the centripetal force tending to the several particles

DE^ X PS

remain always the same at all distances, and DN be made as ^5^^^ ;

the whole force with which the corpuscle is attracted by the sphere is as the area ANB.

Cor. 2. If the centripetal force of the particles be reciprocally as the

DE2 X PS

distance of the corpuscle attracted by it, and DN be made as — 'Wc^ ?

the force with which the corpuscle P is attracted by the whole sphere will

be as the area ANB.

Cor. 3. If the centripetal force of the particles be reciprocally as the

cube of the distance of the corpuscle attracted by it, and DN be made as

DE2 X PS

— :^- , the force with which the corpuscle is attracted by the whole

sphere will be as the area ANB.

Cor. 4. And universally if the centripetal force tending to the several particles of the sphere be supposed to be reciprocally as the quantity V ;

DE^ X PS

and DN be made as —-^ ; the force with which a corpuscle is at-

tracted by the whole sphere will be as the area ANB.

PROPOSITION LXXXI. PROBLEM XLI.

The things remaining as above, it is required to measure the area

ANB.

From the point P let there be drawn the right line PH touching the sphere in H ; and to the axis PAB, letting fall the perpendicular HI, bisect PI in L ; and (by Prop. XII, Book II, Elem.) PE^ is equal to

Sec. XII.1

OF NATURAL PHILOSOPHY.

229

PS2 + SE^ + 2PSD. But because the triangles SPH, SHI are alike, SE^ or SH^ is equal to the rectan- gle PSI, Therefore PE^ is equal to the rectangle contained under PS and PS -f SI + 2SD ; that is, under PS and 2LS + 2SD ;' that is, under PS and 2LD. Moreover DE^

IS

SD^ or SE^

equal to SE=

LS^ +2SLD— LD^ that is, 2SLD — LD^ -. ALB. For LS^ — SE2 or LS2 _SA2 (by Prop. Y% Book II, Elem.) is equal to the rectan- gle ALB. Therefore if instead of "DE ^ we write 2SLD — LD ^ — ALB,

DE^ X PS

the quantity —~^^ .^— ^ which (by Cor. 4 of the foregoing Prop.) is as

PEX V

the length of the ordinate 2SLD X PS LD^ X PS

DN, will now resolve itself into three parts r- ; where if instead of V we write

PE X V PE X V PE X V

the inverse ratio of the centripetal force, and instead of PE the mean pro- portional between PS and 2LD, those three parts will become ordinates to so many curve lines, whose areas are discovered by the common methods. Q.E.D.

Example 1. If the centripetal force tending to the several particles of the sphere be reciprocally as the distance ; instead of V write PE the dis- tance, then 2PS X LD for PE^ ; and DN will become as SL — | LD — ■

-gy-y^- Suppose DN equal to its double 2SL — LD rj-^r ; and 2SL

the given part of the ordinate drawn into the length AB will describe the rectangular area 2SL X AB ; and the indefinite part LD, drawn perpen- dicularly into the same length with a continued motion, in such sort as in its motion one way or another it may either by increasing or decreasing re-

LB2-LA2

main always equal to the length LD, will describe the area ^ ,

that is, the area SL X AB ; which taken from the former area 2SL X

AT B

AB, leaves the area SL X AB. But the third part -y-fr-j drawn after the

LD,

same manner with a continued motion perpendicularly into the sa,me length,

will describe the area of an hyperbola, which subducted ^ ^

from the area SL X AB will leave ANB the area sought.

Whence arises this construction of the Problem. At

the points, L, A, B, erect the perpendiculars LZ, Aa, Bb ;

making Aa equal to LB, and Bb equal to LA. Making

lul and LB asymptotes, describe through the points a, b,i-

m

THE MATHEMATICAL PRINCIPLES

[Book I

tlie hyperoolic curve ab. And the chord ba being drawn, will inclose the area aba equal to the area sought ANB.

Example 2. If the centripetal force tending to the several particles of the sphere be reciprocally as the cube of the distance, or (which is the same

PE3

thing) as that cube applied to any given plane ; write^-^ for Y, and

SL X AS^ AS^

2PS X LD for PE ALB X AS 2

and DN will become as

2PS X LD^

LSI LD

iSI

PS X LD 2PS

that is (because PS, AS, SI are continually proportional), as

If we draw then these three parts into the

2LD^

LSI

length AB, the first |r— - will generate the area of an hyperbola ; the sec- LU

, , . , ALB X SI , ALB X SI

ond iSI the area |AB X SI ; the third —^Try^ — ^^^ ^^^^

2LA

— --— , that is, iAB X SL From the first subduct the sum of tlie

2LB

second and third, and there will remain ANB, the area sought. Whence

arises this construction of the problem. At the points L, A, S, B, erect

I CO the perpendiculars L/ A« S^, Bb, of which suppose S5

equal to SI ; and through the point s, to the asymptotes

hi, LB, describe the hyperbola asb meeting the

perpendiculars Aa, Bb, in a and b ; and the rectangle

-? 2ASI, subducted from the hyberbolic area Aa^^B, will

B leave ANB the area sought.

Example 3. If the centripetal force tending to the several particles of

the spheres decrease in a quadruplicate ratio of the distance from the par-

PE'

tides ; write pToi fo^* ^? ^^^^ v^ ^^^

LD for PE, and DN will become

SP X SL

X

SL

1

SP X ALB

1

rr X

V2Sl '^^/LD=^ 2v^2Sr^ VLD 2^/281 ^/LD5•

These three parts drawn into the length AB, produce so many areas, viz.

2SP X SL

T^SI SI^

into

1

1

^/ LA ^/ LB '

V/2SI

into y/ LB — ^/ LA: and

bSP X ALB .

into

1

3v/2SI v/LA^ v/LB3'

And these after due reduction come

2SP^ X SL ^^

J- , SP, and SI2 +

forth

Sec. XIL

OF NATURAL PHILOSOPHY.

231

2SP . 4SI»

-oj-r. And these by subducting the last from the first, become ^t t»

Therefore the entire force with which the corpuscle P is attracted towards

SP the centre of the sphere is as^, that is, reciprocally as PS^ X PI.

Q.E.L

By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following Theorem.

PROPOSITION LXXXII. THEOREM XLI.

In. a spJiere described about the centre S loith the interval SA, if there be taken SI, S A, SP continually proportional ; I say^ that the attraction of a corpuscle within the sphere in any place I is to its attraction without the sphere in the place P in a ratio compounded of the subduplicate ratio of IS, PS, the distances from the centre, and the subduplicate ratio of the centripetal forces tending" to the centre in those places P and I.

As if the centripetal forces of the particles of the sphere be reciprocally as the distances of the corpuscle at- tracted by them ; the force with which the corpuscle situate in I is attracted P by the entire sphere will be to the force with which it is attracted in P in a ratio compounded of the subdu- plicate ratio of the distance SI to the distance SP, and the subduplicate ratio of the centripetal force in the place I arising from any particle in the centre to the centripetal force in the place P arising from the same particle in the centre ; that is, in the subduplicate ratio of the distances SI, SP to each other reciprocally. These two subduplicate ratios compose the ratio of equality, and therefore the attractions in I and P produced by the whole sphere are equal. By the like calculation, if the forces of the particles of the sphere are reciprocally in a duplicate ratio of the distances, it will be found that the attraction in I is to the attraction in P as the distance SP to the semi -diameter SA of the sphere. If those forces are reciprocally in a triplicate ratio of the distances, the attractions in I and P will be to each other as SP^ to SA^ ; if in a quadruplicate ratio, as SP^ to SA^. There- fore since the attraction in P was found in this last case to be reciprocally as PS 3 X PI, the attraction in I will be reciprocally as S A ^ x PI, that is, because S A ^ is given reciprocally as PL And the progression is the same in infinitum. The demonstration of this Theorem is as follows :

The things remaining as above constructed, and a corpuscle being in any

232

THE MATHEMATICAL PRINCIPLES

[Book I.

DE^ X PS

place P, the ordinate DN was found to be as v7"i7^' Therefore if

IE be drawn, that ordinate for any other place of the corpuscle, as I, will

become [mutatis mutandis) as —^^ ^. Suppose the centripetal forces

flowing from any point of the sphere, as E, to be to each other at the dis- tances IE and PE as PE° to IE" (where the number n denotes the index

DE^ X PS

of the powers of PE and IE), and those ordinates will become as ^^

and

DE^ X IS

^^ ^^ whose ratio to each other is as PS X IE X IE° to IS X IE X IE"

PE X PE". Because SI, SE, SP are in continued proportion, the tri- angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA. For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of the ordinates becomes that of PS X IE" to SA X PE". But the ratio of PS to SA is subduplicate of that of the distances PS, SI ; and the ratio of IE" to PE" (because IE is to PE as IS to SA) is subduplicate of that of the forces at the distances PS, IS. Therefore the ordinates, and conse- quently the areas which the ordinates describe, and the attractions propor- tional to them, are in a ratio compounded of those subduplicate ratios. Q.E.D.

PROPOSITION LXXXIII. PROBLEM XLU.

To find the force toith which a corpuscle placed in the centre of a sphere is attracted toioards any segment of that spliere whatsoever.

R[^ Let P be a body in the centre of that sphere and

RBSD a segment thereof contained under the plane RDS, and the spherical superficies RBS. Let DB be cut in F by a sphaerical superficies EFG described from the centre P, and let the segment be divided into the parts B BREFGS, FEDG. Let us suppose that segment to be not a purely mathematical but a physical superficies, having some, but a perfectly inconsiderable thickness. '^ Let that thickness be called O, and (by what Archi- medes has demonstrated) that superficies will be as PF X DF X O. Let us suppose besides the attrac- tive forces of the particles of the sphere to be reciprocally as that power of the distances, of which n is index ; and the force with which the superficies

EFG attracts the body P will be (by Prop. LXXIX) as — :f^„ — , that

DF^ X O

PF"

js, as

2DF_X_0

PF^

Let the perpendicular FN drawn into

Sec. XIIL] of natural philosophy. 233

O be proportional to this quantity ; and the curvilinear area BDI, which the ordinate FN, drawn through the length DB with a continued motion will describe, will be as the whole force with which the whole segment RBSD attracts the body P. Q.E.I.

PROPOSITION LXXXIY. PROBLEM XLIII.

To find the force with ivhich a corpuscle, placed loithout the centre of a sphere in the axis of any segment, is attracted hy that segment. Let the body P placed in the axis ADB of

the segment ElU-C be attracted by that seg- ment. About the centre P, witli tlie interval

PE, let the sphierical superficies EFK be de-^

scribed; and let it divide the segment into

two parts EBKFE and EFKDE. Find the

force of the . first of those parts by Prop.

LXXXI, and the force of the latter part by

Prop. LXXXIII, and the sum of the forces will be the force of the whole

segment EBKDE. Q.E.I.

SCHOLIUM.

The attractions of sphaerical bodies being now explained, it comes next in order to treat of the laws of attraction in other bodies consisting in like manner of attractive particles ; but to treat of them particularly is not neces- sary to my design. It will be sufficient to subjoin some general proposi- tions relating to the forces of such bodies, and the motions thence arising, because the knowledge of these will be of some little use in philosophical inquiries.

SECTION XIII.

Of the attractive forces of bodies which are not of a sphmrical figure,

PROPOSITION LXXXY. THEOREM XLIL

If a body be attracted by another, and its attraction be vastly stronger wheyi it is contiguous to the attracting body than when they are sepa- rated from 07ie afiother by a very small interval ; the forces of the particles of the attracting body decrease, in the recess of the body at- tracted, in more than a duplicate ratio of the distance of the particles. For if the forces decrease in a duplicate ratio of the distances from the particles, the attraction towards a sphsbrical body being (by Prop. LXXIV) reciprocally as the square of the distance of the attracted body from the centre of the sphere, will not be sensibly increased by the contact, and it

234 THE MATHEMATICAL PRINCIPLES [BoOK 1.

will be still less increased hj it, if the attraction, in the recess of the body attracted, decreases in a still less proportion. The proposition, therefore, is evident concerning attractive spheres. And the case is the same of con- cave sphasrical orbs attracting external bodies. And much more does it appear in orbs that attract bodies placed within them, because there the attractions diffused through the cavities of those orbs are (by Prop. LXX) destroyed by contrary attractions, and therefore have no effect even in the place of contact. Now if from these spheres and sphasrical orbs we take away any parts remote from the place of contact, and add new parts any where at pleas are, we may change the figures of the attractive bodies at pleasure ; but the parts added or taken away, being remote from the place of contact, will cause no remarkable excess of the attraction arising from the contact of the two bodies. Therefore the proposition holds good in bodies of all figures. Q.E.D.

PROPOSITION LXXXYl. THEOREM XLIII.

If the forces of the particles of which an attractive body is composed de- crease, in the recess of the attractive body, in a triplicate or more than a triplicate ratio of the distance from the particles, the attraction will he vastly strong-er in the point of contact than when the attracting and attracted bodies are separated from each other ^ though by never so small an interval.

For that the attraction is infinitely increased when the attracted corpus- cle comes to touch an attracting sphere of this kind, appears, by the solu- tion of Problem XLI, exhibited in the second and third Examples. The same will also appear (by comparing those Examples and Theorem XLI together) of attractions of bodies made towards concavo-convex orbs, whether the attracted bodies be placed without the orbs, or in the cavities within them. And by adding to or taking from those spheres and orbs any at- tractive matter any where without the place of contact, so that the attrac- tive bodies may receive any assigned figure, the Proposition will hold good of all bodies universally. Q,.E.D.

PROPOSITION LXXXVII. THEOREM XLIY.

If tioo bodies similar to each other, and consisting of m^atter equally at- tractive.^ attract separately two corpuscles proportional to those bodies, and in a like situation to them, the accelerative attractions of the cor- puscles towards the entire bodies will be as the accelerative attractions of the corpuscles towards particles of the bodies proportional to the wholes, and alike situated in them.

For if the bodies are divided into particles proportional to the whoks, and alike situated in them, it will be, as the attraction towards any parti- cle of one of the bodies to the attraction towards the correspondent particle

Sec. XIIL] of natural phil'osophy. 235

in the other body, so are the attractions towards the several particles of the first body, to the attractions towards the several correspondent particles of the other body ; and, by composition, so is the attraction towards the first whole body to the attraction towards the second whole body. Q,.E.D.

Cor. 1. Therefore if, as the distances of the corpuscles attracted increase, the attractive forces of the particles decrease in the ratio of any power of the distances, the accelerative attractions towards the whole bodies will be as the bodies directly, and those powers of the distances inversely- As if the forces of the particles decrease in a duplicate ratio of the distances from the corpuscles attracted, and the bodies are as A^ and B^, and there- fore both the cubic sides of the bodies, and the distance of the attracted corpuscles from the bodies, are as A and B ; the accelerative attractions

A^ g3

towards the bodies -vvill be as ;.— and ^, that is, as A and B the cubic

sides of those bodies. If the forces of the particles decrease in a triplicate ratio of the distances from the attracted corpuscles, the accelerative attrac-

A^ g3

tions towards the whole bodies will be as t-^ and ^— , that is, equal. If the

forces decrease in a quadruplicate ratio, the attractions towards the bodies

A^ B=^ will be as-^ and — , that is, reciprocally as the cubic sides A and B.

And so in other cases.

Cor. 2. Hence, on the other hand, from the forces with which like bodies attract corpuscles similarly situated, may be collected the ratio of the de- crease of the attractive forces of the particles as the attracted corpuscle recedes from them ; if so be that decrease is directly or inversely in any ratio of the distances.

PROPOSITION LXXXVIII. THEOREM XLV.

If the attractive forces of the equal particles of any body be as the dis- tance of the places from the particles, the force of the whole body will tend to its centre of gravity ; and will be the same with the force of a globe, consisting of similar and equal ?natter, a7id having its centra in the centre of gravity. Let the particles A, B, of the body RSTV at- tract any corpuscle Z with forces which, suppos-| ing the particles to be equal between themselves, ■>>:; are as the distances AZ, BZ ; but, if they are supposed unequal, are as those particles and their distances AZ, BZ, conjunctly, or (if I may so speak) as those particles drawn into their dis- tances AZ; BZ respectively. And let those forces be expressed by the

236 THE MATHEMATICAL PRINCIPLES [BoOK I.

contents under A X AZ, and B X BZ. Join AB, and let it be cut in G, so that AG may be to BG as the particle B to the particle A ; and G will be the common centre of gravity of the particles A and B. The force A X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZ and A X AG ; and the force B X BZ into the forces B X GZ and B X BG. Now the forces A X AG and B X BG, because A is proportional to B, and BG to AG, are equal, and therefore having contrary directions de- stroy one another. There remain then the forces A X GZ and B X GZ. These tend from Z towards the centre G, and compose the force A + B X GZ ; that is, the same force as if the attractive particles A and B were placed in their common centre of gravity G, composing there a little globe.

By the same reasoning, if there be added a third particle C, and the force of it be compounded with the force A + B X GZ tending to the cen- tre G, the force thence arising will tend to the common centre of gravity of that globe in G and of the particle 0 ; that is, to the common centre of gravity of the three particles A, B, C ; and will be the same as if that globe and the particle C were placed in that common centre composing a greater globe there; and so we may go on m infinitv/m. Therefore the whole force of all the particles of any body whatever RSTV is the same as if that body, without removing its centre of gravity, were to put on the form of a globe. Q,.E.D.

Cor. Hence the motion of the attracted body Z will be the same as if the attracting body RSTV were sphaerical ; and therefore if that attract- ing body be either at rest, or proceed uniformly in a right line, the body attracted will move in an ellipsis having its centre in the centre of gravity of the attracting body.

PROPOSITION LXXXIX. THEOREM XLYI.