As if the centripetal forces of theparticles of the sphere be reciprocally;is the distances of the corpuscle attracted by them ; the force with whichthe corpuscle situate in I is attractedby the entire sphere will be to theforce with which it is attracted in Pin a ratio compounded of the subduplicate ratio of the distance SI to the distance SP, and the subduplicateratio of the centripetal force in the place I arising from any particle in thecentre to the centripetal force in the place P arising from the same particle inthe centre ; that is, in the subduplicate ratio of the distances SI, SP to eachother reciprocally. These two subduplicate ratios compose the ratio ofequality, and therefore the attractions in I and P produced by the wholesphere are equal. By the like calculation, if the forces of the particles ofthe sphere are reciprocally in a duplicate ratio of the distances, it will befound that the attraction in I is to the attraction in P as the distance SPto the semi -diameter SA of the sphere. If those forces are reciprocally ina triplicate ratio of the distances, the attractions in I and P will be to eachother as SP 2 to SA3;if in a quadruplicate ratio, as SP 3 to SA 3. Therefore since the attraction in P was found in this last case to be reciprocallyas PS 3 X PI, the attraction in I will be reciprocally as SA 3 X PI, that is,because SA 3 is given reciprocally as PI. And the progression is the samein injinitnm. The demonstration of this Theorem is as follows :The things remaining as above constructed, and a corpuscle being in anj332 THE MATHEMATICAL PRINCIPLES [BOOK I.place P. the ordinate DN was found to be asT)F 2。" PS00 ^~- Therefore ifI Cj X VIE be drawn, that ordinate for any other place of the corpuscle, as I, willDE2 X ISbecome (mutatis mutandis] as~T~p~rry~- Suppose the centripetalsorcesflowing from any point of the sphere, as E, to be to each other at the distances IE and PE as PE 1 to IE11(where the number u denotes the indexDE 2 X PSof the powers of PE and IE), and those ordinates will become as ^p--57^7,2 。x ISand~"---TT7, whose ratio to each other is as PS X IE X IEn to IS XIE X IE"PE X PEn. Because SI, SE, SP are in continued proportion, the triangles 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 ofthe ordinates becomes that of PS X IE" to SA X PEn. But the ratio ofPS to SA is snbduplicate of that of the distances PS, SI ; and the ratio ofIE" to PE 1(because IE is to PE as IS to SA) is subduplicate of that ofthe forces at the distances PS, IS. Therefore the ordinates, and consequently the areas whioifi the ordinates describe, and the attractions proportional to them, are in a ratio compounded of those subduplicate ratios.Q.E.D.PROPOSITION LXXXIII. PROBLEM XLII.To find the force with which a corpuscle placed in the centre of a sphereis attracted towards any segment of that sphere whatsoever.Let P be a body in the centre of that sphere andRBSD a segment thereof contained under the planeRDS, and thesphrcrical superficies RBS. Let DB be cutin F by a sphaerical superficies EFG described from thecentre P, and let the segment be divided into the parts_B BREFGS, FEDG. Let us suppose that segment tobe not a purely mathematical but a physical superficies,having some, but a perfectly inconsiderable thickness.* Let that thickness be called O, and (by what Archimedes has demonstrated) that superficies will be asPF X DF X O. Let us suppose besides the attractive forces of the particles of the sphere to be reciprocally as that power ofdistances, of which n is index ; and the force with which the superficiesDE2 X OEFG attracts the body P will be (by Prop. LXXIX) as -- that,2DF X Ois, as ---? -,-DF 2 X O~"~ppn*ppnthe perpendicular FN drawn intoSEC. XJ11.I OF NATURAL PHILOSOPHY. 233O be proportional to this quantity ; and the curvilinear area BDI, whichthe ordinate FN, drawn through the length DB with a continued motionwill describe, will be as the whole force with which the whole segmentRBSD attracts the body P. Q.E.I.PROPOSITION LXXXIV. PROBLEM XLIII.To find the force with which a corpuscle, placed without the centre of asphere iti the axis of any segment, is attracted by that segment.Let the body P placed in. the axis ADB ofthe segment KBK be attracted by that segment. About the centre P, with the intervalPE, let the spherical superficies EFK be described;and let it divide the segment intotwo parts EBKFE and EFKDE. Find theforce of the first of those parts by Prop.LXXXI, and the force of the latter part byProp. LXXXIII, and the sum of the forces will be the force of the wholesegment EBKDE. Q.E.I.SCHOLIUM.The attractions of sphaerical bodies being now explained, it comes nextin order to treat of the laws of attraction in other bodies consisting in likemanner of attractive particles ; but to treat of them particularly is not necessary to my design. It will be sufficient to subjoin some general propositions relating to the forces of such bodies, and the motions thence arising,because the knowledge of these will be of some little use in philosophicalinquiries.SECTION XIII.Of the attractive forces of bodies which are not of a sphcerical figure.PROPOSITION LXXXV. THEOREM XLILIf a body be attracted by another, and its attraction be vastly strongerwhen it is contiguous to the attracting body than when they are separated from one another by a very small interval ; the forces of theparticles of the attracting body decrease, in the recess of the body attracted, 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 theparticles, the attraction towards a sphaerical body being (by Prop. LXXIV)reciprocally as the square of the distance of the attracted body from thecentre of the sphere, will not be sensibly increased by the contact, and it234 THE MATHEMATICAL PRINCIPLES [BOOK 1。vill be still less increased by it, if the attraction, in the recess of the bodyattracted, decreases in a still less proportion. The proposition, therefore,is evident concerning attractive spheres. And the case is the same of concave sphaerical orbs attracting external bodies. And much more does itappear in orbs that attract bodies placed within them, because there theattractions diffused through the cavities of those orbs are (by Prop. LXX)destroyed by contrary attractions, and therefore have no effect even in theplace of contact. Now if from these spheres and sphoerical orbs we takeaway any parts remote from the place of contact, and add new parts anywhere at pleasore, we may change the figures of the attractive bodies atpleasure ; but the parts added or taken away, being remote from the placeof contact, will cause no remarkable excess of the attraction arising fromthe contact of the two bodies. 1 herefore the proposition holds good inbodies of all figures. Q.E.I).PROPOSITION LXXXV1. THEOREM XLIII.If the forces of the particles of which an attractive body is composed decrease, in. the recess of the attractive body, in a triplicate or more thana triplicate ratio of the distancefrom the particles, the attraction willbe vastly stronger in the point of contact than when the attracting andattracted bodies are separated from each other, though by never sosmall an interval.For that the attraction is infinitely increased when the attracted corpuscle comes to touch an attracting sphere of this kind, appears, by the solution of Problem XLI, exhibited in e second and third Examples. Thesame will also appear (by comparing those Examples and Theorem XLItogether) of attractions of bodies made towards concavo-convex orbs, whetherthe attracted bodies be placed without the orbs, or in the cavities withinthem. And by aiding to or taking from those spheres and orbs any attractive matter any where without the place of contact, so that the attractive bodies may receive any assigned figure, the Proposition will hold goodof all bodies universally. Q.E.D.PROPOSITION LXXXVII. THEOREM XI. IV.If two bodies similar to each other, and consisting of matter equally attractive attract separately two corpuscles proportional to those bodies,and in a like situation to them, the accelerative attractions of the corpuscles towards the entire bodies will be as the acccleratire attractionsof the corpuscles towards particles of the bodies proportional to thewholes, and alike situated in them.For if the bodies are divided into particles proportional to the wholes,and alike situated in them, it will be, as the attraction towards any particle of one of the bodies to the attraction towards the correspondent particleSEC. A III.] OF NATURAL PHILOSOPHY. 235in the other body, so are the attractions towards the several particles of theiirst body, to the attractions towards the several correspondent particles ofthe other body ; and, by composition, so is the attraction towards the firstwhole body to the attraction towards the second whole body. Q,.E.U.COR. 1 . Therefore if, as the distances of the corpuscles attracted increase,the attractive forces of the particles decrease in the ratio of any powerof the distances, the accelerative attractions towards the whole bodies willbe as the bodies directly, and those powers of the distances inversely. A*if the forces of the particles decrease in a duplicate ratio of the distancesfrom the corpuscles attracted, and the bodies are as A 3 and B 3, and therefore both the cubic sides of the bodies, and the distance of the attractedcorpuscles from the bodies, are as A and B ; the accelerative attractionsA 3 B 3towards the bodies will be as and , that is, as A and B the cubicsjides of those bodies. If the forces of the particles decrease in a triplicateratio of the distances from the attracted corpuscles, the accelerative attrac-A3 B 3tions towards the whole bodies will be as and 5--, that is, equal. If theA. tjforces decrease in a quadruplicate ratio, the attractions towards the bodiesA 3 B 3will be as- an^ 04 *^at 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 bodiesattract corpuscles similarly situated, may be collected the ratio of the decrease of the attractive forces of the particles as the attracted corpusclerecedes from them ;if so be that decrease is directly or inversely in anyratio of the distances.PROPOSITION LXXXVIII. THEOREM XLV.If the attractive forces of the equal particles of any body be as the distance of the places from the particles, the force of the whole body willtend to its centre of gravity ; and will be the same with the force ofa globe, consisting of similar and equal matter, and having its centrein the centre of gravity.Let the particles A, B, of the body RSTV attract any corpuscle Z with forces which, suppos-|ing the particles to be equal between themselves,are as the distances AZ, BZ ; but, if they aresupposed unequal, are as those particles andtheir distances AZ, BZ, conjunctly, or (if I maygo speak) as those particles drawn into their distances AZ, BZ respectively. And let those forces be expressed by the236 THE MATHEMATICAL PRINCIPLES [BOOK 1.contents er 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 Gwill be the common centre of gravity of the particles A and B. The forceA X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZand A X AG ; and the force B X BZ into the forces B X GZ and B XBG. Now the forces A X AG and B X BG, because A is proportional toB, and BG to AG, are equal, and therefore having contrary directions destroy 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 + BX GZ ; that is, the same force as if the attractive particles A and B wereplaced in their common centre of gravity G, composing there a little globe.By the same reasoning, if there be added a third particle G, and theforce of it be compounded with the force A -f B X GZ tending to the centre G, the force thence arising will tend to the common centre of gravityof that globe in G and of the particle C ; that is, to the common centre oigravity of the three particles A, B, C ; and will be the same as if thatglobe and the particle C were placed in that common centre composing agreater globe there ; and so we may go on in injinitum. Thereforethe whole force of all the particles of any body whatever RSTV is thesame as if that body, without removing its centre of gravity, were to puton the form of a globe. Q,.E.D.COR. Hence the motion of the attracted body Z will be the same as ifthe attracting body RSTV were sphaerical ; and therefore if that attracting body be either at rest, or proceed uniformly in a right line, the bodyattracted will move in an ellipsis having its centre in the centre of gravityof the attracting body.PROPOSITION LXXXIX. THEOREM XLVI.If there be several bodies consisting of equal particles whose jorces areas the distances of the places from each, the force compounded of allthe forces by which any corpuscle is attracted will tend to the commoncentre of gravity of the attracting bodies ; and will be the same as ifthose attracting bodies, preserving their common centre of gravity,should unite there, and be formed into a globe.This is demonstrated after the same manner as the foregoing Proposition.COR. Therefore the motion of the attracted body will be the same as ifthe attracting bodies, preserving their common centre of gravity, shouldunite there, and be formed into a globe. And, therefore, if the commoncentre of gravity of the attracting bodies be either at rest, or proceed uniformly in a right line, the attracted body will move in an ellipsis havingUs centre in the common centre of gravity of the attracting bodies.SEC. XlII.j OF NATURAL PHILOSOPHY. 237PROPOSITION XC. PROBLEM XLIV.If to the several points of any circle there tend equal centripeta forces,increasing or decreasing in any ratio of the distances ; it is requiredto Jind the force with which a corpuscle is attracted, that is, situateany where in a right line which stands at right angles to the plantof the circle at its centre.Suppose a circle to be described about the centre A with any interval AD in a plane to which ;the right line AP is perpendicular ; and let it berequired to find the force with which a corpuscleP is attracted towards the same. From any pointE of the circle, to the attracted corpuscle P, letthere be drawn the right line PE. In the rightline PA take PF equal to PE, and make a perpendicularFK, erected at F, to be as the forcewith which the point E attracts the corpuscle P.And let the curve line IKL be the locus of the point K. Let that cu/, femeet the plane of the circle in L. In PA take PH equal to PD, and p/^ctthe perpendicular HI meeting that curve in I; and the attraction of thecorpuscle P towards the circle will be as the area AHIL drawn into thealtitude AP Q.E.I.For let there be taken in AE a very small line Ee. Join Pe, and in PE,PA take PC, Pf equal to Pe. And because the force, with which anypoint E of the annulus described about the centre A with the interval ASin the aforesaid plane attracts to itself the body P, is supposed to be asFK ; and, therefore, the force with which that point attracts the body PAP X FKtowards A is as -^p ; and the force with which the whole annulusAP X FKattracts tne body P towards A is as the annulus and p^ conjunctly; and that annulus also is as the rectangle under the radius AE aad thebreadth Ee, and this rectangle (because PE and AE, Ee and CE are proportional) is equal to the rectangle PE X CE or PE X F/; the force*-ith which that annulus attracts the body P towards A will be as PE XAP X FKFf and pp~~~ conjunctly ; that is, as the content under F/ X FK XAP, or as the area FKkf drawn into AP. And therefore the sum of theforces with which all the annuli, in the circle described about the centre Awith the interval AD, attract the body P towards A, is as the whole areaAHIKL drawn into AP. Q.E.D.COR. 1. Hence if the forces of the points decrease in the duplicate ratio238 THE MATHEMATICAL PRINCIPLES [BOOK Iof the distances, that is, if FK be as rfFK, and therefore the area AHIKLasp-7 p- ; the attraction of the corpuscle P towards the circle willPA AHbe as 1 ; that is, asCOR. 2. And universally if the forces of the points at the distances D b(reciprocally as any power Dn of the distances; that is, if FK be as .and therefore the area AHIKL as1 1" lPH"1 PA, ; the attractionof the corpuscle P towards the circle will be asPA"2PH"lCOR. 3. And if the diameter of the circle be increased in itifinitum, andthe number n be greater than unity ; the attraction of the corpuscle P towards the whole infinite plane will be reciprocally as PA"2, because thePAother term vanishes.PROPOSITION XCI. PROBLEM XLV.To find the attraction of a corpuscle situate in the axis of a round solid,to whose several points there tend equal centripetal forces decreasingin any ratio of the distances whatsoever.Let the corpuscle P, situate in the axis ABof the solid DECG, be attracted towards thatsolid. Let the solid be cut by any circle asRFS, perpendicular to the axis ; and in itssemi-diameter FS, in any plane PALKB passing through the axis, let there be taken (byProp. XC) the length FK proportional to theforce with which the corpuscle P is attractedtowards that circle. Let the locus of the pointK be the curve line LKI, meeting the planes of the outermost circles ALand BI in L and I; and the attraction of the corpuscle P towards thesolid will be as the area LABI. Q..E.I.COR. 1. Hence if the solid be a cylinder described by the parallelogramADEB revolved about the axis AB, and the centripetal forces tending tothe several points be reciprocally as the squares of the distances from thepoints ; the attraction of the corpuscle P towards this cylinder will be asAB PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will bePFas 1 --. The part 1 of this quantity, drawn into the length AB, deSEC. XIII. OF NATURAL PHILOSOPHY 239scribes the area 1 X AB ; and the other partPF, drawn into the length PB describes theixarea 1 into PE AD (as may be easilyshewn from the quadrature of the curveLKI); and, in like manner, the same partdrawn into the length PA describes the areaL into PD AD. and drawn into AB, the"AtGIvS13 M7J" 1difference of PB and PA, describes 1 into PE PD, the difference of theareas. From the first content 1 X AB take away the last content 1 intoPE PD, and there will remain the area LABI equal to 1 intoAB PE -h PD. Therefore the force, being proportional to this area,is as AB PE + PD.COR. 2. Hence also is known the forceby which a spheroid AGBC attracts anybody P situate externally in its axis AB.Let NKRM be a conic section whose ordinateKR perpendicular to PE may be 。always equal to the length of the line PD,continually drawn to tlie point D inwhich that ordinate cuts the spheroid.From the vertices A, B, of the spheriod,let there be erected to its axis AB the perpendiculars AK, BM, respectivelyequal to AP. BP, and therefore meeting the conic section in K and M; andjoin KM cutting offfrom it the segment KMRK. Let S be the centre of thespheroid, and SC its greatest semi-diameter : and the force with which thespheroid attracts the body P will be to the force with which a sphere describ-, ....,,. ASxCS 2 -PSxKMRKed with the diameter AhJ attracts the same body as prrr ^ r-=1 o -f- Go2 AoAS 3is to fkT^,. And by a calculation founded on the same principles may befound the forces of the segments of the spheroid.COR. 3. If the corpuscle be placed within the spheroid and in its axis,the attraction will be as its distance from the centre. This may be easilycollected from the following reasoning, whetherthe particle be in the axis or in any other given